Quantum Mechanics by Konstantin K. Likharev - HTML preview

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Essential Graduate Physics

QM: Quantum Mechanics

2

  

 Z E 

(0)

E

 2 (0)

  2

0





 2

H



 4

 E  

eV.

109

(8.28)

g

g

 2 2



n 

2

H



n ,

1 Z 2



 Z2

This is still somewhat far (though not terribly far!) from the experimental value E g  –78.8 eV – see the bottom level in Fig. 1a.

(a)

(b)

Δ E

singlet state

( s )

(eV)

(“parahelium”)



“parahelium” “orthohelium”

E ex

0

3 p 3 d

3 s

3 s 3 p 3 d

2 p

2 p

E

2 s

ex

triplet state

2 s

-5

( s )



(“orthohelium”)

-20

E dir

B  0

1 s (ground state)

-25

l

l



 

100

nlm

Fig. 8.1. The lower energy levels of a helium atom: (a) experimental data and (b) a schematic structure of an excited state in the first order of the perturbation theory. On panel (a), all energies are referred to that (-2 E H  –55.4 eV) of the ground state of the positive ion He+1, so their magnitudes are the (readily measurable) energies of the atom’s single ionization starting from the corresponding state of the neutral atom. Note that the “spin direction” nomenclature on panel (b) is rather crude: it does not reflect the difference between the entangled states s+ and s–.

Making a minor (but very useful) detour from our main topic, let us note that we can get a much better agreement with experiment by accounting for the electron interaction energy in the 1st order of the perturbation theory. Indeed, in application to our system, Eq. (6.14) reads

(1)

ˆ

E

 g U g

3

3

*

 d r d r  (r , r ) U (r , r ) (r , r ).

(8.29)

g

int

 1 2 g 1 2 int 1 2 g 1 2

Plugging in Eqs. (25)-(27), we get

2





e



r  r 

(1)

1 4

2

3

3

2(

)

E



d r d r

exp

1

2







.

(8.30)

g



 4 3

1

2



 r

4 r 



r

r

0 

0 1

2



0



As may be readily evaluated analytically (this exercise is left for the reader), this expression equals (5/4) E H, so the corrected ground state energy,

(0)

(1)

E  E

 E   

E  

,

(8.31)

g

g

g

 4 5/ 4

74. eV

8

H

is much closer to experiment.

There is still room here for a ready improvement by using the variational method discussed in Sec. 2.9. For our particular case of the 4He atom, we may try to use, as the trial state, the orbital wavefunction given by Eqs. (26)-(27), but with the atomic number Z considered as an adjustable Chapter 8

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parameter Z ef < Z = 2 rather than a fixed number. The physics behind this approach is that the electric charge density (r) = – e(r)2 of each electron forms a negatively charged “cloud” that reduces the effective charge of the nucleus, as seen by the other electron, to Z ef e, with some Z ef < 2. As a result, the single-particle wavefunction spreads further in space (with the scale r 0 = r B/ Z ef > r B/ Z), while keeping its functional form (27) nearly intact. Since the kinetic energy T in the system’s Hamiltonian (25) is proportional to r –2

2

–1

1

0  Z ef , while the potential energy is proportional to r 0

 Z ef , we can write

2

 Z 

Z

ef

ef

E ( Z )  

 T



U

.

(8.32)

g

ef

g

g

Z 2

Z 2

 2 

2

Now we can use the fact that according to Eq. (3.212), for any stationary state of a hydrogen-like atom (just as for the classical circular motion in the Coulomb potential),  U = 2 E, and hence  T = E –

 U = – E. Using Eq. (30), and adding the correction (31) to the potential energy, we get 2

  Z  

 Z 

ef

5

E ( Z )  4

   8

ef

 

 E .

(8.33)

g

ef

 2 



4  2

H





This expression allows an elementary calculation of the optimal value of Z ef, and the corresponding minimum of the function E g( Z ef):



5 

( Z )

 21

 

,

6875

.

1

E

 

E  

.

(8.34)

ef opt

 g

85

.

2

eV

5

.

77



32

H

min



Given the trial state’s crudeness, this number is in surprisingly good agreement with the experimental value cited above, with a difference of the order of 1%.

Now let us return to the main topic of this section – the effects of the particle (in this case, electron) indistinguishability. As we have just seen, the ground-level energy of the helium atom is not affected directly by this fact; the situation is different for its excited states – even the lowest ones. The reasonably good precision of the perturbation theory, which we have seen for the ground state, tells us that we can base our analysis of wavefunctions (e) of the lowest excited state orbitals, on products like

100(r k) nlm(r k’), with n > 1. To satisfy the fermion permutation rule, Pj = –1, we have to take the orbital factor of the state in either the symmetric or the antisymmetric form:

Orthohelium

1

and

 (r , r ) 



r 

r 

r 

r

,

(8.35) parahelium:

e

1

2



( )

( )

( )

( )

100

1

nlm

2

nlm

1

100

2 

2

orbital

wavefunctions

with the proper total permutation asymmetry provided by the corresponding spin factor (18) or (21), so the upper/lower sign in Eq. (35) corresponds to the singlet/triplet spin state. Let us calculate the expectation values of the total energy of the system in the first order of the perturbation theory. Plugging Eq. (35) into the 0th-order expression

(0)

3

3

*

ˆ

ˆ

E

 d r d r  r , r h

h  r , r ,

(8.36)

e





1 

2

e  1

2   1

2 

e  1

2 

we get two groups of similar terms that differ only by the particle index. We can merge the terms of each pair by changing the notation as (r1  r, r2  r ’ ) in one of them, and (r1  r ’, r2  r) in the counterpart term. Using Eq. (25), and the mutual orthogonality of the wavefunctions 100(r) and  nlm(r), we get the following result:

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2

2

(0)

*

  

2 2

2

2

e



  

e



r

3

*

2 2

E

  (r) 



 (r) d r   ( '

r ) 

'

r 

 ( '

r ) 3





d r'

e

100



2 m

4

100



nlm



 r

2 m

4





 r'

nlm

(8.37)

0





0



 

  ,

with n  .

1

100

nlm

It may be interpreted as the sum of eigenenergies of two separate single particles, one in the ground state 100, and another in the excited state nlm – although actually the electron states are entangled. Thus, in the 0th order of the perturbation theory, the electrons’ entanglement does not affect their total energy.

However, the potential energy of the system also includes the interaction term U int, which does not allow such separation. Indeed, in the 1st approximation of the perturbation theory, the total energy E e of the system may be expressed as 

(1)

100 +  nlm + E int

, with

)

1

(

3

3

*

E

 U

 d r d r  (r , r ) U (r , r ) (r , r ) , (8.38)

int

int

 1 2 e 1 2 int 1 2 e 1 2

Plugging Eq. (35) into this result, using the symmetry of the function U int with respect to the particle number permutation, and the same particle coordinate re-numbering as above, we get

)

1

(

E

 E  E ,

(8.39)

int

dir

ex

with the following, deceivingly similar expressions for the two components of this sum/difference: Direct

interaction

3

3

*

E

 d r d r'  (r) *

 ( '

r ) U (r, '

r )

(r)

( '

r ),

(8.40)

dir

energy

 

100

nlm

int

100

nlm

Exchange

interaction

3

3

*

E  d r d r' 

(r) *

 ( '

r ) U (r, '

r )

(r)

( '

r ).

(8.41)

ex

energy

 

100

nlm

int

nlm

100

Since the single-particle orbitals can be always made real, both components are positive – or at least non-negative. However, their physics and magnitude are different. The integral (40), called the direct interaction energy, allows a simple semi-classical interpretation as the Coulomb energy of interacting electrons, each distributed in space with the electric charge density (r) = – e*(r)(r):14



r 

'

r

3

3

( )

( )

100

E  d r d r'

nlm

  (r) (r) 3

d r   (r) (r) 3

d r,

(8.42)

dir

 





4

100

nlm

nlm

100

 r  '

r

0

where (r) are the electrostatic potentials created by the electron “charge clouds”:15

1





3

( '

r )

1

100

3

( '

r )



r

( ) 

d r'

,



r

( ) 

d r' nlm

.

(8.43)

100





4

r  '

nlm

r

4

r  '

r

0

0

However, the integral (41), called the exchange interaction energy, evades a classical interpretation, and (as it is clear from its derivation) is the direct corollary of electrons’

indistinguishability. The magnitude of E ex is also very much different from E dir because the function under the integral (41) disappears in the regions where the single-particle wavefunctions 100(r) and

 nlm(r) do not overlap. This is in full agreement with the discussion in Sec. 1: if two particles are identical but well separated, i.e. their wavefunctions do not overlap, the exchange interaction disappears, 14 See, e.g., EM Sec. 1.3, in particular Eq. (1.54).

15 Note that the result for E dir correctly reflects the basic fact that a charged particle does not interact with itself, even if its wavefunction is quantum-mechanically spread over a finite space volume. Unfortunately, this is not true for some popular approximate theories of multiparticle systems – see Sec. 4 below.

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i.e. measurable effects of particle indistinguishability vanish. (In contrast, the integral (40) decreases with the growing separation of the electrons only slowly, due to their long-range Coulomb interaction.) Figure 1b shows the structure of an excited energy level, with certain quantum numbers n > 1, l, and m, given by Eqs. (39)-(41). The upper, so-called parahelium 16 level, with the energy E

 

 

 E  E  

 

(8.44)

para

 100 nlm 

,

dir

ex

100

nlm

corresponds to the symmetric orbital state and hence to the singlet-spin state (18), while the lower, orthohelium level, with

E

 

 

 E  E  E

(8.45)

orth

 100 nlm 

,

dir

ex

para

corresponds to the degenerate triplet-spin state (21).

This degeneracy may be lifted by an external magnetic field, whose effect on the electron spins17

is described by the following evident generalization of the Pauli Hamiltonian (4.163), ˆ

e



H

 

 ˆs B  ˆ

ˆ

B

s  B  

 S B ,

with     

 2



,

(8.46)

field

1

2

e

m e



where

ˆS  ˆs  ˆs ,

(8.47)

1

2

is the operator of the (vector) sum of the system of two spins.18 To analyze this effect, we need first to make one more detour, to address the general issue of spin addition. The main rule19 here is that in a full analogy with the net spin of a single particle, defined by Eq. (5.170), the net spin operator (47) of any system of two spins, and its component S âlong the (arbitrarily selected) z-axis, obey the same z

commutation relations (5.168) as the component operators, and hence have the properties similar to those expressed by Eqs. (5.169) and (5.175):

S ˆ 2 S, M

 2

 S



ˆ

1 ,

,

,

 

,

,



with



  , (8.48)

S

 S  S M

S S M

M S M

S

M

S

S

z

S

S

S

S

where the ket vectors correspond to the coupled basis of joint eigenstates of the operators of S 2 and Sz (but not necessarily all component operators – see again the Venn shown in Fig. 5.12 and its discussion, with the replacements S, L  s1,2 and J  S). Repeating the discussion of Sec. 5.7 with these replacements, we see that in both the coupled and the uncoupled bases, the net magnetic number MS is simply expressed via those of the components

16 This terminology reflects the historic fact that the observation of two different hydrogen-like spectra, corresponding to the opposite signs in Eq. (39), was first taken as evidence for two different species of 4He, which were called, respectively, the “orthohelium” and the “parahelium”.

17 As we know from Sec. 6.4, the field also affects the orbital motion of the electrons, so the simple analysis based on Eq. (46) is strictly valid only for the s excited state ( l = 0, and hence m = 0). However, the orbital effects of a weak magnetic field do not affect the triplet-level splitting we are analyzing now.

18 Note that similarly to Eqs. (22) and (25), here the uppercase notation of the component spins is replaced with the lowercase notation, to avoid any possibility of confusion with the total spin of the system.

19 Since we already know that the spin of a particle is physically nothing more than some (if specific) part of its angular momentum, the similarity of the properties (48) of the sum (47) of spins of different particles to those of the sum (5.170) of different spin components of the same particle it very natural, but still has to be considered as a new fact – confirmed by a vast body of experimental data.

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M  m

 m

.

(8.49)

S

 s   

1

s 2

However, the net spin quantum number S (in contrast to the Nature-given spins s 1,2 of its elementary components) is not universally definite, and we may immediately say only that it has to obey the following analog of the relation  l – s   j  ( l + s) discussed in Sec. 5.7: s  s  S  s  s .

(8.50)

1

2

1

2

What exactly S is (within these limits), depends on the spin state of the system.

For the simplest case of two spin-½ components, each with s = ½ and ms = ½, Eq. (49) gives three possible values of MS, equal to 0 and 1, while Eq. (50) limits the possible values of S to just either 0 or 1. Using the last of Eqs. (48), we see that the possible combinations of the quantum numbers are

 S  ,

0



S  ,

1





and

(8.51)

 M  ,

0

M

S



 ,

0 1.

S

It is virtually evident that the singlet spin state s– belongs to the first class, while the simple (separable) triplet states  and  belong to the second class, with MS = +1 and MS = –1, respectively. However, for the entangled triplet state s+, evidently with MS = 0, the value of S is less obvious. Perhaps the easiest way to recover it20 to use the “rectangular diagram”, similar to that shown in Fig. 5.14, but redrawn for our case of two spins, i.e., with the replacements ml  ( ms)1 = ½, ms  ( ms)2 = ½ – see Fig. 2.

 ms 2

 ½





S  1

Fig. 8.2. The “rectangular diagram”

s



showing the relation between the

S  0, 1

uncoupled-representation states (dots)

 ½

0

 ½  ms 1

and the coupled-representation states

(straight lines) of a system of two spins-





½ – cf. Fig. 5.14.

 ½

S  1

Just as at the addition of various angular momenta of a single particle, the top-right and bottom-left corners of this diagram correspond to the factorable triplet states  and , which participate in both the uncoupled-representation and coupled-representation bases, and have the largest value of S, i.e.

1. However, the entangled states s, which are linear combinations of the uncoupled-representation states  and , cannot have the same value of S, so for the triplet state s+, S has to take the value different from that (0) of the singlet state, i.e. 1. With that, the first of Eqs. (48) gives the following expectation values for the square of the net spin operator:

 2 2

 ,

state,

et

each tripl

for

2

S

 

(8.52)

 0,

state.

singlet

for the

20 Another, a bit longer but perhaps more prudent way is to directly calculate the expectation values of 2

ˆ S for the

states s, and then find S by comparing the results with the first of Eqs. (48); it is highly recommended to the reader as a useful exercise.

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Note that for the entangled triplet state s+, whose ket-vector (20) is a linear superposition of two kets of states with opposite spins, this result is highly counter-intuitive, and shows how careful we should be interpreting entangled quantum states. (As will be discussed in Chapter 10, quantum entanglement brings even more surprises for measurements.)

Now we may return to the particular issue of the magnetic field effect on the triplet state of the 4He atom. Directing the z-axis along the field, we may reduce Eq. (46) to S ˆ

ˆ

ˆ

z

H

 

 S

 

.

(8.53)

field

e

B

2

z

BB 

Since all three triplet states (21) are eigenstates, in particular, of the operator S ˆ , and hence of the z

Hamiltonian (53), we may use the second of Eqs. (48) to calculate their energy change simply as

 ,

1



state

triplet

factorable

for the

,



 E

 2

M

B



(8.54)

S

2 B

s

field

B





B

 ,

0

state

triplet

entangled

for the

,



 ,1



state

triplet

factorable

for the

.

This splitting of the “orthohelium” level is schematically shown in Fig. 1b.21

8.3. Multiparticle systems

Leaving several other problems on two-particle systems for the reader’s exercise, let me proceed to the discussion of systems with N > 2 indistinguishable particles, whose list notably includes atoms, molecules, and condensed-matter systems. In this case, Eq. (7) for fermions is generalized as ˆ

P     ,

k

all

for

, k'  ,

1 ,...,

2

N ,

(8.55)

kk '





where the operator ˆ

P permutes particles with numbers k and k’. As a result, for systems with non-kk '

directly-interacting fermions, the Pauli principle forbids any state in which any two particles have similar single-particle wavefunctions. Nevertheless, it permits two fermions to have similar orbital wavefunctions, provided that their spins are in the singlet state (18), because this satisfies the permutation requirement (55). This fact is of paramount importance for the ground state of the systems whose Hamiltonians do not depend on spin because it allows the fermions to be in their orbital single-particle ground states, with two electrons of the spin singlet sharing the same orbital state. Hence, for the limited (but very important!) goal of finding ground-state energies of multi-fermion systems with negligible direct interaction, we may ignore the actual singlet spin structure, and reduce the Pauli 21 It is interesting that another very important two-electron system, the hydrogen (H2) molecule, which was briefly discussed in Sec. 2.6, also has two similarly named forms, parahydrogen and orthohydrogen. However, their difference is due to two possible (respectively, singlet and triplet) states of the system of two spins of the two hydrogen nuclei – protons, which are also spin-½ particles. The resulting ground-state energy of the parahydrogen is lower than that of the orthohydrogen by only ~15 meV per molecule – the difference lower than k B T at room temperature (~26 meV). As a result, at very low temperatures, hydrogen at equilibrium is dominated by parahydrogen, but at ambient conditions, the orthohydrogen is nearly three times more abundant, due to its triple nuclear spin degeneracy. Curiously, the theoretical prediction of this effect by W. Heisenberg (together with F.

Hund) in 1927 was cited in his 1932 Nobel Prize award as the most noteworthy application of quantum theory.

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exclusion principle to the rudimentary picture of single-particle orbital energy levels, each “occupied with two fermions”.

As a very simple example, let us find the ground energy of five fermions confined in a hard-wall, cubic-shaped 3D volume of side a, ignoring their direct interaction. From Sec. 1.7, we know the single-particle energy spectrum of the system:

2

2



 n





,

(8.56)

x , ny ,



n  n  n



n n n 

n

0  2

2

2

x

y

z 



,

with

,

and

,

,

,

1 ,

2

0

z

2

x

y

z



2 ma

so the lowest-energy states are:

– one ground state with { nx,ny,nz} = {1,1,1}, and energy 111= (12+12+12)0 = 30, and

– three excited states, with { nx,ny,nz} equal to either {2,1,1}, or {1,2,1}, or {1,1,2}, with equal energies 211= 121 = 112 = (22+12+12)0 = 60.

According to the above simple formulation of the Pauli principle, each of these orbital energy levels can accommodate up to two fermions. Hence the lowest-energy (ground) state of the five-fermion system is achieved by placing two of them on the ground level 111 = 30, and the remaining three particles, in any of the degenerate “excited” states of energy 60, so the ground-state energy of the system is 12 2 2



E  2  3  3 6  24



 

.

(8.57)

g

0

0

0

2

ma

Moreover, in many cases, relatively weak interaction between fermions does not blow up such a simple quantum state classification scheme qualitatively, and the Pauli principle allows tracing the order of single-particle state filling. This is exactly the simple approach that was used in our discussion of atoms in Sec. 3.7. Unfortunately, it does not allow for a more specific characterization of the ground states of most atoms, in particular the evaluation of the corresponding values of the quantum numbers S, L, and J that characterize the net angular momenta of the atom, and hence its response to an external magnetic field. These numbers are defined by relations similar to Eqs. (48), each for the corresponding vector operator of the net angular momenta:

N

N

N

ˆS  ˆ

ˆ

s ,

L 

ˆ

ˆ

ˆ

l ,

J

j ;

(8.58)

k





k

 k

k 1

k 1

k 1

note that these definitions are consistent with Eq. (5.170) applied both to the angular momenta s k, l k, and j k of each particle, and to the full vectors S, L, and J. When the numbers S, L, and J for a state are known, they are traditionally recorded in the form of the so-called Russell-Saunders symbols:22

2 S 1

 L ,

(8.59)

J

where S and J are the corresponding values of these quantum numbers, while L is a capital letter, encoding the quantum number L – via the same spectroscopic notation as for single particles (see Sec.

3.6): L = S for L = 0, L = P for L = 1, L = D for L = 2, etc. (The reason why the front superscript of the Russel-Saunders symbol lists 2 S + 1 rather than just S, is that according to the last of Eqs. (48), it 22 Named after Henry Russell and Frederick Saunders, whose pioneering (circa 1925) processing of experimental spectral-line data has established the very idea of the vector addition of the electron spins, described by the first of Eqs. (58).

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shows the number of possible values of the quantum number MS, which characterizes the state’s spin degeneracy, and is called its multiplicity.)

For example, for the simplest, hydrogen atom ( Z = 1), with its single electron in the ground 1 s state, L = l = 0, S = s = ½, and J = S = ½, so its Russell-Saunders symbol is 2 S 1/2. Next, the discussion of the helium atom ( Z = 2) in the previous section has shown that in its ground state L = 0 (because of the 1 s orbital state of both electrons), and S = 0 (because of the singlet spin state), so the total angular momentum also vanishes: J = 0. As a result, the Russell-Saunders symbol for this state is 1 S 0. The structure of the next atom, lithium ( Z = 3) is also easy to predict, because, as was discussed in Sec. 3.7, its ground-state electron configuration is 1 s 22 s 1, i.e. includes two electrons in the “helium shell”, i.e. on the 1 s orbitals (now we know that they are actually in an entangled singlet spin state), and one electron in the 2 s state, of higher energy, also with zero orbital momentum, l = 0. As a result, the total L in this state is evidently equal to 0, and S is equal to ½, so J = ½, meaning that the Russell-Saunders symbol of the lithium’s ground state is 2 P 1/2. Even in the next atom, beryllium ( Z = 4), with the ground-state configuration 1 s 22 s 2, the symbol is readily predictable, because none of its electrons has non-zero orbital momentum, giving L = 0. Also, each electron pair is in the singlet spin state, i.e. we have S = 0, so J = 0

– the quantum number set described by the Russell-Saunders symbol 1 S 0 – just as for helium.

However, for the next, boron atom ( Z = 5), with its ground-state electron configuration 1 s 22 s 22 p 1

(see, e.g., Fig. 3.24), there is no obvious way to predict the result. Indeed, this atom has two pairs of electrons, with opposite spins, on its two lowest s-orbitals, giving zero contributions to the net S, L, and J. Hence these total quantum numbers may be only contributed by the last, fifth electron with s = ½ and l = 1, giving S = ½, L = 1. As was discussed in Sec. 5.7 for the single-particle case, the vector addition of the angular momenta S and L enables two values of the quantum number J: either L + S = ³/2 or L – S

= ½. Experiment shows that the difference between the energies of these two states of boron is very small (~2 meV), so at room temperature (with k B T  26 meV) they are both partly occupied, with the genuine ground state having J = ½, so its Russell-Saunders symbol is 2 P 1/2.

Such energy differences, which become larger for heavier atoms, are determined both by the Coulomb and spin-orbit23 interactions between the electrons. Their quantitative analysis is rather involved (see below), but the results tend to follow simple phenomenological Hund rules, with the following hierarchy:

Rule 1. For a given electron configuration, the ground state has the largest possible S, and hence the largest possible multiplicity 2 S + 1.

Rule 2. For a given S, the ground state has the largest possible L.

Rule 3. For given S and L, J has its smallest possible value,  L – S , if the given sub-shell { n, l}

is filled not more than by half, while in the opposite case, J has its largest possible value, L + S.

Let us see how these rules work for the boron atom we have just discussed. For it, the Hund Rules 1 and 2 are satisfied automatically, while the sub-shell { n = 2, l = 1}, which can house up to 2(2 l

+ 1) = 6 electrons, is filled with just one 2 p electron, i.e. by less than a half of the maximum value. As a result, Rule 3 predicts the ground state’s value J = ½, in agreement with experiment. Generally, for 23 In light atoms, the spin-orbit interaction is so weak that it may be reasonably well described as an interaction of the total momenta L and S of the system – the so-called LS (or “Russell-Saunders”) coupling. On the other hand, in very heavy atoms, the interaction is effectively between the net momenta j k = l k + s k of the individual electrons

– the so-called jj coupling. This is the reason why in such atoms the Hund Rule 3 may be violated.

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lighter atoms, the Hund rules are well obeyed. However, the lower down the Hund rule hierarchy, the less “powerful” the rules are, i.e. the more often they are violated in heavier atoms.

Now let us discuss possible approaches to a quantitative theory of multiparticle systems – not only atoms. As was discussed in Sec. 1, if fermions do not interact directly, the stationary states of the system have to be the antisymmetric eigenstates of the permutation operator, i.e. to satisfy Eq. (55). To understand how such states may be formed from the single-electron ones, let us return for a minute to the case of two electrons, and rewrite Eq. (11) in the following compact form:

state

1 state

2





(8.60a)

1



β '

 



 '

 '





 









1,

number

particle

1



2

2 

β ' 

2,

number

particle

where, in the last form, the direct product signs are just implied. In this way, the Pauli principle is mapped on the well-known property of matrix determinants: if any two columns of a matrix coincide, its determinant vanishes. This Slater determinant approach24 may be readily generalized to N fermions occupying any N (not necessarily the lowest-energy) single-particle states ,  ’,  ’’, etc: state



list

 β'  "



  particle

Slater

 β'  "

determinant

1

 

 

N

list

(8.60b)



 N 1/2



!

 β'  "  





    



















N

The Slater determinant form is extremely nice and compact – in comparison with direct writing of a sum of N! products, each of N ket factors. However, there are two major problems with using it for practical calculations:

(i) For the calculation of any bra-ket product (say, within the perturbation theory) we still need to spell out each bra- and ket-vector as a sum of component terms. Even for a limited number of electrons (say N ~ 102 in a typical atom), the number N! ~ 10160 of terms in such a sum is impracticably large for any analytical or numerical calculation.

(ii) In the case of interacting fermions, the Slater determinant does not describe the eigenvectors of the system; rather the stationary state is a superposition of such basis functions, i.e. of the Slater determinants – each for a specific selection of N states from the full set of single-particle states – that is generally larger than N.

For atoms and simple molecules, whose filled-shell electrons may be excluded from an explicit analysis (by describing their effects, approximately, with effective pseudo-potentials), the effective number N may be reduced to a smaller number N ef of the order of 10, so N ef! < 106, and the Slater determinants may be used for numerical calculations – for example, in the Hartree-Fock theory – see the 24 It was suggested in 1929 by John C. Slater.

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next section. However, for condensed-matter systems, such as metals and semiconductors, with the number of free electrons is of the order of 1023 per cm3, this approach is generally unacceptable, though with some smart tricks (such as using the crystal’s periodicity) it may be still used for some approximate (also mostly numerical) calculations.

These challenges make the development of a more general theory that would not use particle numbers (which are superficial for indistinguishable particles to start with) a must for getting any final analytical results for multiparticle systems. The most effective formalism for this purpose, which avoids particle numbering at all, is called the second quantization.25 Actually, we have already discussed a particular version of this formalism, for the case of the 1D harmonic oscillator, in Sec. 5.4. As a reminder, after the definition (5.65) of the “creation” and “annihilation” operators via those of the particle’s coordinate and momentum, we have derived their key properties (5.89), 1/ 2

†

1/ 2

ˆ

a n  n

n 1 ,

ˆ a n   n  

1

n 1 ,

(8.61)

where n are the stationary (Fock) states of the oscillator. This property allows an interpretation of the operators’ actions as the creation/annihilation of a single excitation with the energy 0 – thus justifying the operator names. In the next chapter, we will show that such excitation of an electromagnetic field mode may be interpreted as a massless boson with s = 1, called the photon.

In order to generalize this approach to arbitrary bosons, not appealing to a specific system, we may use relations similar to Eq. (61) to define the creation and annihilation operators. The definitions look simple in the language of the so-called Dirac states, described by ket-vectors N , N ,

N

,

(8.62) Dirac

1

2 

,

j 

state

where Nj is the state occupancy, i.e. the number of bosons in the single-particle state j. Let me emphasize that here the indices 1, 2, … j,… number single-particle states (including their spin parts) rather than particles. Thus the very notion of an individual particle’s number is completely (and for indistinguishable particles, very relevantly) absent from this formalism. Generally, the set of single-particle states participating in the Dirac state may be selected arbitrarily, provided that it is full and orthonormal in the sense

'

'

N ,N , '

N ,

N ,N ... N

 





(8.63)

j' 

,

,

j 



,

1

2

1

2

N N'

N N'

N N'

1

1

2

2

j

j

though for systems of non- (or weakly) interacting bosons, using the stationary states of individual particles in the system under analysis is almost always the best choice.

Now we can define the particle annihilation operator as follows:

Boson

ˆ a N , N ,

N

N

N N

N

(8.64) annihilation

j



,

1/ 2

j  

,

,

j



 ,

1

j

 .

1

2

1

2

operator

Note that the pre-ket coefficient, similar to that in the first of Eqs. (61), guarantees that any attempt to annihilate a particle in an initially unpopulated state gives the non-existing (“null”) state: 25 It was invented (first for photons and then for arbitrary bosons) by P. Dirac in 1927, and then (in 1928) adjusted for fermions by E. Wigner and P. Jordan. Note that the term “second quantization” is rather misleading for the non-relativistic case discussed here, but finds certain justification in the quantum field theory.

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ˆ a N , N ,0 ,

,

(8.65)

j

j   0

1

2

where the symbol 0 j means zero occupancy of the j th state. According to Eq. (63), an equivalent way to write Eq. (64) is

'

N , '

N , , '

N ,... ˆ a . N , N , , N ,...  N 





(8.66)

1

2 

j

j

1

2 

1/ 2

j

j

N N'

N N'  N' , N 1

 

1

1

2

2

j

j

According to the general Eq. (4.65), the matrix element of the Hermitian-conjugate operator †

ˆ a is

j

'

'

'

†

'

'

'

*

N ,N ,, N ,

a N N

N

 N N

N

a N ,N

N

j  ˆ

,

,

j

,

,

j 

,

,,

,

j  ˆ

,

j

,

,

1

2

1

2

1

2

1

2

j 

 N , N ,, N ,

'

'

'

'

'

N

N ,N

N 

 N







(8.67)

1

2

j  

j 1/ 2

,,

,

1

1

2

j



 j 1/2 N N' N N'  N N'  

,

1

1

1

2

2

j

j

  N 







j

11/2 N N' N N'  N  N' ,

,

1

1

1

2

2

j

j

meaning that

Boson

creation

ˆ†

a N , N ,

N

N

N N

N

(8.68)

j

,

,







1

2

j 

 j 11/2 , ,,

,

1

j

 ,

1

2

operator

in total compliance with the second of Eqs. (61). In particular, this particle creation operator allows a description of the generation of a single particle from the vacuum (not null!) state  0, 0, …: ˆ†

a

,

0 ,

0



(8.69)

j

 0

, ,

j  0

,

,

0 ,

0  1

, ,

j 0 ,

and hence a product of such operators may create, from vacuum, a multiparticle state with an arbitrary set of occupancies: 26

ˆ†

a ˆ†

a  ˆ†

a ˆ†

a ˆ†

a  ˆ†

a

 ,

0 ,

0

 N N

N N

(8.70)

1

1

1

2

2

2



 ! !

1

2

1/2



,

, .

1

2







  





 

N

N

1 times

2 times

Next, combining Eqs. (64) and (68), we get

ˆ†

a ˆ a N , N ,

N

 N N N

N

(8.71)

j

j



,

j 

,

,

j

,

,

j  ,

1

2

1

2

so, just as for the particular case of the harmonic-oscillator excitations, the operator Number-counting

N ˆ

a ˆ†



a ˆ

(8.72)

j

j

j

operator

“counts” the number of particles in the j th single-particle state, while preserving the whole multiparticle state. Acting on a state by the creation-annihilation operators in the reverse order, we get ˆ a ˆ†

a N , N ,

N

 N  N N

N

(8.73)

j

j

,

,

1

2

j 

 j 1 , ,, , j .

1

2

Eqs. (71) and (73) show that for any state of a multiparticle system (which may be represented as a linear superposition of Dirac states with all possible sets of numbers Nj), we may write 26 The resulting Dirac state is not an eigenstate of every multiparticle Hamiltonian. However, we will see below that for a set of non-interacting particles it is a stationary state, so the full set of such states may be used as a good basis in perturbation theories of systems of weakly interacting particles.

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ˆ a ˆ†

a  ˆ†

a ˆ a



 ˆ a , ˆ†

a   I ,

ˆ

(8.74)

j

j

j

j

j

j





again in agreement with what we had for the 1D oscillator – cf. Eq. (5.68). According to Eqs. (63), (64), and (68), the creation and annihilation operators corresponding to different single-particle states do commute, so Eq. (74) may be generalized as

 ˆ a , ˆ† 

ˆ

a

 I ,

(8.75)

j

j'

jj'





Bosonic

operators:

while similar operators commute, regardless of which states they act upon:

commutation

relations

 ˆ†

a , ˆ† 



a

 ˆ a ,



a

 0ˆ

ˆ



.

(8.76)

 j

j' 

 j j' 

 



As was mentioned earlier, a major challenge in the Dirac approach is to rewrite the Hamiltonian of a multiparticle system, that naturally carries particle numbers k (see, e.g., Eq. (22) for k = 1, 2), in the second quantization language, in which there are no these numbers. Let us start with single- particle components of such Hamiltonians, i.e. operators of the type

N

Single-

F ˆ   f ˆ .

(8.77) particle

k

operator

k 1

where all N operators f âre similar, besides that each of them acts on one specific ( k th) particle, and N

k

is the total number of particles in the system, which is evidently equal to the sum of single-particle state occupancies:

N   N .

(8.78)

j

j

The most important examples of such operators are the kinetic energy of N similar single particles and their potential energy in an external field:

N

p 2

ˆ

N

T ˆ   k , ˆ U 

ˆ

 u(r .)

(8.79)

k

k 1 2 m

k 1



For bosons, instead of the Slater determinant (60), we have to write a similar expression, but without the sign alternation at permutations:

 N

1/ 2

!

!

1  N j  

N , ,

,

 

,

(8.80)

1  N

 



' "

j



 





N!









 

P

N operands

sometimes called the permanent. Note again that the left-hand side of this relation is written in the Dirac notation (that does not use particle numbering), while on its right-hand side, just in formulas of Secs. 1

and 2, the particle numbers are coded with the positions of the single-particle states inside the state vectors, and the summation is over all different permutations of the states in the ket – cf. Eq. (10).

(According to the basic combinatorics,27 there are N!/( N 1! Nj!) such permutations, so the front coefficient in Eq. (80) ensures the normalization of the Dirac state, provided that the single-particle 27 See, e.g., MA Eq. (2.3).

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states ,  ’, are normalized.) Let us use Eq. (80) to spell out the following matrix element for a system with ( N –1) particles:

ˆ

 N ,, N  ,...

1

F  N  ,

1 , N ,

j

j'

j

j' 

N !( N  )!

1 ( N  )!

1

N 

(8.81)

1

j

j'



  N N

 '  " 

f  '  "

j

j' 

1

1/ 2

ˆ

 



 ,

( N  )!

1

k

P N 1



P N 1



k 1



where all non-specified occupation numbers in the corresponding positions of the bra- and ket-vectors are equal to each other. Each single-particle operator f ˆ participating in the operator sum acts on the k

bra- and ket-vectors of states only in one ( k th) position, giving the following result, independent of the position number:



f ˆ 

  f ˆ 

 f .

(8.82)

j

th

k

j'

th

j

j'

jj'

i k

n

position

k

in

position

Since in both permutation sets participating in Eq. (81), with ( N – 1) state vectors each, all positions are equivalent, we can fix the position (say, take the first one) and replace the sum over k with the multiplication by of the bracket by ( N – 1). The fraction of permutations with the necessary bra-vector (with number j) in that position is Nj/( N – 1), while that with the necessary ket-vector (with number j’) in the same position is Nj’/( N – 1). As a result, the permutation sum in Eq. (81) reduces to N

N

( N  )

1

j

j'

f

 '   '  "

(8.83)

jj

 

 ,

N 1 N 1

'

P N 2 P N 2

where our specific position k is now excluded from both the bra- and ket-vector permutations. Each of these permutations now includes only ( Nj – 1) states j and ( Nj’ – 1) states j’, so using the state orthonormality, we finally arrive at a very simple result:

ˆ

 N ,, N  ,

1  F  N  ,

1  N ,

j

j'

j

j' 

N !( N  )!

1 ( N  )!

1 

N

N

1

j

j'

1/ 2

j

j'

( N  2)!



 N N

N 

f

(8.84)

j

j' 

(

)

1

( N  )!

1

N 1 N 1 jj' N !( N  )!

1 ( N  )!

1

1

j

j'



  N N

f

j

j' 1/ 2

.

jj'

On the other hand, let us calculate the matrix elements of the following operator:



†

f

ˆ a ˆ a .

(8.85)

jj'

j

j'

j, j'

A direct application of Eqs. (64) and (68) shows that the only non-vanishing elements are

 N ,, N  ,

1  f a†

ˆ a ˆ  N  ,1, N ,  N N 1/2 f .

(8.86)

j

j'

jj '

j

j'

j

j'

 j j' jj'

But this is exactly the last form of Eq. (84), so in the basis of Dirac states, the operator (77) may be represented as

Single-

particle

operator

F ˆ   f a ˆ† a ˆ .

(8.87)

jj'

j

j'

in Dirac

j, j'

representation

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This beautifully simple relation is the key formula of the second quantization theory and is essentially the Dirac-representation analog of Eq. (4.59) of the single-particle quantum mechanics. Each term of the sum (87) may be described by a very simple mnemonic rule: for each pair of single-particle states j and j’, first, annihilate a particle in the state j’, then create one in the state j, and finally weigh the result with the corresponding single-particle matrix element. One of the corollaries of Eq. (87) is that the expectation value of an operator whose eigenstates coincide with the Dirac states is ˆ

F   N , F  N , 

f N

(8.88)

j

j



,

jj

j

j

with an evident physical interpretation as the sum of single-particle expectation values over all states, weighed by the occupancy of each state.

Proceeding to fermions, which have to obey the Pauli principle, we immediately notice that any occupation number Nj may only take two values, 0 or 1. To account for that, and also make the key relation (87) valid for fermions as well, the creation-annihilation operators are defined by the following relations:

ˆ a N , N ,



a N N

 j

 

N N

(8.89) Fermion

j

,0 , j

,

0

ˆ

,

,

j

 1

, , j ( )

1

,

1

(

)

1

,

,,0 , j ,

1

2

1

2

1

2

creation-

annihilation

ˆ†

a N , N ,

j

N N

a N N

(8.90) operators

j

,0 , j  ( )

1  ,

1

(

 )

1

,

, 1

, , j ,

ˆ†

,

,

j

 1

, , j  ,

0

1

2

1

2

1

2

where the symbol ( J, J’) means the sum of all occupancy numbers in the states with numbers from J to J’, including the border points:

J'

( J , J' )   N ,

(8.91)

j

j J

so the sum participating in Eqs. (89)-(90) is the total occupancy of all states with the numbers below j.

(The states are supposed to be numbered in a fixed albeit arbitrary order.) As a result, these relations may be conveniently summarized in the following verbal form: if an operator replaces the j th state’s occupancy with the opposite one (1 with 0 and vice versa), it also changes the sign before the result if (and only if) the total number of particles in the states with j’ < j is odd.

Let us use this (perhaps somewhat counter-intuitive) sign alternation rule to spell out the ket-vector 11 of a completely filled two-state system, formed from the vacuum state 00 in two different ways. If we start by creating a fermion in state 1, we get

ˆ†

a

,

0 0  ( )

1 0 ,

1 0  ,

1 0 ,

ˆ†

a ˆ†

a

,

0 0  ˆ†

a

,

1 0  ( )

1 1 ,

1 1   ,

1 1 ,

(8.92a)

1

2

1

2

while if the operator order is different, the result is

ˆ†

a

,

0 0  ( )

1 0 ,

0 1  ,

0 1 ,

ˆ†

a ˆ†

a

,

0 0  ˆ†

a

,

0 1  ( )

1 0 ,

1 1  ,

1 1 ,

(8.92b)

2

1

2

1

so

 ˆ†

a ˆ†

a  ˆ†

a ˆ†

a 



,

0 0  0 .

(8.93)

1

2

2

1





Since the action of any of these operator products on any initial state rather than the vacuum one also gives the null ket, we may write the following operator equality:

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ˆ†

a ˆ†

a  ˆ†

a ˆ†

a

ˆ†

a , †

a

 .

0ˆ

ˆ

(8.94)

1

2

2

1

1

2 





It is straightforward to check that this result is valid for Dirac vectors of an arbitrary length, and does not depend on the occupancy of other states, so we may generalize it as

ˆ†

a , ˆ†



a

 a a

;

(8.95)

j

j'

 ˆ ,



j

j'   0

ˆ

ˆ









Fermionic

operators: these equalities hold for j = j’ as well. On the other hand, an absolutely similar calculation shows that commutation the mixed creation-annihilation commutators do depend on whether the states are different or not:28

relations

 a ˆ , a ˆ†  I

ˆ

.

(8.96)

j

j'

jj'





These equations look very much like Eqs. (75)-(76) for bosons, “only” with the replacement of commutators with anticommutators. Since the core laws of quantum mechanics, including the operator compatibility (Sec. 4.5) and the Heisenberg equation (4.199) of operator evolution in time, involve commutators rather than anticommutators, one might think that all the behavior of bosonic and fermionic multiparticle systems should be dramatically different. However, the difference is not as big as one could expect; indeed, a straightforward check shows that the sign factors in Eqs. (89)-(90) just compensate those in the Slater determinant, and thus make the key relation (87) valid for the fermions as well. (Indeed, this is the very goal of the introduction of these factors.)

To illustrate this fact on the simplest example, let us examine what the second quantization formalism says about the dynamics of non-interacting particles in the system whose single-particle properties we have discussed repeatedly, namely two nearly similar potential wells, coupled by tunneling through the separating potential barrier – see, e.g., Figs. 2.21 or 7.4. If the coupling is so small that the states localized in the wells are only weakly perturbed, then in the basis of these states, the single-particle Hamiltonian of the system may be represented by the 22 matrix (5.3). With the energy reference selected in the middle between the energies of unperturbed states, the coefficient b vanishes, this matrix is reduced to

 c

c 

h

z

 c  σ 

,

with c  c  ic ,





(8.97)



x

y

c

 c

 

z 

and its eigenvalues to

   c,

with c  c 

(8.98)



 2 2 2

c  c  c

x

y

z 1/ 2 .

Using the key relation (87) together with Eq. (97), we may represent the Hamiltonian of the whole system of particles in terms of the creation-annihilation operators:

†

†

†

†

ˆ

H  c ˆ a ˆ a  c ˆ a ˆ a  c ˆ a ˆ a  c ˆ a ˆ a , z 1

1



(8.99)

1

2

 2

1

z

2

2

where †

ˆ a and ˆ a are the operators of creation and annihilation of a particle in the corresponding

,

1 2

,

1 2

potential well. (Again, in the second quantization approach the particles are not numbered at all!) As 28 A by-product of this calculation is proof that the operator defined by Eq. (72) counts the number of particles Nj (now equal to either 1 or 0), just as it does for bosons.

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Eq. (72) shows, the first and the last terms of the right-hand side of Eq. (99) describe the particle energies 1,2 =  c z in uncoupled wells,

c ˆ†

a ˆ

ˆ

ˆ

a  c N   N ,

 c ˆ†

a ˆ

ˆ

ˆ

a   c N   N ,

(8.100)

z 1

1

z

1

1

1

z

2

2

z

2

2

2

while the sum of the middle two terms is the second-quantization description of tunneling between the wells.

Now we can use the general Eq. (4.199) of the Heisenberg picture to spell out the equations of motion of the creation-annihilation operators. For example,

i ˆ

















a

  a H  c a a a  c a a a  c a a a  c a a (8.101)

1

ˆ ˆ,

1



†

†

†

†

ˆ , ˆ ˆ

ˆ , ˆ ˆ

ˆ , ˆ ˆ

ˆ , ˆ ˆ .

z

1

1

1



a

1

1

2



1

2

1

z

1

2

2

































Since the Bose and Fermi operators satisfy different commutation relations, one could expect the right-hand side of this equation to be different for bosons and fermions. However, it is not so! Indeed, all commutators on the right-hand side of Eq. (101) have the following form:

 ˆ a , ˆ†

a ˆ a   ˆ a ˆ†

a ˆ a  ˆ†

a ˆ a ˆ a .

(8.102)

j

j'

j"

j

j'

j"

j'

j"

j





As Eqs. (74) and (94) show, the first pair product of operators on the right-hand side may be recast as ˆ a ˆ†

ˆ

a  I  ˆ†

a ˆ a ,

(8.103)

j

j'

jj'

j'

j

where the upper sign pertains to bosons and the lower one to fermions, while according to Eqs. (76) and (95), the very last pair product in Eq. (102) is

ˆ a ˆ a   ˆ a ˆ a ,

(8.104)

j"

j

j

j"

with the same sign convention. Plugging these expressions into Eq. (102), we see that regardless of the particle type, there is a universal (and generally very useful) commutation relation

 a ˆ , a ˆ† a ˆ   a ˆ  ,

(8.105)

j

j'

j"

j"

jj'





valid for both bosons and fermions. As a result, the Heisenberg equation of motion for operator ˆ a , and 1

the equation for ˆ a (which may be obtained absolutely similarly), are also universal:29

2

i ˆ a

  c ˆ a  c ˆ a ,

1

z 1

 2

(8.106)

i ˆ a



 c ˆ a  c ˆ a .

2

 1

z

2

This is a system of two coupled linear differential equations, which is similar to the equations for the c-number probability amplitudes of single-particle wavefunctions of a two-level system – see, e.g., Eq. (2.201) and the model solution of Problem 4.25. Their general solution is a linear superposition ˆ a ( t) 

ˆ ()



 t

(8.107)

,

1 2

,

1 2



exp  .



†

29 Equations of motion for the creation operators ˆ a are just the Hermitian conjugates of Eqs. (106), and do not

,

1 2

add any new information about the system’s dynamics.

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As usual, in order to find the exponents , it is sufficient to plug a particular solution a ˆ

t

( )  înto Eq. (106) and require that the determinant of the resulting linear

,

1 2

,

1 2



exp  t

system for the “coefficients” (actually, time-independent operators) ˆ

 equals zero. This gives us the

,

1 2

following characteristic equation

c  i

z



c



 0 ,

(8.108)

c

 c  i

z





with two roots  =  i/2, where   2 c/ – cf. Eq. (5.20). Now plugging each of the roots, one by one, into the system of equations for ˆ

 , we can find these operators, and hence the general solution of

,

1 2

system (98) for arbitrary initial conditions.

Let us consider the simple case cy = cz = 0 (meaning in particular that the wells are exactly aligned, see Fig. 2.21), so /2  c = cx; then the solution of Eq. (106) is

 t

 t

 t

t



ˆ a ( t)  ˆ a (0) cos

 i ˆ a (0)sin

,

ˆ a ( t)   i ˆ a (0)sin

 ˆ a (0) cos

.

(8.109)

1

1

2

2

2

2

1

2

2

2

Multiplying the first of these relations by its Hermitian conjugate, and ensemble-averaging the result, we get

t



t



Quantum

N  ˆ†

a ( t) ˆ a ( t)  ˆ†

a ( )

0 ˆ a ( )

0 cos2

 ˆ†

a ( )

0 ˆ a ( )

0 sin 2

1

1

1

1

1

2

2

oscillations:

second

2

2

(8.110)

quantization

t



t



 i ˆ†

a ( )

0 ˆ a ( )

0  ˆ†

a ( )

0 ˆ a ( )

0 sin

cos

.

form

1

2

2

1

2

2

Let the initial state of the system be a single Dirac state, i.e. have a definite number of particles in each well; in this case, only the two first terms on the right-hand side of Eq. (110) are different from zero, giving:30

t



t



N  N ( )

0 cos2

 N ( )

0 sin 2

.

(8.111)

1

1

2

2

2

For one particle, initially placed in either well, this gives us our old result (2.181) describing the usual quantum oscillations of the particle between two wells with the frequency . However, Eq. (111) is valid for any set of initial occupancies; let us use this fact. For example, starting from two particles, with initially one particle in each well, we get  N 1 = 1, regardless of time. So, the occupancies do not oscillate, and no experiment may detect the quantum oscillations, though their frequency  is still formally present in the time evolution equations. This fact may be interpreted as the simultaneous quantum oscillations of two particles between the wells, exactly in anti-phase. For bosons, we can go on to even larger occupancies by preparing the system, for example, in the state with N 1(0) = N, N 2(0) = 0.

The result (111) says that in this case, we see that the quantum oscillation amplitude increases N-fold; this is a particular manifestation of the general fact that bosons can be (and in time, stay) in the same quantum state. On the other hand, for fermions we cannot increase the initial occupancies beyond 1, so the largest oscillation amplitude we can get is if we initially fill just one well.

30 For the second well’s occupancy, the result is complementary, N 2( t) = N 1(0) sin2 t + N 2(0) cos2 t, giving a good sanity check: N 1( t) + N 2( t) = N 1(0) + N 2(0) = const.

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The Dirac approach may be readily generalized to more complex systems. For example, Eq. (99) implies that an arbitrary system of potential wells with weak tunneling coupling between the adjacent wells may be described by the Hamiltonian

H ˆ   a ˆ† a ˆ 

 a ˆ† a ˆ

,

h.c.

(8.112)

j

j

j





jj'

j

j'

j

 j, j' 

where the symbol { j, j’} means that the second sum is restricted to pairs of next-neighbor wells – see, e.g., Eq. (2.203) and its discussion. Note that this Hamiltonian is still a quadratic form of the creation-annihilation operators, so the Heisenberg-picture equations of motion of these operators are still linear, and its exact solutions, though possibly cumbersome, may be studied in detail. Due to this fact, the Hamiltonian (112) is widely used for the study of some phenomena, for example, the very interesting Anderson localization effects, in which a random distribution of the localized-site energies  j prevents tunneling particles, within a certain energy range, from spreading to unlimited distances.31

8.4. Perturbative approaches

The situation becomes much more difficult if we need to account for explicit interactions between the particles. Let us assume that the interaction may be reduced to that between their pairs (as in the case at the Coulomb forces and most other interactions32), so it may be described by the following

“pair-interaction” Hamiltonian

1 N

ˆ

U



ˆ

 u (r , r ,)

(8.113)

int

2

int

k

k'

k , k ' 1



k  k '

with the front factor of ½ compensating the double-counting of each particle pair by this double sum. Pair-interaction

The translation of this operator to the second-quantization form may be done absolutely similarly to the Hamiltonian: derivation of Eq. (87), and gives a similar (though naturally more involved) result two forms

1

ˆ

U 

u

ˆ†

a ˆ†

a ˆ a ˆ



a ,

(8.114)

int

2

jj'll'

j

j' l' l

j, j' , l, l'

where the two-particle matrix elements are defined similarly to Eq. (82):

u

   ˆ u   .

(8.115)

jj'll'

j

j'

int

l

l'

The only new feature of Eq. (114) is a specific order of the indices of the creation operators. Note the mnemonic rule of writing this expression, similar to that for Eq. (87): each term corresponds to moving a pair of particles from states l and l’ to states j’ and j (in this order!) factored with the corresponding two-particle matrix element (115).

However, with the account of this term, the resulting Heisenberg equations of the time evolution of the creation/annihilation operators become nonlinear, so solving them and calculating observables from the results is usually impossible, at least analytically. The only case when some general results 31 For a review of the 1D version of this problem, see, e.g., J. Pendry, Adv. Phys. 43, 461 (1994).

32 A simple but important example from the condensed matter theory is the so-called Hubbard model, in which particle repulsion limits their number on each of localized sites to either 0, or 1, or 2, with negligible interaction of the particles on different sites – though the next-neighbor sites are still connected by tunneling, as in Eq. (112).

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may be obtained is the weak interaction limit. In this case, the unperturbed Hamiltonian contains only single-particle terms such as (79), and we can always (at least as a matter of principle :-) find such a basis of orthonormal single-particle states  j in which that Hamiltonian is diagonal in the Dirac representation:

H ˆ (0)   (0)

 a ˆ† a ˆ .

(8.116)

j

j

j

j

Now we can use Eq. (6.14), in this basis, to calculate the interaction energy as a first-order perturbation:

)

1

(

1

ˆ

E

 N , N , U N , N , 

N , N ,  u ˆ†

a ˆ†

a ˆ a ˆ a N , N ,

int

1

2

int

1

2

jj'll'

j

j'

l'

l



2

1

2

1

2

j, j' , l, l'

(8.117)

1



 u N , N ,

jj'll'

 ˆ†

a ˆ†

a ˆ a ˆ a N , N ,

j

j'

l'

l

 .

2

1

2

1

2

j, j' , l, l'

Since, according to Eq. (63), the Dirac states with different occupancies are orthogonal, the last long bracket is different from zero only for three particular subsets of its indices: (i) j  j’, l = j, and l’ = j’. In this case, the four-operator product in Eq. (117) is equal to ˆ†

a ˆ†

a ˆ a ˆ a , and applying the proper commutation rules twice, we can bring it to the so-called normal j

j'

j'

j

ordering, with each creation operator standing to the right of the corresponding annihilation operator, thus forming the particle number operator (72):

a ˆ† a ˆ† a ˆ a ˆ   a ˆ† a ˆ† a ˆ a ˆ   a ˆ†  a ˆ a ˆ† 



a ˆ  a ˆ† a ˆ a ˆ† a ˆ  N ˆ N ˆ , (8.118)

j

j'

j'

j

j

j'

j

j'

j

j

j'

j'

j

j

j'

j'

j

j'





with a similar sign of the final result for bosons and fermions.

(ii) j  j’, l = j’, and l’ = j. In this case, the four-operator product is equal to a ˆ† a ˆ† a ˆ a ˆ , and j

j'

j

j'

bringing it to the form N ˆ N ˆ requires only one commutation: j

j'

a ˆ† a ˆ† a ˆ a ˆ  a ˆ†  a ˆ a ˆ† 



a ˆ   a ˆ† a ˆ a ˆ† a ˆ   N ˆ N ˆ , (8.119)

j

j'

j

j'

j

j

j'

j'

j

j

j'

j'

j

j'





with the upper sign for bosons and the lower sign for fermions.

(iii) All indices are equal to each other, giving a ˆ† a ˆ† a ˆ a ˆ  a ˆ† a ˆ† a ˆ a ˆ . For fermions, such an j

j' l' l

j

j

j

j

operator (that “tries” to create or to kill two particles in a row, in the same state) immediately gives the null vector. In the case of bosons, we may use Eq. (74) to commute the internal pair of operators, getting ˆ†

a ˆ†

a ˆ a ˆ a  ˆ†

a  ˆ a ˆ†

ˆ

a  I 



ˆ

ˆ

ˆ

ˆ

a  N ( N  I ) .

(8.120)

j

j

j

j

j

j

j

j

j

j





Note, however, that this expression formally covers the fermion case as well (always giving zero). As a result, Eq. (117) may be rewritten in the following universal form:

Particle

interaction:

)

1

(

1

1

E



energy

 N N u  u   N N  u

(8.121)

int

j

j'  jj'jj'

jj'j'j 

(

)

1

.

2

j

j

jjjj

correction

j, j'

2 j

j j'

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The corollaries of this important result are very different for bosons and fermions. In the former case, the last term usually dominates, because the matrix elements (115) are typically the largest when all basis functions coincide. Note that this term allows a very simple interpretation: the number of the diagonal matrix elements it sums up for each state ( j) is just the number of interacting particle pairs residing in that state.

In contrast, for fermions, the last term is zero, and the interaction energy is proportional to the difference between the two terms inside the first parentheses. To spell them out, let us consider the case when there is no direct spin-orbit interaction. Then the vectors  j of the single-particle state basis may be represented as direct products  o j  m j  of their orbital and spin-orientation parts. (Here, for the brevity of notation, I am using m instead of ms.) For spin-½ particles, including electrons, mj may equal only either +½ or –½; in this case, the spin part of the first matrix element proportional to ujj’jj’ equals m  m' m  m' ,

(8.122)

where, as in the general Eq. (115), the position of a particular state vector in each direct product encodes the particle’s number. Since the spins of different particles are defined in different Hilbert spaces, we may swap their state vectors to get

m  m' m  m'   m m    m' m'   1, (8.123)

1

2

for any pair of j and j’. On the other hand, the second matrix element, ujj’j’j, is factored as m  m' m'  m   m m'    m' m   

.

(8.124)

1

2

mm'

In this case, it is convenient to rewrite Eq. (121) in the coordinate representation, by using single-particle wavefunctions called spin-orbitals

 r

( )  r 

 r o  m .

(8.125) Spin-

j

j



 j

orbital

They differ from the spatial parts of the usual orbital wavefunctions of the type (4.233) only in that their index j should be understood as the set of the orbital-state and the spin-orientation indices.33 Also, due to the Pauli-principle restriction of the numbers Nj to either 0 or 1, Eq. (121) may be also rewritten without the explicit occupancy numbers, with the understanding that the summation is extended only over the pairs of occupied states. As a result, it becomes

 *

*



 r 

'

r u

r '

r  r 

'

r

Energy

)

1

(

1

( )

( )

( , )

( )

( )

correction

3

3

j

j'

int

E



d r d r' 

j

j'





.

int

 

(8.126)

2

*

due to

j j



*











, '



(r)

( '

r ) u (r, '

r )

(r)

( '

r )

fermion

j j'



j

j'

int

j'

j



interaction

In particular, for a system of two electrons, we may limit the summation to just two states ( j, j’ =

1, 2). As a result, we return to Eqs. (39)-(41), with the bottom (minus) sign in Eq. (39), corresponding to the triplet spin states. Hence, Eq. (126) may be considered as the generalization of the direct and exchange interaction picture to an arbitrary number of orbitals and an arbitrary total number N of 33 The spin-orbitals (125) are also close to spinors (13), besides that the former definition takes into account that the spin s of a single particle is fixed, so the spin-orbital may be indexed by the spin’s orientation m  ms only.

Also, if an orbital index is used, it should be clearly distinguished from j, i.e. the set of the orbital and spin indices. This is why I believe that the frequently met notation of spin-orbitals as  j,s(r) may lead to confusion.

Chapter 8

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electrons. Note, however, that this formula cannot correctly describe the energy of the singlet spin states, corresponding to the plus sign in Eq. (39), and also of the entangled triplet states.34 The reason is that the description of entangled spin states, given in particular by Eqs. (18) and (20), requires linear superpositions of different Dirac states. (Proof of this fact is left for the reader’s exercise.) Now comes a very important fact: the approximate result (126), added to the sum of unperturbed energies  (0)

j

, equals the sum (over j) of exact eigenenergies of the so-called Hartree-Fock equation:35

2

 



2



  u

Hartree-

r  (r)



 j

Fock

2 m





(8.127)

equation

*

   ' u

' 



' 



' d r'   

 j' r  r r

r

r

r

r

r

int  ,

 ( ) ( )

( )

( )

j

j'

j'

j

 3

( ),

j

j

j'  j

where u(r) is the external field’s potential acting on each particle separately – see the second of Eqs.

(79). An advantage of this equation in comparison with Eq. (126) is that it allows the (approximate) calculation of not only the energy spectrum of the system but, simultaneously, a more exact calculation of the corresponding spin-orbitals  j(r), which takes into account the electron-electron interaction. Of course, Eq. (127) describes a system of mutually coupled integro-differential equations. There are, however, efficient methods of numerical solution of such systems, typically based on iterative approaches. One more important practical trick is the exclusion of the filled internal electron shells (see Sec. 3.7) from the explicit calculations because the shell states are virtually unperturbed by the valence electrons involved in typical atomic phenomena and chemical reactions. In this approach, the Coulomb field of the shells, described by fixed pre-calculated pseudo-potentials, is added to that of the nuclei.

This approach dramatically cuts the computing resources necessary for systems of relatively heavy atoms, enabling a pretty accurate simulation of electronic and chemical properties of rather complex molecules, with thousands of electrons.36 As a result, the Hartree-Fock approximation has become the de facto baseline of all so-called ab initio (“first-principle”) calculations in the very important field of quantum chemistry.37

In departures from this baseline, there are two opposite trends. For larger accuracy (and typically smaller systems), several “post-Hartree-Fock methods”, notably including the configuration interaction method , 38 that are more complex but may provide higher accuracy, have been developed. There is also a strong opposite trend of extending such ab initio (“first-principle”) methods to larger systems while sacrificing some of the results’ accuracy and reliability. The ultimate limit of this trend is applicable when the single-particle wavefunction overlaps are small and hence the exchange interaction is 34 Indeed, due to the condition j’ j, and Eq. (124), the calculated negative exchange interaction is limited to electron state pairs with the same spin direction – such as the factorable triplet states ( and ) of a two-electron system, in which the contribution of the E ex given by Eq. (41), to the total energy is also negative.

35 This equation was suggested in 1929 by Douglas Hartree for the direct interaction and extended to the exchange interaction by Vladimir Fock in 1930. It may be derived by variational methods, but to verify its compliance with Eq. (126), it is sufficient to multiply all terms of Eq. (127) by * j(r), integrate them over all r-

space (so the right-hand side would give  j), and then sum the single-particle energies over all occupied states j.

36 For condensed-matter systems, this and other computational methods are applied to single elementary spatial cells, with a limited number of electrons in them, using cyclic boundary conditions.

37 See, e.g., A. Szabo and N. Ostlund, Modern Quantum Chemistry, Revised ed., Dover, 1996.

38 That method, in particular, allows the calculation of proper linear superpositions of the Dirac states (such as the entangled states for N = 2, discussed above) which are missing in the generic Hartree-Fock approach.

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negligible. In this limit, the last term in the square brackets in Eq. (127) may be ignored and the multiplier  j(r) taken out of the integral, resulting in the Schrödinger equation for a single particle but in a self-consistent effective potential:

u (r)  u(r)  u (r), u (r) 

*

 (r )

(r, r ) (r )

.

(8.128) Hartree

ef

dir

dir



' u

'

' d 3 r'

j'

int

j'

approximation

j'  j

This is the so-called Hartree approximation – which gives reasonable results for some systems,39

especially those with low electron density.

However, in dense electron systems (such as typical atoms, molecules, and condensed matter), the exchange interaction described by the second term in the square brackets of Eqs. (126)-(127) may be as high as ~30% of the direct interaction, and frequently cannot be ignored. The tendency to take this interaction in the simplest possible form is currently dominated by the so-called Density-Functional Theory,40 universally known by its acronym DFT. In this approach, the equation solved for each eigenfunction  j(r) is a Schrödinger-like Kohn-Sham equation 2





Kohn-

2

    u(r)

KS

 u (r)  u (r)  (r)    (r)





,

(8.129) Sham

2

dir

xc

j

j

j

m





equation

where



KS

1

3

(r ' )

u (r)   e(r),

(r) 

d r'

,

(r)   en(r),

(8.130)

dir





4

r  r '

0

and n(r) is the total electron density in a particular point, calculated self-consistently as n(r)

*

  (r) (r .)

(8.131)

j

j

j

The most important feature of the Kohn-Sham Hamiltonian is the simplified description of the exchange and correlation effects by the effective exchange-correlation potential u xc(r). This potential is calculated in various approximations, most of them valid only in the limit when the number of electrons in the system is very high. The simplest of them (proposed by Kohn et al. in the 1960s) is the Local Density Approximation (LDA) in which the effective exchange potential at each point r is a function only of the electron density n at the same point, taken from the theory of a uniform gas of free electrons.41 However, for many tasks of quantum chemistry, the accuracy given by the LDA is insufficient because inside molecules, the density n typically changes very fast, so the DFT has become widely accepted in that field only after the introduction, in the 1980s, of more accurate though more cumbersome models for u xc(r), notably the so-called Generalized Gradient Approximations (GGAs).

Due to its relative simplicity, the so-modified DFT enables calculation of some properties of much 39 An example of the Hartree approximation is the Thomas-Fermi model of heavy atoms (with Z >> 1), in which the atom’s electrons, at each distance r from the nucleus, are treated as an ideal, uniform Fermi gas, with a certain density n( r) corresponding to the local value u ef( r), but a global value of their highest full single-particle energy, 

= 0, to ensure the equilibrium. (The analysis of this model is left for the reader’s exercise.) 40 It had been developed by Walter Kohn and his associates (notably Pierre Hohenberg) in 1965-66, and eventually (in 1998) was marked with a Nobel Prize in Chemistry for W. Kohn.

41 Just for the reader’s reference: for a uniform, degenerate Fermi-gas of electrons (with the Fermi energy F >> k B T), the most important, exchange part u x of u xc may be calculated analytically: u x = –(3/4) e 2 k F/40, where the Fermi momentum k

3

3

F = (2 m eF)1/2/ is defined by the electron density: n = 2(4/3) k F /(2)3  k F /32.

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larger systems than the methods based on the Hartree-Fock theory, with the same computing resources and reasonable precision. As a result, it has become a very popular tool for ab initio calculations. This popularity is enhanced by the availability of several advanced DFT software packages, some of them in the public domain.

Please note, however, that despite this undisputable success, this approach has its problems.

From my personal point of view, the most offensive of them is the implicit assumption of unphysical Coulomb interaction of an electron with itself – by dropping, on the way from Eq. (128) to Eq. (130), the condition j’  j at the calculation of u KS

dir

). As a result, all the available DFT packages I am aware of

are either unable to account for some charge transfer effects or require substantial artificial tinkering.42

Unfortunately, because of a lack of time/space, for details I have to refer the interested reader to specialized literature.43

8.5. Quantum computation and cryptography

Now I have to review the emerging fields of quantum computation and cryptography.44 These fields are currently the subject of intensive research and development efforts, which have already brought, besides much hype, some results of general importance. My coverage will focus on these results, referring the reader interested in details to special literature.45 Because of the very active stage of the field, the style of this section is closer to a brief literature review than to a textbook’s section.

Presently, most work on quantum computation and encryption is based on systems of spatially separated (and hence distinguishable) two-level systems – in this context, universally called qubits.46

Due to this distinguishability, the issues that were the focus of the previous sections of this chapter, including the second quantization approach, are irrelevant here. On the other hand, systems of qubits have some interesting properties that have not been discussed in this course yet.

First of all, a system of N >> 1 qubits may contain much more information than the same number of N classical bits. Indeed, according to the discussions in Chapter 4 and Sec. 5.1, an arbitrary pure state of a single qubit may be represented by its ket vector (4.37) – see also Eq. (5.1):



  u   u ,

(8.132)

1

1

2

2

N 1



42 For just a few examples, see N. Simonian et al., J. Appl. Phys. 113, 044504 (2013); M. Medvedev et al., Science 335, 49 (2017); A. Hutama et al., J. Phys. Chem. C 121, 14888 (2017).

43 See, e.g., either the monograph by R. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford U. Press, 1994, or the later textbook J. A. Steckel and D. Sholl, Density Functional Theory: Practical Introduction, Wiley, 2009. A popular review and references to more recent work in this still-developing field was given by A. Zangwill, Phys. Today 68, 34 (July 2015).

44 Since these fields are much related, they are often referred to under the common title of “quantum information science”, though this term is rather misleading, de-emphasizing physical aspects of the topic.

45 Despite the recent flood of new books on the field, one of its first surveys, by M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge U. Press, 2000, is perhaps still the best one.

46 In some texts, the term qubit (or “Qbit”, or “Q-bit”) is used instead for the information contents of a two-level system – very much like the classical bit of information (in this context, frequently called “Cbit” or “C-bit”) describes the information contents of a classical bistable system – see, e.g., SM Sec. 2.2.

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where { uj} is any orthonormal two-state basis. (It is natural and common to employ, as uj, the eigenstates of the observable that is eventually measured in the particular physical implementation of the qubit.) It is also common to write the kets of these base states as 0 and 1, so Eq. (132) takes the form



 a 0  a 1 

a j .

(8.133) Qubit state’s



0

1

1

 j

N

representation

j

(Here, and in the balance of this section, the letter j is used to denote an integer equal to either 0 or 1.) According to this relation, any state  of a qubit is completely defined by two complex c-numbers aj, i.e. by 4 real numbers. Moreover, due to the normalization condition  a 12 +  a 22 = 1, we need just 3

independent real numbers – say, the Bloch sphere coordinates  and  (see Fig. 5.3), plus the common phase , which becomes important only when we consider states of a several-qubit system.

This is a good time to note that a qubit is very much different from any classical bistable system used to store single bits of information – such as two possible voltage states of the usual SRAM cell (essentially, a positive-feedback loop of two transistor-based inverters). Namely, the stationary states of a classical bistable system, due to its nonlinearity, are stable with respect to small perturbations, so they may be very robust to unintentional interactions with their environment. In contrast, the qubit’s state may be disturbed (i.e. its representation point on the Bloch sphere shifted) by even minor perturbations, because it does not have such an internal state stabilization mechanism.47 Due to this reason, qubit-based systems are rather vulnerable to environment-induced drifts, including the dephasing and relaxation discussed in the previous chapter, creating major experimental challenges – see below.

Now, if we have a system of two qubits, the vectors of its arbitrary pure state may be represented as a sum of 22 = 4 terms,48



 a 00  a 01  a 10  a 11 

a

j j ,

(8.134)

N 

00

01

10

11

2

 j j 1 2

1 2

j , j

1 2

with four complex coefficients, i.e. eight real numbers, subject to just one normalization condition that follows from the requirement    = 1:

2

 a

 1

j j

.

(8.135)

1 2

j ,12

The evident generalization of Eqs. (133)-(134) to an arbitrary pure state of an N-qubit system is a sum of 2 N terms:





a

j j ... j

,

(8.136)

N



j j ... j

1 2

N

1 2

N

j , j 

,

j

1 2

N

including all possible combinations of 0s and 1s for N indices j, so the state is fully described by 2 N

complex numbers, i.e. 22 N  2 N+1 real numbers, with only one constraint, similar to Eq. (135), imposed by the normalization condition. This exponential growth of the information contents would not be 47 In this aspect as well, the information processing systems based on qubits are much closer to classical analog computers (which were popular once, but nowadays are used for a few special applications only) rather than classical digital ones.

48 Here and in most instances below I use the same shorthand notation as was used at the beginning of this chapter

– cf. Eq. (1b). In this short form, the qubit’s number is coded by the order of its state index inside a full ket-vector, while in the long form, such as in Eq. (137), by the order of its single-qubit vector in a full direct product.

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possible without the qubit state entanglement. Indeed, in the particular case when qubit states are not entangled, i.e. are factorable:



   ... 

,

(8.137)

1

2

N

N

where each  n is described by an equality similar to Eq. (133) with its individual expansion coefficients, the system state description requires only 3 N – 1 real numbers – e.g., N sets {, , } less one common phase.

However, it would be wrong to project this exponential growth of information contents directly on the capabilities of quantum computation, because this process has to include the output information readout, i.e. qubit state measurements. Due to the fundamental intrinsic uncertainty of quantum systems, the measurement of a single qubit even in a pure state (133) generally may give either of two results, with probabilities W 0 =  a 02 and W 1 =  a 12. To comply with the general notion of computation, any quantum computer has to provide certain (or virtually certain) results, and hence the probabilities Wj have to be very close to either 0 or 1, so before the measurement, each measured qubit has to be in one of the basis states – either 0 or 1. This means that the computational system with N output qubits, just before their final readout, has to be in one of the factorable states



 j j ... j

 j j ... j ,

(8.138)

1

2

N

1 2

N

N

which is a very small subset even of the set of all unentangled states (137), and whose maximum information contents is just N classical bits.

Now the reader may start thinking that this constraint strips quantum computations of any advantages over their classical counterparts, but such a view is also superficial. To show that, let us consider the scheme of the “baseline” type of quantum computation, shown in Fig. 3.

j

j

1 in

 j

1 out

1 in

1

qubit

 j 1out

classical

classical

bits

j

j

2

2 out

bits

in

of the

 j

 j 2

2 

2

qubit

in

U

out

of the

input

output

  

  

     

number

number

j

j

N in



N

j

out

 jN 

N 

qubit N

in

out

qubit state



unitary



qubit state

in

out

preparation

transform

measurement

Fig. 8.3. The baseline scheme of quantum computation.

Here each horizontal line (sometimes called a “wire”49) corresponds to a single qubit, tracing its time evolution in the same direction as at the usual time function plots: from left to right. This means 49 The notion of “wires” stems from the similarity between such quantum schemes and the schematics of classical computation circuits – see, e.g., Fig. 4a below. In the classical case, the lines may be indeed understood as physical wires connecting physical devices: logic gates and/or memory cells. In this context, note that classical computer components also have non-zero time delays, so even in that case, the left-to-right device ordering is useful to indicate the timing of (and frequently the causal relation between) the signals.

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that the left column in of ket-vectors describes the initial state of the qubits,50 while the right column

out describes their final (but pre-measurement) state. The box labeled U represents the qubit evolution in time due to their specially arranged interactions between each other and/or external drive “forces”.

These forces are assumed to be noise-free, and the system, during this evolution, is supposed to be ideally isolated from any dephasing and energy-dissipating environment, so the process may be described by a unitary operator defined in the 2 N-dimensional Hilbert space of N qubits: ˆ



 U  .

(8.139)

out

in

With the condition that the input and output states have the simple form (138), this equality reads



ˆ

j

j

... j

 U j

j

... j

.

(8.140)

1 



out

2 

 N 



out

out

1  

in

2 

 N 

in

in

The art of quantum computer design consists of selecting such unitary operators U ˆ that would:

– satisfy Eq. (140),

– be physically implementable, and

–enable substantial performance advantages of the quantum computation over its classical counterparts with similar functionality, at least for some digital functions (algorithms).

I will have time/space to demonstrate the possibility of such advantages on just one, perhaps the simplest example – the so-called Deutsch problem,51 discussing several common notions and issues of this field on the way. Let us consider the family of single-bit classical Boolean functions j out = f( j in).

Since both j are Boolean variables, i.e. may take only values 0 and 1, there are evidently only 22 = 4

such functions – see the first four columns of the following table:

f

f(0) f(1)

class

F

f(1)– f(0)

f 1

0

0

constant

0

0

(8.141)

f 2

0

1

balanced

1

+1

f

3

1

0

balanced

1

–1

f 4

1

1

constant

0

0

Of them, the functions f 1 and f 4, whose values are independent of their arguments, are called constants, while the functions f 2 (called “YES” or “IDENTITY”) and f 3 (“NOT” or “INVERSION”) are called balanced. The Deutsch problem is to determine the class of a single-bit function, implemented in a “black box”, as being either constant or balanced, using just one experiment.

50 As was discussed in Chapter 7, the preparation of a pure state (133) is (conceptually :-) straightforward. Placing a qubit into a weak contact with an environment of temperature T << / k B, where  is the difference between energies of the eigenstates 0 and 1, we may achieve its relaxation into the lowest-energy state. Then, if the qubit must be set into a different pure state, it may be driven there by the application of a pulse of a proper external classical “force”. In most physical implementations of qubits, the most practicable way for that step is to use the proper part of the Rabi oscillation period – see Sec. 6.5.

51 It is named after David Elieser Deutsch, whose 1985 paper (motivated by an inspirational but not very specific publication by Richard Feynman in 1982) launched the whole field of quantum computation.

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Classically, this is clearly impossible, and the simplest way to perform the function’s classification involves two similar black boxes f – see Fig. 4a.52 It also uses the so-called exclusive-OR

(XOR for short) gate whose output is described by the following function F of its two Boolean arguments j 1 and j 2:53

0, if j  j ,

F( j , j )  j  j 

(8.142)

1

2

1

2



1

2

1, if j  j .

1

2

In the particular circuit shown in Fig. 4a, the gate produces the following output: F  f ( )

0  f )

1

( ,

(8.143)

which is equal to 1 if f(0)  f(1), i.e. if the function f is balanced, and to 0 in the opposite case – see column F in Eq. (141).

(a)

1

1

(b)

0  1 F



1 

 0  1 

 

f ( )

0

2

2

0

f

0

H

F

H

1

1

XOR

F

 0  1 

0  1 

f )

1

(