Quantum Mechanics by Konstantin K. Likharev - HTML preview
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ˆ
ˆ
u ( t, t ) exp
H ( t' ) dt' .
(4.181)
0
explicit
t
expression
0
This replacement means that the first form of Eq. (176) should be replaced with k
1
k
t
i
1
k t
t
t
i
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
u ( t, t ) I
H ( t' ) dt'
I
dt dt ... dt H( t ) H( t )... H( t ). (4.182) 0
k
k
k 1
k !
k 1
k !
1
2
1
2
t
0
t 0
t 0
t 0
The proof that Eq. (182) satisfies Eq. (158) is absolutely similar to the one carried out above.
Chapter 4
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We may now use Eq. (181) to show that the time-evolution operator remains unitary at any moment, even for a time-dependent Hamiltonian, if it satisfies Eq. (179). Indeed, Eq. (181) yields
†
t
i
t
i
u ˆ t
( , t u
) ˆ t
( , t )
.
(4.183)
0
0
H êxp
t'
( ) dt'
H êxp
t"
( ) dt"
t
t
0
0
Since each of these exponents may be represented with the Taylor series (182), and, thanks to Eq. (179), different components of these sums may be swapped at will, the expression (183) may be manipulated exactly as the product of c-number exponents, for example rewritten as
i t
t
ˆ u ( t, t ) ˆ †
ˆ
ˆ
u ( t, t ) exp H ( t' ) dt' H ( t" ) dt"
I.ˆ
}
0êxp{
(4.184)
0
0
t
0
t 0
This property ensures, in particular, that the system state’s normalization does not depend on time:
( t) ( t) ( t ) ˆ †
u ( t,t ) ˆ u ( t,t ) ( t ) ( t ) ( t ) .
(4.185)
0
0
0
0
0
0
The most difficult cases for the explicit solution of Eq. (158) are those where Eq. (179) is violated.44 It may be proved that in these cases, Eqs. (181)-(182) should be replaced with the following Dyson series using the so-called time-ordering operator T ˆ :
i t
1 i k t
t
t
ˆ u ( t, t ˆ
) T
H t' dt'
I
dt dt
dt T H t H t
H t
0
êxp
(
ˆ
)
1 2
ˆ
...
k
ˆ ˆ
ˆ
( ) ( )... ( )
1
2
k ,
t
k 1 k !
0
t 0
t 0
t 0
(4.186)
ˆ
H t H t
t
t
where T ˆ
H t H t
1 ˆ 2
ˆ 1 ˆ 2 ,
for ,
2
1
ˆ H t H t
t
t
2 ˆ 1 ,
for .
1
2
Since we would not have time/space to use this relation in this course, I will skip its proof.45
Let me now return to the general discussion of quantum dynamics to outline its alternative, the Heisenberg picture. For its introduction, let us recall that according to Eq. (125), in quantum mechanics the expectation value of any observable A is a long bracket. Let us explore the even more general form of such a bracket:
A ˆ ,
(4.187)
because in some applications, the states and may be different. As was discussed above, in the Schrödinger picture the bra- and ket-vectors of the states evolve in time, while the operators of observables remain time-independent (if they do not explicitly depend on time). As a result, Eq. (187) applied to the moment t, may be represented as
ˆ
( t) A ( t) ,
(4.188)
S
where the index “S” is added to emphasize the Schrödinger picture. Let us apply the evolution law (157a) to the bra- and ket-vectors in this expression:
t ˆ
†
A t t
u t t A u t t
t
(4.189)
S
ˆ
( ) ˆ ( , )
ˆ ( , ) ( ) .
0
0
S
0
0
44 We will run into such situations in Chapter 7, but will not need to apply Eq. (186) there.
45 It may be found, for example, in Chapter 5 of J. Sakurai’s textbook – see References.
Chapter 4
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This equality means that if we form a long bracket with bra- and ket-vectors of the initial-time states, together with the following time-dependent Heisenberg operator 46
Heisenberg
operator
ˆ
†
ˆ
†
ˆ
A ( t) ˆ u ( t, t ) A ˆ u ( t, t ) ˆ u ( t, t ) A ( t ) ˆ u ( t, t ) , (4.190)
H
0
S
0
0
H
0
0
all experimentally measurable results will remain the same as in the Schrödinger picture: Heisenberg
t ˆ A t
ˆ
( t ) A ( t, t ) ( t ) .
(4.191)
0
H
0
0
picture
For full clarity, let us see how the Heisenberg picture works for the same simple (but very important!) problem of the spin-½ precession in a z-oriented magnetic field, described (in the z-basis) by the Hamiltonian matrix (164). In that basis, Eq. (157b) for the time-evolution operator becomes
u
u
1
0
u
u
u
u
11
12
11
12
11
12
i
.
(4.192)
t u
u
2 0
1 u
u
2
u
u
21
22
21
22
21
22
We see that in this simple case, the differential equations for different matrix elements of the evolution operator matrix are decoupled, and readily solvable by using the universal initial conditions (178):47
i t
/ 2
e
t
t
0
u( t )
0
,
Icos
i σ sin
.
(4.193)
z
0
i t / 2
e
2
2
Now let us use them in Eq. (190) to calculate the Heisenberg-picture operators of spin components – still in the z-basis. Dropping the index “H” for the notation brevity (the Heisenberg-picture operators are clearly marked by their dependence on time anyway), we get S ( t) u† ( t 0
, S
) (0)u( t,0) u† ( t,0)σ u( t,0)
x
x
2
x
i t / 2
0 1
e
0
i t / 2
e
0
(4.194)
2
i t / 2
1 0
0
e
0
i t / 2
e
0
ei t
σ cos t σ sin t
x
y
S (0)cos t S (0)sin t .
2 i t
2
x
y
e
0
Absolutely similar calculations of the other spin components yield
i t
0
ie
S t
( )
σ cos σ sin S (0)cos S ( )
0 sin , (4.195)
y
t
t
y
x
t
t
y
x
2
ie i t
0
2
46 Note that this relation is similar in structure to the first of Eqs. (94), with the state bases { v} and { u} loosely associated with the time moments, respectively, t and t 0.
47 We could of course use this solution, together with Eq. (157), to obtain all the above results for this system within the Schrödinger picture. In our simple case, the use of Eqs. (161) for this purpose was more straightforward, but in some cases, e.g., for some time-dependent Hamiltonians, an explicit calculation of the time-evolution matrix may be the best (or even the only practicable) way to proceed.
Chapter 4
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1
0
S ( t)
σ S ( )
0 .
(4.196)
z
2 0 1 2 z
z
One practical advantage of these formulas is that they describe the system’s evolution for arbitrary initial conditions, thus making the analysis of initial state effects very simple. Indeed, since in the Heisenberg picture, the expectation values of observables are calculated using Eq. (191) (with =
), with time-independent bra- and ket-vectors, such averaging of Eqs. (194)-(196) immediately returns us to Eqs. (170), (173), and (174), which were obtained above in the Schrödinger picture. Moreover, these equations for the Heisenberg operators formally coincide with the classical equations of the torque-induced precession for c-number variables. (Below we will see that the same exact correspondence is valid for the Heisenberg picture of the orbital motion.)
In order to see that the last fact is by no means a coincidence, let us combine Eqs. (157b) and (190) to form an explicit differential equation of the Heisenberg operator’s evolution. For that, let us differentiate Eq. (190) over time:
†
ˆ
d
ˆ u
† A
†
ˆ u
ˆ
ˆ
S
ˆ
A
A ˆ u ˆ u
ˆ u ˆ u A
.
(4.197)
H
S
S
dt
t
t
t
Plugging in the derivatives of the time evolution operator from Eq. (157b) and its Hermitian conjugate, and multiplying both sides of the equation by i, we get
ˆ
d
A
†
†
S
†
ˆ
i
A ˆ ˆ ˆ
ˆ
u
A
H ˆ u i ˆ u
ˆ u ˆ
ˆ
u A
ˆ u
H .
(4.198a)
H
S
S
dt
t
If for the Schrödinger-picture’s Hamiltonian, the condition (179) is satisfied, then, according to Eqs.
(177) or (182), the Hamiltonian commutes with the time evolution operator and its Hermitian conjugate, and may be swapped with any of them.48 Hence, we may rewrite Eq. (198a) as
d
†
† A ˆ
S
†
† A ˆ
i
A ˆ
ˆ
u
H ˆ A ˆ u ˆ i u
ˆ
u ˆ u ˆ A ˆ u ˆ H ˆ i u
ˆ
S u ˆ
†
,
. (4.198b)
H
S
S
u A ˆ
ˆ
S u
ˆ H ˆ
dt
t
t
Now using the definition (190) again, for both terms on the right-hand side, we may write d
A ˆ
Heisenberg
i
A ˆ i
,
.
(4.199)
H
A ˆH H ˆ
equation
dt
t
of motion
H
This is the so-called Heisenberg equation of motion.
Let us see how this equation looks for the same problem of the spin-½ precession in a z-oriented, time-independent magnetic field described in the z-basis by the Hamiltonian matrix (164), which does not depend on time. In this basis, Eq. (199) for the operator vector of spin reads49
†
†
48
ˆ
H ˆ ˆ
u H ˆ u ˆ u ˆ ˆ
ˆ
Due to the same reason,
H
u
H ; this is why the Hamiltonian operator’s index may be
H
S
S
S
dropped in Eqs. (198)-(199).
49 Using the commutation relations (155), this equation may be readily generalized to the case of an arbitrary magnetic field B( t) and an arbitrary state basis – the exercise highly recommended to the reader.
Chapter 4
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S
S
Ω
S
S
- S
11
12
11
12 1
0
0
12
i
,
Ω
.
(4.200)
S
S
S
S
S
21
22
2 21
22 0
1
0
21
Once again, the equations for different matrix elements are decoupled, and their solution is elementary: S t S
S t S
11
(0) const,
11
22
(0) const,
22
(4.201)
i t
i
t
S t S
e
S t S
e
12
(0)
,
12
21
(0)
.
21
According to Eq. (190), the initial values of the Heisenberg-picture matrix elements are just the Schrödinger-picture ones, so using Eq. (117) we may rewrite this solution in either of two forms:
i t
i t
1 0
0
e
0
S( t)
ie
n
n
n
2
x
y
z
i t
i t
0 1
e
0
ie
0
(4.202)
n
n e i t
z
,
where n n in .
2
x
y
n e i t
n
z
The simplicity of the last expression is spectacular. (Remember, it covers any initial conditions and all three spatial components of spin!) On the other hand, for some purposes the previous form may be more convenient; in particular, its Cartesian components give our earlier results (194)-(196).50
One of the advantages of the Heisenberg picture is that it provides a more clear link between classical and quantum mechanics, found by P. Dirac. Indeed, analytical classical mechanics may be used to derive the following equation of time evolution of an arbitrary function A( qj, pj, t) of the generalized coordinates qj and momenta pj of the system, and time t: 51
dA
A
,
A H ,
(4.203)
P
dt
t
where H is the classical Hamiltonian function of the system, and {..,..}P is the so-called Poisson bracket defined, for two arbitrary functions A( qj, pj, t) and B( qj, pj, t), as Poisson
A B
A B
bracket
A,
B
.
(4.204)
P
j p q
q p
j
j
j
j
Comparing Eq. (203) with Eq. (199), we see that the correspondence between the classical and quantum mechanics (in the Heisenberg picture) is provided by the following symbolic relation 50 Note that the “values” of the same Heisenberg operator at different moments of time may or may not commute.
For example, consider a free 1D particle, with the time-independent Hamiltonian H ˆ
p ˆ2
/ 2 m . In this case, Eq.
(199) yields the following equations: i x ˆ
[ x ˆ H ˆ
, ] i p ˆ / m and i ˆ p
[ ˆ ˆ
p, H ] 0 , with simple solutions
(similar to those for the classical motion): ˆ p( t) const ˆ p(0) and x ˆ t ( ) x ˆ(0) p ˆ(0) t / m , so
[ x ˆ( ),
0 x ˆ t
( )] [ x ˆ( ),
0 p ˆ( )]
0 t / m [ x ˆ , p ˆ ] t / m i t / m 0, for t 0.
S
S
51 See, e.g., CM Eq. (10.17). The notation there does not use the subscript “P” that is employed in Eqs. (203)-
(205) to distinguish the classical Poisson bracket (204) from the quantum anticommutator (34).
Chapter 4
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i
Classical
A
B
.
(4.205) vs.
P
A ˆ
,
B ˆ
,
quantum
mechanics
This relation may be used, in particular, for finding appropriate operators for some observables, if their form is not immediately evident from the correspondence principle.
Finally, let us discuss one more alternative picture of quantum dynamics. It is attributed to P. A.
M. Dirac, and is called either the “Dirac picture”, or (more frequently) the interaction picture. The last name stems from the fact that this picture is very useful for perturbative (approximate) approaches to systems whose Hamiltonians may be partitioned into two parts,
ˆ
ˆ
ˆ
H H H ,
(4.206)
0
int
where ˆ
H is the sum of relatively simple Hamiltonians of the component subsystems, while the second 0
term in Eq. (206) represents their weak interaction.52 (Note, however, that all relations in the balance of this section are exact and not directly based on the interaction weakness.) In this case, it is natural to consider, together with the full operator ˆ u t, t of the system’s evolution, which obeys Eq. (157b), a 0
similarly defined unitary operator ˆ u t, t of the “unperturbed” evolution described by ˆ
H alone:
0
0
0
ˆ
i
ˆ u H ˆ u ,
(4.207)
0
0 0
t
and also the following interaction evolution operator,
Interaction
u ˆ u ˆ† u ˆ .
(4.208) evolution
I
0
operator
The motivation for these definitions becomes more clear if we insert the reciprocal relation, ˆ u ˆ u ˆ†
u ˆ u ˆ u ˆ u ,
(4.209)
0
0
0
I
and its Hermitian conjugate,
†
ˆ u ˆ u ˆ u
ˆ u ˆ u ,
(4.210)
0
I †
† †
I
0
into the basic Eq. (189):
ˆ
†
ˆ
A ( t ) ˆ u ( t, t ) A ˆ u ( t, t ) ( t ) 0
0
S
0
0
(4.211)
( t ) ˆ†
†
u t t u t t A u t t u t t t
0
I , 0 ˆ0
0 ˆ
,
ˆ
S 0 , 0 Î , 0
( ) .
0
This relation shows that any long bracket (187), i.e. any experimentally verifiable result of quantum mechanics, may be expressed as
ˆ
ˆ
A ( t) A ( t) ( t) ,
(4.212)
I
I
I
if we assume that both the state vectors and the operators depend on time, with the vectors evolving only due to the interaction operator ˆ u ,
I
Interaction
( t) ( t ) ˆ†
u ( t, t ),
( t) ˆ u ( t, t ) ( t ) ,
(4.213) picture:
I
0
I
0
I
I
0
0
state vectors
52 This picture may also useful in more standard problems of the perturbation theory (see Ch. 6 below) where ˆ
H describes a weak perturbation of a single system described by a relatively simple Hamiltonian ˆ
H .
int
0
Chapter 4
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while the operators’ evolution being governed by the unperturbed operator ˆ u : 0
Interaction
picture:
ˆ
†
ˆ
A ( t) ˆ u t, t A ˆ u t, t .
(4.214)
I
0
0 S 0
0
operators
These relations describe the interaction picture of quantum dynamics. Let me defer an example of its use until the perturbative analysis of open quantum systems in Sec. 7.6, and end this section with proof that the interaction evolution operator (208) satisfies the following natural equation,
i
ˆ
ˆ
u H ˆ u ,
(4.215)
I
I I
t
where ˆ
H is the interaction Hamiltonian formed from ˆ
H in accordance with the same rule (214):
I
int
ˆ
H t ˆ†
ˆ
u t, t H ˆ u t, t .
(4.216)
I
0
0
int 0
0
The proof is very straightforward: first using the definition (208), and then Eqs. (157b) and the Hermitian conjugate of Eq. (207), we may write
†
u
u
i
ˆ
†
†
†
†
†
†
u i
u u i
u u i
H u u u
u
H H u u u H H
u
I
ˆ ˆ0
ˆ
ˆ
0 ˆ
ˆ
ˆ ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
0
0 0
0
0 0
0 ˆ
ˆ
0
int ˆ
t
t
t
t
(4.217)
ˆ
H ˆ†
u ˆ u ˆ† ˆ
u H ˆ u ˆ† ˆ
u H ˆ
†
†
†
u H u u H u u H u
0 0
0
0
0
int
ˆ ˆ ˆ ˆ
0 0
0
0 ˆ
ˆ ˆ ˆ.
0
int
Since †
ˆ u may be represented as an integral of an exponent of ˆ
H over time (similar to Eq. (181) relating
0
0
u ând H ˆ ), these operators commute, so the parentheses in the last form of Eq. (217) vanish. Now plugging u ˆ from the last form of Eq. (209), we get the equation,
i
ˆ u ˆ† ˆ
u H ˆ u u ˆ† ˆ
u H ˆ u ˆ u ,
(4.218)
I
0
int 0 I
0 int 0 I
t
which is clearly equivalent to the combination of Eqs. (215) and (216).
As Eq. (215) shows, if the energy scale of the interaction H int is much smaller than that of the background Hamiltonian H 0, the interaction evolution operators ˆ u and †
ˆ u , and hence the state vectors
I
I
(213) evolve relatively slowly, without fast background oscillations. This is very convenient for the perturbative approaches to complex interacting systems, in particular to the “open” quantum systems that weakly interact with their environment – see Sec. 7.6.
4.7. Coordinate and momentum representations
Now let me show that in application to the orbital motion of a particle, the bra-ket formalism naturally reduces to the notions and postulates of wave mechanics, which were discussed in Chapter 1.
For that, we first have to modify some of the above formulas for the case of a basis with a continuous spectrum of eigenvalues. In that case, it is more appropriate to replace discrete indices, such as j, j’, etc.
broadly used above, with the corresponding eigenvalue – just as it was done earlier for functions of the wave vector – see, e.g., Eqs. (1.88), (2.20), etc. For example, the key Eq. (68), defining the eigenkets and eigenvalues of an operator, may be conveniently rewritten in the form
Chapter 4
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A ˆ a
A a .
(4.219)
A
A
More substantially, all sums over such continuous eigenstate sets should be replaced with integrals. For example, for a full and orthonormal set of the continuous eigenstates aA, the closure relation (44) should be replaced with
Continuous
dA a
a
I ˆ
,
(4.220) spectrum:
A
A
closure
relation
where the integral is over the whole interval of possible eigenvalues of the observable A.53 Applying this relation to the ket-vector of an arbitrary state , we get the following replacement of Eq. (37):
I ˆ dA a a dA a a .
(4.221)
A
A
A
A
For the particular case when = aA’, this relation requires that
Continuous
a a
( A A' );
(4.222) spectrum:
A
A'
state ortho-
normality
this formula replaces the orthonormality condition (38).
According to Eq. (221), in the continuous case the bracket aA still plays the role of probability amplitude, i.e. a complex c-number whose modulus squared determines the state aA’s probability – see the last form of Eq. (120). However, for a continuous observable, the probability of finding the system exactly in a particular state is infinitesimal; instead (as was already discussed in Sec.
1.2), we should speak about the probability dW = w( A) dA of finding the observable within a small interval dA << A near the value A, with probability density w( A) aA 2. The coefficient of proportionality in this relation may be found by making a similar change from the summation to integration in the normalization condition (121):
dA a
a
.
1
(4.223)
A
A
Since the total probability of the system being in some state should be equal to w( A) dA, this means that Continuous
2
(
w )
A a
a a
.
(4.224) spectrum:
A
A
A
probability
density
Now let us see how we can calculate the expectation values of continuous observables, i.e. their ensemble averages. If we speak about the same observable A whose eigenstates are used as the continuous basis (or any compatible observable), everything is simple. Indeed, inserting Eq. (224) into the general statistical relation
A w( A) AdA
(4.225)
that is the obvious continuous version of Eq. (1.37), we get
A a A a
.
dA
(4.226)
A
A
Inserting a delta function to represent this expression formally as a double integral, A dA dA' a
A ( A A' ) a ,
(4.227)
A
A'
53 The generalization to cases when the eigenvalue spectrum consists of both a continuous interval plus some set of discrete values, is straightforward, though leads to somewhat bulky formulas.
Chapter 4
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and using the continuous-spectrum version of Eq. (98),
ˆ
a A a
A ( A A' ) ,
(4.228)
A
A'
we may write
ˆ
ˆ
A dA dA' a
a A a
a A ,
(4.229)
A
A
A'
A'
so Eq. (4.125) remains valid in the continuous-spectrum case without any changes. This formula is very convenient for applications because it does not require the calculation of the eigenstates aA, and its matrix form is valid in any basis.
Now we are ready for a discussion of the relationship between the bra-ket formalism and wave mechanics. (For the notation simplicity I will discuss its 1D version; its generalization to 2D and 3D
cases is straightforward.) Let us start with postulating the (intuitively, almost evident) existence of a quantum state basis, whose ket-vectors will be called x, corresponding to a certain definite value x of the particle’s coordinate. Writing the trivial identity x x = x x and comparing it with Eq. (219), we see that they do not contradict each other if we assume that x on the left-hand side of this relation is the Hermitian operator x ôf the particle’s coordinate, in a specific representation when its action on a ket-
(or bra-) vector is just the multiplication by the c-number x:
ˆ x x x x .
(4.230)
In this way, we consider vectors x to be the eigenstates of the operator x ˆ . (This looks like a proof, but is actually a separate, independent postulate, no matter how plausible.)
Let me hope that the reader will excuse me if I do not pursue here strict proof that the set of all x-
states is full and orthogonal,54 so we may apply Eq. (222) to it:
x x' x x' .
(4.231)
Using this basis is called the coordinate representation – the term which was already mentioned several times in this course, but without explanation. In the basis of the x-states, the inner product aA( t)
becomes x( t), and Eq. (223) takes the following form:
(
w x, t) ( t) x x ( t) x ( t) * x ( t) .
(4.232)
Comparing this formula with the basic postulate (1.22) of wave mechanics, we see that they coincide if the wavefunction of a time-dependent state is identified with that short bracket:55
Wave-
function
as inner
( x, t) x ( t)
.
(4.233)
product
This key formula provides the desired connection between the bra-ket formalism and the wave mechanics, and should not be too surprising for the (thoughtful :-) reader. Indeed, Eq. (45) shows that any inner product of two state vectors describing two states is a measure of their similarity – just as the scalar product of two geometric vectors is; the orthonormality condition (38) is a particular 54Such proof is rather involved mathematically, but physically this fact should be evident.
55 I do not quite like expressions like x used in some papers and even textbooks. Of course, one is free to replace with any other letter ( including) to denote a quantum state, but then it is better not to use the same letter to denote the wavefunction, i.e. an inner product of two state vectors, to avoid confusion.
Chapter 4
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QM: Quantum Mechanics
manifestation of this fact. In this language, the particular value (233) of a wavefunction at some point x and moment t characterizes “how much of a particular coordinate x” the state contains at time t. (Of course, this informal language is too crude to reflect the fact that ( x, t) is a complex function, which has not only a modulus but also an argument – the quantum-mechanical phase.) Now let us rewrite the most important formulas of the bra-ket formalism in the wave mechanics notation. Inner-multiplying both parts of Eq. (219), written for an arbitrary operator, by the ket-vector
x, and then inserting into the left-hand side of that relation the identity operator in the form (220) for coordinate x’, we get
dx' x A x' x' a
A x a
ˆ
,
(4.234)
A
A
i.e., using the wavefunction’s definition (233),
ˆ
dx' x A x' ( x' ) A ( x)
,
(4.235)
A
A
where, for the notation brevity, the time dependence of the wavefunction is just implied (with the capital
serving as a reminder of this fact), and will be restored when needed. For a general operator, we would have to stop here, because if it does not commute with the coordinate operator, its matrix in the x-
basis is not diagonal, and the integral on the left-hand side of Eq. (235) cannot be worked out explicitly.
However, virtually all quantum-mechanical operators discussed in this course56 are ( space-) local: they depend on only one spatial coordinate, say x. For such an operator, we may define its coordinate representation by the following equality (valid for an arbitrary wavefunction, not only A): Operator:
A ˆ ( x
ˆ
)
( )
.
(4.236) coordinate
in x
x A x' x' dx'
representation
The explicit form of the coordinate representation still needs to be determined for each operator type. Let us consider, for example, the 1D version of the Hamiltonian (1.41),
ˆ 2
p
ˆ
H
x
U ( ˆ x) ,
(4.237)
2 m
which was the basis of all our discussions in Chapter 2. Its potential-energy part U (even if it is time-dependent as well) commutes with the operator x ˆ , i.e. its matrix in the x-basis is diagonal. For such an operator, the long bracket in Eq. (236) may be transformed using Eq. (231): x U x' U x x x' , so the right-hand part of this equality becomes just U( x)( x). Comparing it with the left-hand part, we see that the coordinate representation of such an operator is given merely by the c-number function U( x). (Eq. (230) may be viewed as just a particular manifestation of this rule.) The situation with the momentum operator p ˆ (and hence the kinetic energy p ˆ2 / 2 m ), which do x
x
not commute with x ˆ , is less evident. Let me show that its coordinate representation is given by the 1D
version of Eq. (1.26), if we postulate that the commutation relation (2.14),
x ˆ, p ˆ i I,ˆ
i.e. ˆ p
x ˆ p ˆ x ˆ i I ˆ ,
(4.238)
x
x
56 The only substantial exception is the statistical operator w ˆ ( x, x’), to be discussed separately in Chapter 7.
Chapter 4
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is valid in any representation.57 For that, let us consider the following matrix element: x ˆ p x ˆ p ˆ x ˆ x' .
x
x
On one hand, we may use Eq. (238), and then Eq. (231), to write
x ˆˆ p
x
ˆ p ˆ
ˆ
x x' x i I x' i x x' i ( x x' ) .
(4.239)
x
x
On the other hand, since x ˆ x' x' x' and x x ˆ x x , we may represent the same matrix element as x ˆ p
x ˆ p ˆ x ˆ x' x p
x ˆ p ˆ x' x'
'
ˆ
.
(4.240)
x
x
x
x
x x x p x'
x
Comparing Eqs. (239) and (240), we get
( x x'
x p ˆ x'
)
i
x
.
(4.241)
x x'
As it follows from the definition of the delta function,58 all expressions involving it acquire final sense only at their integration, in our current case, as described by Eq. (236). Plugging Eq. (241) into the right-hand side of that relation, we get
x x'
x p ˆ x' ( x' ) dx' i
( )
.
(4.242)
x
x' dx'
x x'
Since the right-hand-part integral is contributed only by an infinitesimal vicinity of the point x’ = x, we may calculate it by expanding the continuous wavefunction ( x’) into the Taylor series in small ( x’ – x), and keeping only two leading terms of the series, so Eq. (242) is reduced to
x x'
x'
x p ˆ x' ( x' ) dx' i ( x)
dx'
.
(4.243)
x
x x'
x' dx'
x
x x'
x'
Since the delta function may be always understood as an even function of its argument, in our case of ( x
– x’), the first term on the right-hand side is proportional to an integral of an odd function in symmetric limits and is equal to zero, and we get59
x p ˆ x' ( x' ) dx' i
.
(4.244)
x
x
Comparing this expression with the left-hand side of Eq. (236) with A ˆ p ˆ , we see that in the x
coordinate representation, we indeed get the 1D version of Eq. (1.26), which was used so much in Chapter 2,60
p ˆ
i
.
(4.245)
x in x
x
57 Another possible approach to the axiomatics of wave mechanics is to derive Eg. (238) by postulating the form, ˆ
T exp{ i ˆ p X / }, of the operator that shifts any wavefunction by distance X along the x-axis. In my X
x
approach, this expression will be derived when we need it (in Sec. 5.5), while Eq. (238) is postulated.
58 If necessary, please revisit MA Sec. 14.
59 One more useful expression of this type, which may be proved similarly, is (/ x)( x – x’) = ( x – x’)/ x’.
60 This means, in particular, that in the sense of Eq. (236), the operator of differentiation is local, despite the fact that its action on a function f may be interpreted as the limit of the fraction f/ x, involving two points. (In some axiomatic systems, local operators are defined as arbitrary polynomials of functions and their derivatives.) Chapter 4
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It is virtually evident (and straightforward to prove by using the Taylor expansion just as in Sec.
6) that the coordinate representation of any operator function f ( ˆ p ) is x
f i
.
(4.246)
x
In particular, this pertains to the kinetic energy operator in Eq. (237), so the coordinate representation of this Hamiltonian also takes the very familiar form:
1
2
2
2
ˆ
H
i
U ( x, t)
U ( x, t) .
(4.247)
in x
2 m
x
2
2
m x
Now returning to the discussion of the general Eq. (235), and comparing its last form with that of Eq. (236), we see that for a local operator in the coordinate representation, the eigenproblem (219) takes the form
ˆ A
( x) A ( x),
(4.248) Eigenproblem
in x
A
A
in x-
representation
even if the operator A ˆ does not commute with the operator x ˆ . The most important case of this coordinate-representation form of the eigenproblem (68) is the familiar Eq. (1.60) for the eigenvalues En of the energy of a system with a time-independent Hamiltonian.
The operator locality also simplifies the expression for its expectation value. Indeed, plugging the closure relation in the form (231) into the general Eq. (125) twice (written in the first case for x and in the second case for x’), we get
ˆ
*
ˆ
A dx dx' ( t) x x A x' x' ( t) dx dx' ( x, t) x A x' ( x' , t)
.
(4.249)
Now, Eq. (236) reduces this result to just
A dx dx' *( x, t A ˆ
)
( x, t
*
ˆ
)
( , )
( , )
.
(4.250)
in x
x x'
x t A x t dx
in x
i.e. to Eq. (1.23), which had to be postulated in Chapter 1 where the x-representation of the operators was just implied.
Finally, let us discuss the time evolution of the wavefunction, in the Schrödinger picture. For that, we may use Eq. (233) to calculate the (partial) time derivative of the wavefunction of some state :
i
i
x ( t) .
(4.251)
t
t
Since the coordinate operator x ˆ does not depend on time explicitly, its eigenstates x are stationary, and we can swap the time derivative and the time-independent bra-vector x. Now using the Schrödinger-picture equation (158), and then inserting the identity operator in the continuous form (220) of the closure relation, written for the coordinate eigenstates,
dx' x' x'
I ˆ
,
(4.252)
we may continue to develop the right-hand side of Eq. (251) as
Chapter 4
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x i
( t
ˆ
)
x H ( t
ˆ
)
dx' x H x' x' ( t
ˆ
)
dx' x H x' Ψ ( x' ) ,
(4.253)
t
If the Hamiltonian operator is local, we may apply Eq. (236) to the last expression, to get the familiar form (1.28) of the Schrödinger equation:
i
H ˆ
.
(4.254)
in x
t
So, for the local operators that obey Eq. (236), we have been able to derive all the basic notions and postulates of the wave mechanics from the bra-ket formalism. Moreover, the formalism has allowed us to get a very useful equation (248) for an arbitrary local operator, which will be repeatedly used below. (In the first three chapters of this course, we have only used its particular case (1.60) for the Hamiltonian operator.)
Now let me deliver on my promise to develop a more balanced view of the de Broglie wave (4.1), which would be more respectful to the evident r p symmetry of the coordinate and momentum.
Let us discuss the 1D case when the wave may be represented as
px
( x) a exp
,
all
for
.
(4.255)
p
p
i
x
(For the sake of brevity, from this point to the end of the section, I am dropping the index x in the notation of the momentum – just as it was done in Chapter 2.) Let us have a good look at this function.
Since it satisfies Eq. (248) for the 1D momentum operator (245),
ˆ p
p ,
(4.256)
in x
p
p
p is an eigenfunction of that operator. But this means that we can also write Eq. (219) for the corresponding ket-vector:
p ˆ p p p ,
(4.257)
and according to Eq. (233), the wavefunction (255) may be represented as
( x) x p ,
so
*
( x) p x .
(4.258)
p
p
These expressions are quite remarkable in their x p symmetry – which may be pursued further on. Before doing that, however, we have to discuss the normalization of such wavefunctions. Indeed, in this case, the probability density w( x) of the wave (255) is constant, so its integral
w( x) dx ( x) *
( x) dx
(4.259)
p
p
diverges if ap 0. Earlier in the course, we discussed two ways to avoid this divergence. One is to use a very large but finite integration volume – see Eq. (1.31). Another way is to work with wave packets of the type (2.20), possibly of a very large length and hence a very narrow spread of the momentum values.
Then the integral (259) may be required to equal 1 without any conceptual problem.
However, both these methods, while being convenient for the solution of many particular problems, violate the x p symmetry and hence are unfit for our current conceptual discussion.
Chapter 4
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Instead, let us continue to identify the eigenvectors p and p of the momentum with the bra- and ket-vectors aA and aA of the general theory described at the beginning of this section. Then the normalization condition (222) becomes
p p' ( p p' ).
(4.260)
Inserting the identity operator in the form (252), with the integration variable x’ replaced by x, into the left-hand side of this equation, and using Eq. (258), we can translate this normalization rule to the wavefunction language:
*
dx p x x p' dx ( x) ( x) ( p p' ).
(4.261)
p
p'
For the particular wavefunction (255), this requirement turns into the following condition:
*
( p' p) x
2
a a
exp i
dx a
2
( p p' ) ( p p' ),
(4.262)
p
p'
p
so, finally, ap = ei/(2)1/2, where is an arbitrary (real) phase, and Eq. (255) becomes61
1
px
( x) x p
exp
.
(4.263)
p
i
2 1/2
Now let us represent an arbitrary wavefunction ( x) as a wave packet of the type (2.20), based on the wavefunctions (263), taking = 0 for the notation brevity, because the phase may be incorporated into the (generally, complex) envelope function ( p):
1
px
x-
( x)
( )exp
.
(4.264) representation:
2
p
i
1/ 2
dp
wavefunctions
From the mathematical point of view, this is just a 1D Fourier spatial transform, and its reciprocal is 1
px
p-
( p)
( )exp
.
(4.265) representation:
2
x
1/ 2
i
dx
wavefunctions
These expressions are completely symmetric, and represent the same wave packet; this is why the functions ( x) and ( p) are frequently called the reciprocal representations of a quantum state of the particle: respectively, its coordinate ( x-) and momentum ( p-) representations. Using Eq. (258), and Eq.
(263) with = 0, they may be recast into simpler forms,
( x) ( p) x p dp,
( p)
( x) p x dx
,
(4.266)
in which the inner products satisfy the basic postulate (14) of the bra-ket formalism: 1
px
*
p x
.
(4.267)
exp i
x p
2
1/ 2
61 Repeating such calculation for each Cartesian component of a plane monochromatic wave of arbitrary dimensionality d, we get p = (2)– d/2exp{ i(pr/ + )}.
Chapter 4
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Next, we already know that in the x-representation, i.e. in the usual wave mechanics, the coordinate operator x îs reduced to the multiplication by x, and the momentum operator is proportional to the partial derivative over the coordinate:
x-
representation:
ˆ x
x,
ˆ p
i
.
(4.268)
in x
in x
operators
x
It is natural to guess that in the p-representation, the expressions for operators would be reciprocal: p-
representation:
ˆ x
i
,
ˆ p
p,
(4.269)
in p
in
operators
p
p
with the only difference of one sign, which is due to the opposite signs of the Fourier exponents in Eqs.
(264) and (265). The proofs of Eqs. (269) are straightforward; for example, acting by the momentum operator on the arbitrary wavefunction (264), we get
1
px
1
px
ˆ p ( x) i ( x)
(4.270)
x
p
i
i
dp
p
p
i
dp
2
( )
exp
1/ 2
x
2
( ) exp
,
1/ 2
and similarly for the operator x âcting on the function ( p). Comparing the final form of Eq. (270) with the initial Eq. (264), we see that the action of the operators (268) on the wavefunction (i.e. the state’s x-representation) gives the same results as the action of the operators (269) on the function (i.e. its p-
representation).
It is also illuminating to have a different look at this coordinate-momentum duality. For that, notice that according to Eqs. (82)-(84), we may consider the bracket x p as an element of the (infinite-size) matrix Uxp of the unitary transform from the x-basis to the p-basis. Let us use this fact to derive the general operator transform rule that would be a continuous version of Eq. (92). Say, we want to calculate the general matrix element of some operator known in the x-representation, in the p-
representation:
p A ˆ p' .
(4.271)
Inserting two identity operators (252) written for x and x’ into this bracket, and then using Eq. (258) and its complex conjugate, and also Eq. (236) (again, valid only for space-local operators!), we get ˆ
ˆ
*
ˆ
p A p' dx dx' p x x A x' x' p' dx dx' ( x) x A x' ( x' )
p
p'
1
px
(4.272)
ˆ
p'x
dx exp
i
A
exp
x
i
.
2
in
As a sanity check, for the momentum operator itself, this relation yields:
1
px
p'x
p'
( p' p) x
p ˆ p p'
dx exp
i
i
exp i
exp i
dx p' ( p' p). (4.273)
2
x
2
Due to Eq. (257), this result is equivalent to the second of Eqs. (269).
From a thoughtful reader, I anticipate the following natural question: why is the momentum representation used much less often than the coordinate representation – i.e. wave mechanics? The answer is purely practical: with an important exception of the 1D harmonic oscillator (to be revisited in Chapter 4
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Sec. 5.4), in most systems, the orbital-motion Hamiltonian (237) is not x p symmetric, with the potential energy U(r) typically being a more complex function than the kinetic energy p 2/2 m. Because of that, it is easier to analyze such systems treating the potential energy operator just as a c-number multiplier, as it is in the coordinate representation – and as this was done in Chapters 1-3.
The most significant exception from this practice is the motion in a periodic potential in the presence of a coordinate-independent external force F( t). As was discussed in Secs. 2.7 and 3.4, in such periodic systems the eigenenergies En(q), playing the role of the effective kinetic energy of the particle, may be rather involved functions of its quasimomentum q, while its effective potential energy U ef = –
F( t)r due to the additional force F( t) is a very simple function of coordinates. This is why detailed analyses of the quantum effects that were briefly discussed in Sec. 2.8 (the Bloch oscillations, etc.) and also such statistical phenomena as drift, diffusion, etc.62 in solid-state theory are typically based on the momentum (or rather quasimomentum) representation.
4.8. Exercise problems
4.1. Prove that if A ând B âre linear operators, and C is a c-number, then: (i) A ˆ† † A ˆ;
(ii) ˆ
CA†
* †
ˆ
C A ;
(iii) ˆ ˆ
AB†
† †
ˆ ˆ
B A ;
(iv) the operators
†
ˆ ˆ A
A
and A ˆ† A âre Hermitian.
4.2. Prove that for any linear operators ˆ ˆ ˆ
ˆ
,
A B, C
and
,
D,
ˆ B
A ˆ C ˆ
, D ˆ ˆ
A B ˆ C ˆ
, D ˆ ˆ
C
A ˆ B ˆ ˆ
,
D A ˆ C ˆ
, ˆ B
D ˆ C ˆ A ˆ ˆ
,
D B ˆ .
4.3. Calculate all possible binary products j j’ (for j, j’ = x, y, z) of the Pauli matrices defined by Eqs. (105), and their commutators and anticommutators (defined similarly to those of the corresponding operators). Summarize the results by using the Kronecker delta and Levi-Civita permutation symbols.63
4.4. Calculate the following expressions,
(i) (c) n, and then
(ii) ( b I + c) n,
for the scalar product c of the Pauli vector’s matrix n x x + n y y + n z z by an arbitrary c-number geometric vector c, where n is a non-negative integer c-number and b is an arbitrary scalar c-number.
Hint: For Task (ii), you may like to use the binomial theorem64 and then transform the result to a form enabling you to use the same theorem backward.
4.5. Use the solution of the previous problem to derive Eqs. (2.191) for the transparency T of the Dirac comb – a system of N similar, equidistant, delta-functional potential barriers.
62 In this series, a brief discussion of these effects may be found in SM Chapter 6.
63 See, e.g., MA Eqs. (13.1) and (13.2).
64 See, e.g. MA Eq. (2.9).
Chapter 4
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4.6. Use the solution of Problem 4(i) to spell out the following matrix: exp{ i n}, where is the 3D vector (117) of the Pauli matrices, n is a c-number geometric vector of unit length, and is a c-
number scalar.
4.7. Use the solution of Problem 4(ii) to calculate exp{A}, where A is an arbitrary 22 matrix.
4.8. Express all elements of the matrix B exp{A} explicitly via those of the 22 matrix A.
Spell out your result for the following matrices:
a a
i i
A
,
A '
,
a a
i i
with real a and .
4.9. Prove that for arbitrary square matrices A and B,
Tr (
)
AB Tr (BA) .
Is each diagonal element ( AB) jj necessarily equal to ( BA) jj?
4.10. Calculate the trace of the following 22 matrix:
A a σb σc σ,
where is the Pauli vector’s matrix, while a, b, and c are arbitrary c-number vectors.
4.11. Prove that the matrix trace of an arbitrary operator does not change at its unitary transformation.
4.12. Prove that for any two full and orthonormal bases { u} and { v} of the same Hilbert space, Tr u v
v u
j
j'
.
j'
j
4.13. Is the 1D scattering matrix S, defined by Eq. (2.124), unitary? What about the 1D transfer matrix T defined by Eq. (2.125)?
4.14. Calculate the trace of the following matrix:
exp ia
σ
exp ib
σ ,
where is the Pauli vector’s matrix, while a and b are c-number geometric vectors.
4.15. Prove the following operator-vector identity:
σ ˆrσ ˆp I ˆr ˆp iσ ˆr ˆp, where is the Pauli vector’s matrix, and I is the 22 identity matrix.
Hint: Take into account that the operator vectors rând pâre defined in the orbital-motion Hilbert space, different from that of the Pauli vector σˆ , and hence commute with it – even though they do not commute with each other.
Chapter 4
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4.16. Let Aj be the eigenvalues of some operator A ˆ . Express the following two sums,
and
,
1
A
A 2
j
2
j
j
j
via the matrix elements Ajj’ of this operator in an arbitrary basis.
4.17. Calculate of a spin–½ in the quantum state with the following ket-vector: z
const ,
where (, ) and (, ) are the eigenstates of the Pauli matrices z and x, respectively.
Hint: Double-check whether your solution is general.
4.18. A spin-½ is fully polarized in the positive z-direction. Calculate the probabilities of the alternative outcomes of a perfect Stern-Gerlach experiment with the magnetic field oriented in an arbitrarily different direction.
4.19. In a certain basis, the Hamiltonian of a two-level system is described by the matrix
E
0
1
H
,
with E E
,
1
2
0
E
2
while the operator of some observable A of this system, by the matrix
1
1
A
.
1
1
For the system’s state with the energy definitely equal to E 1, find the possible results of measurements of the observable A and the probabilities of the corresponding measurement outcomes.
4.20. Three states u 1,2,3 form a full and orthonormal basis of a system with the following Hamiltonian
ˆ
H u u u u u u
1
2
2
3
3
1
,
h.c.
where is a real constant, while h.c. means the Hermitian conjugate of the previous expression.
Calculate its stationary states and energy levels. Can you relate this system to any other(s) discussed earlier in the course?
4.21. Guided by Eq. (2.203), and by the solutions of the previous problem and also of Problem 3.15, suggest a Hamiltonian describing particle’s dynamics in an infinite 1D chain of similar potential wells within the tight-binding approximation, in the bra-ket formalism. Verify that its eigenstates and eigenvalues correspond to those discussed in Sec. 2.7.
4.22. In a certain full and orthonormal basis of three states u 1,2,3, operators A ând B âre defined by the following equalities:
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
A u u , A u u , A u u ;
B u u , B u ,
0
B u u .
1
3
2
2
3
1
1
1
2
3
3
Chapter 4
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(i) Prove that the operators 2
ˆ A and B ˆ commute and form an orthonormal basis of their common eigenstates.
(ii) Give the most general expression for the matrix (in the u-basis) of an operator that would commute with B ˆ .
4.23. Calculate the eigenvectors and eigenvalues of the following matrices:
0 0 0 1
0 1 0
0 0 1 0
A 1 0 1,
B
0 1 0 0
0 1 0
1 0 0 0
4.24. A certain state is an eigenstate of each of two operators, A ând B ˆ . What can be said about the corresponding eigenvalues a and b, if the operators anticommute?
4.25. An operator A ˆ commutes with each of two other operators B ând ˆ
C , but these two operators
do not commute: B ˆ C ˆ
, 0. Prove that the full set of eigenvalues of the operator A încludes some degenerate ones.
4.26. Derive the differential equation for the time evolution of the expectation value of an observable, by using (i) the Schrödinger picture and (ii) the Heisenberg picture of quantum dynamics.
4.27. At t = 0, a spin-½ whose interaction with an external field is described by the Hamiltonian H ˆ c ˆσ c σ ˆ c σ ˆ c σ ˆ
x
x
y
y
z
z
(where cx,y,z are real c-number constants, and ˆ
are the Pauli operators) was in the state , one of the
x, y, z
two eigenstates of ˆ . In the Schrödinger picture, calculate the time evolution of: z
(i) the ket-vector of the spin (in any time-independent basis you like),
(ii) the probabilities to find the spin in the states and , and
(iii) the expectation values of all three Cartesian components of the spin vector.
Analyze and interpret the results for the particular case cy = cz = 0.
Hint: Think about the best basis to use for the solution.
4.28. For the same system as in the previous problem, use the Heisenberg picture to calculate the time evolution of:
(i) all three Cartesian components of the spin operator ˆS ( t), and H
(ii) the expectation values of the spin components.
Compare the latter results with those of the previous problem.
4.29. For the same system as in the two previous problems, calculate the matrix elements of the operator în the basis of the stationary states of the system.
z
Chapter 4
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4.30. In the Schrödinger picture of quantum dynamics, certain three operators satisfy the following commutation relation:
A ˆ B ˆ, C ˆ .
What is their relation in the Heisenberg picture, at a certain time instant t?
4.31. Prove the Bloch theorem given by either Eq. (3.107) or Eq. (3.108), where R is an arbitrary vector of the Bravais lattice (3.106).
Hint: Analyze the commutation properties of the so-called translation operator T ˆ , defined by R
the following result of its action on an arbitrary function f(r): ˆ
T f (r) f (r R) ,
R
and apply them to an eigenfunction (r) of the stationary Schrödinger equation for a particle moving in the periodic potential described by Eq. (3.105).
4.32. A constant force F is applied to an (otherwise free) 1D particle of mass m. Calculate the stationary wavefunctions of the particle in:
(i) the coordinate representation, and
(ii) the momentum representation.
Discuss the relation between the results.
4.33. Use the momentum representation to re-solve the problem discussed at the beginning of Sec. 2.6, i.e. calculate the eigenenergy of a 1D particle of mass m, localized in a very short potential well of “weight” W.
4.34. The momentum representation of a certain operator of orbital 1D motion is p-1. Use two different approaches to find its coordinate representation.
4.35.* For a particle moving in a 3D periodic potential, develop the bra-ket formalism for the q-
representation, in which a complex amplitude similar to aq in Eq. (2.234) (but generalized to 3D and all energy bands) plays the role of the wavefunction. In particular, calculate the operators r and v in this representation, and use the result to prove Eq. (2.237) for the 1D case in the low-field limit.
4.36. A uniform, time-independent magnetic field B = n z B is induced in z
one semi-space, while the other semi-space is field-free, with a sharp plane B
boundary x = 0 between these two regions – see figure on the right. A monochromatic beam of non-relativistic, electrically-neutral spin-½ particles 0
with a gyromagnetic ratio 0,65 in a certain spin state and with a kinetic
x
energy E, propagating within the [ x, z] plane, is incident on this boundary from the field-free side under angle . Calculate the coefficient of particle reflection E
from the boundary.
65 The fact that may be different from zero even for electrically-neutral particles such as neutrons, is explained by the Standard Model of the elementary particles, in which a neutron “consists” (in a broad sense of this word) of three electrically-charged quarks with zero net charge.
Chapter 4
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Chapter 5. Some Exactly Solvable Problems
The objective of this chapter is to describe several relatively simple but important applications of the bra-ket formalism, including a few core problems of wave mechanics we have already started to discuss in Chapters 2 and 3.
5.1. Two-level systems
The discussion of the bra-ket formalism in the previous chapter was peppered with numerous illustrations of its main concepts on the examples of “spin-½-like” systems with the smallest non-trivial (two-dimensional) Hilbert space. In such a system, the bra- and ket-vectors of an arbitrary quantum state
may be represented as a linear superposition of just two basis vectors, for example
,
(5.1)
where the states and are defined as the eigenstates of the Pauli matrix z – see Eq. (4.105). For the genuine spin-½ particles (such as electrons) placed in a z-oriented time-independent magnetic field, these states are the stationary “spin-up” and “spin-down” stationary states of the Pauli Hamiltonian (4.163), with the corresponding two energy levels (4.167).
However, an approximate but reasonable quantum description of some other important systems may also be given in such Hilbert space. For example, as was discussed in Sec. 2.6, two weakly coupled space-localized orbital states of a spin-free particle are sufficient for an approximate description of its quantum oscillations between two potential wells. A similar coupling of two traveling waves explains the energy band splitting in the weak-potential approximation of the band theory – Sec. 2.7. As will be shown in the next chapter, in systems with time-independent Hamiltonians, such a situation almost unavoidably appears each time when two energy levels are much closer to each other than to other levels. Moreover, as will be shown in Sec. 6.5, a similar truncated description is adequate even in cases when two levels En and En’ of an unperturbed system are not close to each other, but the corresponding states become coupled by an applied ac field of a frequency very close to the difference ( En – En’ )/.
Such two-level systems are nowadays the focus of additional attention in the view of prospects of their use for quantum information processing and encryption.1 This is why I will spend a bit more time reviewing the main properties of an arbitrary two-level system.
The most general form of the Hamiltonian of a two-level system is represented, in an arbitrary basis, by a 22 matrix
H
H
H
11
12
.
(5.2)
H
H
21
22
According to the discussion in Secs. 4.3-4.5, since the Hamiltonian operator has to be Hermitian, the diagonal elements of the matrix H have to be real, and its off-diagonal elements have to be complex 1 In the last context, to be discussed in Sec. 8.5, the two-level systems are usually called qubits.
© K. Likharev
QM: Quantum Mechanics
conjugate: H 21 = H 12*. As a result, we may not only represent H as a linear combination (4.106) of the identity matrix and the Pauli matrices but also reduce it to a more specific form:
b c
c ic b c
c
z
x
y
H I
z
b c σ
,
with c c ic ,
(5.3)
x
y
c ic
b c c
b c
x
y
z
z
where the scalar b and the Cartesian components of the vector c are real c-number coefficients: H H
H H
H H
H H
11
22
b
,
12
21
c
Re H ,
21
12
c
Im H ,
11
22
c
. (5.4)
2
x
2
21
y
2
21
i
z
2
If such a Hamiltonian does not depend on time, the corresponding characteristic equation (4.103) for the system’s energy levels E,
b c E
c
z
0,
(5.5)
c
b c E
z
is a simple quadratic equation, with the following roots:
1/ 2
2
1/ 2
1/ 2
H H
H H
2
E b c b c c
c
b
c
c
c
H
.(5.6)
2
z
2
2
2
x
y
z
11
22
11
22
21
2
2
The parameter b ( H 11 + H 22)/2 evidently gives the average energy E(0) of the system, which does not contribute to the level splitting
1/
1/ 2
2
2
E E E 2 c 2 c
c
c
H
H
H
.
(5.7)
2 2 2
x
y
z
4
11
22
2
21
So, the splitting is a hyperbolic function of the coefficient cz ( H 11 – H 22)/2. A plot of this function is the famous level-anticrossing diagram (Fig. 1), which has already been discussed in Sec. 2.7 in the particular context of the weak-potential limit of the 1D band theory.
(0)
E E
E
H
21
0
H H
11
22
c
z
H
2
21
E
Fig. 5.1. The level-anticrossing diagram
for an arbitrary two-level system.
The physics of the diagram becomes especially clear if the two states of the basis used to spell out the matrix (2) may be interpreted as the stationary states of two potentially independent subsystems, with the energies, respectively, H 11 and H 22. (For example, in the case of two weakly coupled potential wells discussed in Sec. 2.6, these are the ground-state energies of two very distant wells.) Then the off-diagonal elements c
*
– H 12 and c+ H 21 = H 12 describe the subsystem coupling, and the anticrossing Chapter 5
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Essential Graduate Physics
QM: Quantum Mechanics
diagram shows how do the eigenenergies of the coupled system depend (at fixed coupling) on the difference of the subsystem energies. As was already discussed in Sec. 2.7, the most striking feature of the diagram is that any non-zero coupling c
2
2
( cx + cy )1/2 changes the topology of the eigenstate energies, creating a gap of the width E.
As it follows from our discussions of particular two-level systems in Secs. 2.6 and 4.6, the dynamics of such systems also has a general feature – the quantum oscillations. Namely, if we put any two-level system into any initial state different from one of its eigenstates , and then let it evolve on its own, the probability of its finding the system in any of the “partial” states exhibits oscillations with the frequency
c
2
E
E E
,
(5.8)
lowest at the exact subsystem symmetry ( cz = 0, i.e. H 11 = H 22), when it is proportional to the coupling strength: min = 2 c/ 2 H 12/ = 2 H 21/.
In the case discussed in Sec. 2.6, these are the oscillations of a particle between the two coupled potential wells (or rather of the probabilities to find it in either well) – see, e.g., Eqs. (2.181). On the other hand, for a spin-½ particle in an external magnetic field, these oscillations take the form of spin precession in the plane normal to the field, with periodic oscillations of its Cartesian components (or rather their expectation values) – see, e.g., Eqs. (4.173)-(4.174). Some other examples of the quantum oscillations in two-level systems may be rather unexpected; for example, the ammonium molecule NH3
(Fig. 2) has two similar states that differ only by the inversion of the nitrogen atom relative to the common plane of the three hydrogen atoms. These states are weakly coupled due to the quantum-mechanical tunneling of the nitrogen atom through this plane.2 Since for this particular molecule, in the absence of external fields, the level splitting E corresponds to an experimentally convenient frequency
/2 24 GHz, it played an important historic role in the initial development of the atomic frequency standards and microwave quantum generators ( masers) in the early 1950s,3 which paved the way for laser technology.
N
nm
0.102
107.8
H
H
H
Fig. 5.2. An ammonia molecule and its inversion.
Now let us now discuss a very convenient geometric representation of an arbitrary quantum state
of any two-level system. As Eq. (1) shows, such a state is completely described by two complex 2 Since the hydrogen atoms are much lighter, it may be fairer to speak about the tunneling of their triangle around the (nearly immobile) nitrogen atom.
3 In particular, these molecules were used in the demonstration of the first maser by C. Townes’ group in 1954.
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coefficients ( c-numbers) – say, and . If the vectors of the basis states and are normalized, then these coefficients must obey the following restriction:
W
.
(5.9)
*
*
2
2
*
*
1
This requirement is automatically satisfied if we take the moduli of and equal to the sine and cosine of the same real angle. Thus we may write, for example,
i
i(
)
cos e , sin e
.
(5.10)
2
2
Moreover, according to the general Eq. (4.125), if we deal with just one two-level system,4 the common phase factor exp{ i} drops out of the calculation of any expectation value, so we may take = 0, and Eq.
(10) is reduced to
cos , sin
i
e
Bloch
.
(5.11)
2
2
sphere
representation
The reason why the argument of these sine and cosine functions is usually taken in the form /2, is clear from Fig. 3a: Eq. (11) conveniently maps each state of a two-level system onto a certain representation point on a unit-radius Bloch sphere,5 with the polar angle and the azimuthal angle .
z
(a)
z
(b)
z
(c)
y
c
y
c
0
x
0
x
0
x
Fig. 5.3. The Bloch sphere: (a) the representation of an arbitrary state (solid red point) and the eigenstates of the Pauli matrices (black-dotted points), and (b, c) the two-level system’s evolution: (b) in a constant “field” c directed along the z-axis, and (c) in an arbitrarily orientated field.
In particular, the basis state , described by Eq. (1) with = 1 and = 0, corresponds to the North Pole of the sphere ( = 0), while the opposite state , with = 0 and = 1, to its South Pole (
= ). Similarly, the eigenstates and of the matrix x, described by Eqs. (4.122), i.e. having =
4 If you need a reminder of why this condition is crucial, please revisit the discussion at the end of Sec. 1.6. Note also that the mutual phase shifts between different two-level systems are important, in particular, for quantum information processing (see Sec. 8.5 below), so most discussions of these applications have to start from Eq. (10) rather than Eq. (11).
5 This representation was suggested in 1946 by the same Felix Bloch who pioneered the energy band theory discussed in Chapters 2-3.
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1/2 and = 1/2, correspond to the equator ( = /2) points with, respectively, = 0 and = .
Two more special points (denoted in Fig. 3a as ⊙ and ) are also located on the sphere’s equator, at =
/2 and = /2; it is easy to check that they correspond to the eigenstates of the matrix y (in the same z-basis).
To understand why this mutually perpendicular location of these three special point pairs on the Bloch sphere is not occasional, let us plug Eqs. (11) into Eqs. (4.131)-(4.133) for the expectation values of the spin-½ components. In terms of the Pauli vector operator (4.117),
ˆ
ˆσ S / / 2, the result is
sin cos,
sin sin,
cos ,
(5.12)
x
y
z
showing that the radius vector of any representation point is just the expectation value .
Now let us use Eq. (3) to see how the representation point moves in various cases, ignoring the term b I – which, again, describes the offset of the total energy of the system relative to some reference level, and does not affect its dynamics. First of all, according to Eq. (4.158), if c = 0 (when the Hamiltonian operator turns to zero, and hence the state vectors do not depend on time) the point does not move at all, and its position is determined by initial conditions, i.e. by the system’s preparation. If c
0, we may re-use some results of Sec. 4.6, obtained for the Pauli Hamiltonian (4.163a), which coincides with Eq. (3) if6
c
B .
(5.13)
2
In particular, if the field B, and hence the vector c, is directed along the z-axis and is time-independent, Eqs. (4.170) and (4.173)-(4.174) show that the representation point on the Bloch sphere rotates within a plane normal to this axis (see Fig. 3b) with the angular velocity
d
2 c
z
B
.
(5.14)
z
dt
Almost evidently, since the selection of the coordinate axes is arbitrary, this picture should remain valid for any orientation of the vector c, with the representation point rotating, on the Bloch sphere, around its direction, with the angular speed = 2 c/ – see Fig. 3c. This fact may be proved using any picture of the quantum dynamics, discussed in Sec. 4.6. Actually, the reader may already have done that by solving Problems 4.27-4.28, just to see that even for the particular, simple initial state of the system (), the final results for the Cartesian components of the vector are somewhat bulky.
However, this description may be readily simplified, even for an arbitrary time dependence of the
“field” vector c( t) in Eq. (3), by using the geometric vector language.
For that, let us rewrite Eq. (3) (again, with b = 0) in the operator form, ˆ
H c t σˆ ,
(5.15)
valid in an arbitrary basis. According to Eq. (4.199), the corresponding Heisenberg equation of motion for the j th Cartesian components of the vector-operator σˆ (which does not depend on time explicitly,
ˆ
/ t 0 ) is
6 This correspondence justifies using the use of the term “field” for the vector c.
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3
3
i ˆ
H
, c t σ
, c t
c t .
(5.16)
j
ˆ ˆ,
j
ˆ j ˆ ˆ j j' ˆ j' j' ˆ , ˆ
j
j'
j'1
j' 1
Now using the commutation relations (4.155), which remain valid in any basis and in any picture of time evolution,7 we get
3
i ˆ
2 i
c t
,
(5.17)
j
j' ˆ j" jj'j"
j' , j" 1
where jj’j” is the Levi-Civita symbol. But it is straightforward to verify that the usual vector product of two 3D vectors may be represented in a similar Cartesian-component form:
n
n
n
1
2
3
3
ab a
a
a
a b
,
(5.18)
j
1
2
3
j' j" jj'j"
j' , j" 1
b
b
b
1
2
3 j
As a result, the right-hand side of Eq. (17) may be rewritten as i
2 c tσˆ , and that relation may be
j
recast in a vector form – or rather several equivalent forms:
2
i ˆσ
2 ic t ˆσ,
or ˆσ c t ˆσ,
or ˆσ Ω t ˆσ,
(5.19)
where the vector is defined as
Ω t c t
(5.20)
2
– an evident vector generalization of Eq. (14).8 As we have seen in Sec. 4.6, any linear relation between two Heisenberg operators is also valid for the expectation values of the corresponding observables, so the last form of Eq. (19) yields:
σ Ω t σ .
(5.21)
But this is the well-known kinematic formula9 for the rotation of a constant-length classical 3D
vector around the instantaneous direction of the vector ( t), with the instantaneous angular velocity
( t). So, the time evolution of the representation point on the Bloch sphere is quite simple, especially in the case of a time-independent c, and hence – see Fig. 3c.10 Note that it is sufficient to turn off the field to stop the precession instantly. (Since Eq. (21) is the first-order differential equation, the representation point has no effective inertia.11) Hence, changing the direction and the magnitude of the 7 Indeed, if some three operators in the Schrödinger picture are related as [ ˆ
ˆ
A , B ] = ˆ
C , then according to Eq.
S
S
S
(4.190), in the Heisenberg picture:
ˆ
ˆ
† ˆ
[ A , B ] [ ˆ u A ˆ u, ˆ† ˆ
† ˆ
u B ˆ u] ˆ u A ˆ u ˆ† ˆ
u B ˆ u ˆ† ˆ
† ˆ
† ˆ
u B ˆ u ˆ u A ˆ u ˆ
ˆ
† ˆ
ˆ
u [ A , B ] ˆ u ˆ u C ˆ u C .
H
H
H
H
H
H
H
H
S
S
S
H
8 It is also easy to verify that in the particular case = n z, Eqs. (19) are reduced, in the z-basis, to Eqs. (4.200) for the spin-½ vector matrix S = (/2).
9 See, e.g., CM Sec. 4.1, in particular Eq. (4.8).
10 The bulkiness of the solutions of Problems 4.27-4.28 (which were offered just as useful exercises in quantum dynamic formalisms) reflects the awkward expression of the resulting simple circular motion of the vector
(see Fig. 3c) via its Cartesian components.
11 This is also true for the classical angular momentum L at its torque-induced precession – see, e.g., CM Sec. 4.5.
Chapter 5
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effective external field, it is possible to drive the representation point of a two-level system from any initial position to any final position on the Bloch sphere, i.e. make the system take any of its possible quantum states.
In the particular case of a spin-½ in a magnetic field B( t), it is more customary to use Eqs. (13) and (20) to rewrite Eq. (21) as the following equation for the expectation value of the spin vector S =
(/2):
S S B t.
(5.22)
As we know from the discussion in Chapter 4, such a classical description of the spin’s evolution does not give a full picture of the quantum reality; in particular, it does not describe the possible large uncertainties of its components – see, e.g., Eqs. (4.135). The situation, however, is different for a collection of N >> 1 similar, non-interacting spins, initially prepared to be in the same state – for example by polarizing all spins with a strong external field B0, at relatively low temperatures T, with k B T << B0. (A practically important example of such a collection is a set of nuclear spins in macroscopic condensed-matter samples, where the spin interaction with each other and the environment is typically very small.) For such a collection, Eq. (22) is still valid, while the relative uncertainty of the resulting sample’s magnetization M = nm = nS (where n N/ V is the spin density) is proportional to 1/ N 1/2 << 1. Thus, the evolution of magnetization may be described, with good precision, by the essentially classical equation:
M
M
B t.
(5.23)
This equation, or the equivalent set of three Bloch equations 12 for its Cartesian components, with the right-hand side augmented with small terms describing the effects of dephasing and relaxation (to be discussed in Chapter 7), is used, in particular, to describe the magnetic resonance, taking place when the frequency (4.164) of the magnetization’s precession in a strong dc magnetic field approaches the frequency of an additionally applied (and usually weak) ac/rf field. Two species of this effect, the electron paramagnetic resonance (EPR) and the nuclear magnetic resonance (NMR) are broadly used in material science, chemistry, and medicine. Unfortunately, I will not have time to discuss the related technical issues and methods (in particular, interesting ac/rf pulsing techniques, including the so-called spin echo and Ramsey interferometry) in detail, and have to refer the reader to special literature.13
5.2. The Ehrenfest theorem
In Sec. 4.7, we have derived all the basic relations of wave mechanics from the bra-ket formalism, which will also enable us to get some important additional results in that area. One of them is a pair of very interesting relations, together called the Ehrenfest theorem. To derive them, for the simplest case of 1D orbital motion, let us calculate the following commutator:
12 They were introduced by F. Bloch in the same 1946 paper as the Bloch-sphere representation. In the 1950s when the value of Eq. (21) for quantum optics became recognized, this equation and its open-system generalizations became known as optical Bloch equations. Currently, the term ‘Bloch equations’ is frequently used for any two-level systems, regardless of the physical origin of the Hamiltonian (15).
13 For introductions see, e.g., J. Wertz and J. Bolton, Electron Spin Resonance, 2nd ed., Wiley, 2007; J. Keeler, Understanding NMR Spectroscopy, 2nd ed., Wiley, 2010.
Chapter 5
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ˆ x, ˆ 2
p
(5.24)
x
ˆ x ˆ p ˆ p
ˆ p ˆ p ˆ.
x
x
x
x
x
Let us apply the commutation relation (4.238), in the following form:
ˆˆ p
x
ˆ p ˆ x i I,ˆ
(5.25)
x
x
to the first term of the right-hand side of Eq. (24) twice, with the goal to chase the coordinate operator into the rightmost position:
ˆˆ p
x
ˆ p p x i I
p p p
x
i p
p p x i I
i p
p p x i p
(5.26)
x
x
ˆ ˆ ˆ
x
ˆ ˆ ˆˆ
ˆ
ˆ
x
x
x
x
x ˆ ˆ
ˆ
x
ˆ ˆ ˆ ˆ 2 ˆ .
x
x
x
x
The first term of this result cancels with the last term of Eq. (24), so the commutator becomes quite simple:
ˆ x, ˆ2 p i p
(5.27)
x
2 ˆ .
x
Let us use this equality to calculate the Heisenberg-picture equation of motion of the operator x ˆ , by applying the general Heisenberg equation (4.199) to the 1D orbital motion described by the Hamiltonian (4.237), but possibly with a more general, time-dependent potential energy U: d ˆ x
1
1
ˆ 2
p
ˆ ˆ
x, H
ˆ
x, x U ( ˆ x, t) .
(5.28)
dt
i
i
2
m
The potential energy operator is a function of the coordinate operator and hence, as we know, commutes with it. Thus, the right-hand side of Eq. (28) is proportional to the commutator (27), and we get Heisenberg
d ˆ x
ˆ p
x
.
(5.29) equation
dt
m
for
coordinate
In this operator equation, we readily recognize the full analog of the classical relation between the particle’s momentum and its velocity.
Now let us see what a similar procedure gives for the momentum’s derivative:
d ˆ px
1
1
ˆ 2
p
ˆ ˆ
p , H
(5.30)
x
ˆ p , x U(ˆ x, t) .
x
dt
i
i
2
m
The kinetic energy operator commutes with the momentum operator and hence drops from the right-hand side of this equation. To calculate the remaining commutator of the momentum and the potential energy, let us use the fact that any smooth (infinitely differentiable) function may be represented by its Taylor expansion:
k
1 U
k
U ( x ˆ, t)
x ˆ ,
(5.31)
k
0
! ˆ
k
k
x
where the derivatives of U may be understood as c-numbers (evaluated at x = 0, and the given time t), so we may write
1 k
U
1
U
k
k
p ˆ , U ( x ˆ, t)
p ˆ , x ˆ
p ˆ ˆ x
x ˆ.. x ˆ ˆ x
x ...
ˆ x ˆ p ˆ
.
(5.32a)
x
k x
k x
x
0
! ˆ
0
! ˆ
k k
x
k k
x k times k times
Applying Eq. (25) k times to the last term in the parentheses, exactly as we did in Eq. (26), we get Chapter 5
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