Quantum Mechanics by Konstantin K. Likharev - HTML preview
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l
s
BB
. . .
m ,
0 m ½
l
s
Fig. 6.5. The Paschen-Back effect.
m ,
2 m ½
l
s
17 Despite its smallness, the quadratic term is necessary for a description of the negative contribution of the orbital motion to the magnetic susceptibility m (the so-called orbital diamagnetism, see EM Sec. 5.5), whose analysis, using Eq. (63), is left for the reader’s exercise.
18 See, e.g., EM Sec. 5.4, in particular, Eqs. (5.95) and (5.100).
Chapter 6
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Note that all these levels, besides the top and bottom ones, remain doubly degenerate. This limit of the Zeeman effect is sometimes called the Paschen-Back effect – whose simplicity was recognized only in the 1920s, due to the need in very high magnetic fields for its observation.
In the opposite limit of relatively low magnetic fields, the Zeeman effect takes place on the background of the much larger fine-structure splitting. As was discussed in Sec. 3, at B = 0 each split sub-level has a 2(2 j + 1)-fold degeneracy corresponding to (2 j + 1) different values of the half-integer quantum number mj, ranging from – j to + j, and two values of the integer l = j ½ – see Fig. 4.19 The magnetic field lifts this degeneracy. Indeed, in the coupled representation discussed in Sec. 5.7, the perturbation (65) is described by the matrix with elements
)
1
(
e B
ˆ
ˆ
e B
ˆ
ˆ
H
j, m L 2 S j' , m
j, m J S j' , m
2
j
z
z
j'
m
2
j
z
z
j'
m
e
(6.67)
e
B
e
ˆ
m
j, m S j' , m
j
m m
j
z
j' .
2
j
j'
m e
To spell out the second term, let us use the general expansion (5.183) for the particular case s =
½, when (as was discussed at the end of Sec. 5.7) it has at most two non-vanishing terms, with the Clebsh-Gordan coefficients (5.190):
j l ½, m j
l m ½ 1/2
l m
(6.68)
j
½ 1/2
j
m m ½, m ½
m m ½, m ½
.
2 l 1
l
j
s
2 l 1
l
j
s
Taking into account that the operator S ˆ gives non-zero brackets only for m z
s = ms’, the 22 matrix of
elements
ˆ
m m ½, m ½
is diagonal, so we may use Eq. (27) to get
l
j
S m m ½, m ½
s
z
l
j
s
e B
l m ½
j
l m ½
j
(0)
Anomalous
E E
m
Zeeman
2 m
j
2
2 l 1
2
2 l 1
effect
e
(6.69)
for s = 1/2
e B
1
1
m 1
B m 1
, for j m j,
2 m
j
2 l 1
B
j
2 l 1
j
e
where the two signs correspond to the two possible values of l = j
. . .
½ – see Fig. 6.
. . .
3
m
j
2
3
m
j
2
m ½
m
j
½
(0)
j
E
(0)
E
n, j
n, j
l j ½
m ½
l j ½
m ½
j
j
. . .
3
m
. . .
3
m
j
2
j
2
Fig. 6.6. The anomalous Zeeman effect in a hydrogen-like atom/ion.
19 In the almost-hydrogen-like, but more complex atoms (such as those of alkali metals), the degeneracy in l may be lifted by electron-electron Coulomb interaction even in the absence of an external magnetic field.
Chapter 6
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We see that the magnetic field splits each sub-level of the fine structure, with a given l, into 2 j +
1 equidistant levels, with the distance between the levels depending on l. In the late 1890s when this effect was first observed (by T. Preston), there was no notion of spin at all, so this puzzling result was called the anomalous Zeeman effect.20
The strict quantum-mechanical analysis of the anomalous Zeeman effect for arbitrary s (which is important for applications to multi-electron atoms) is conceptually not too complex but requires explicit expressions for the corresponding Clebsch-Gordan coefficients, which are rather bulky. Let me just cite the unexpectedly simple result of this analysis:
Δ E
m
B
g,
(6.70a)
B
j
Anomalous
Zeeman
where g is the so-called Lande factor:21
effect
for arbitrary s
j( j )
1 s( s )
1 l( l )
1
g 1
.
(6.70b)
2 j( j )
1
For s = ½ (and hence j = l ½), this factor is reduced to the parentheses in the last forms of Eq. (69).
It is remarkable that Eqs. (70) may be readily derived using very plausible classical arguments, similar to those used in Sec. 5.7 – see Fig. 5.13 and its discussion. As was discussed in Sec. 5.6, in the absence of spin, the quantization of the observable Lz is a modification of the classical picture of the torque-induced precession of the vector L about the magnetic field’s direction, so the interaction energy, proportional to B Lz = B L, remains constant – see Fig. 7a. On the other hand, at the spin-orbit interaction without an external magnetic field, the Hamiltonian function of the system includes the product SL, so in the stationary state it has to be constant, together with J 2, L 2, and S 2. Hence, this system’s classical image is a joint precession of the vectors S and L about the direction of the vector J =
L + S, in such a manner that the spin-orbit interaction energy, proportional to the product LS, remains constant (Fig. 7b). On this backdrop, the anomalous Zeeman effect in a relatively weak magnetic field B
= Bn z corresponds to a much slower additional precession of the vector J about the z-axis, “dragging”
with it the vectors L and S, rapidly rotating around it.
z
(a)
z
(b)
J
Lz
L
(L )
J z
L J
Fig. 6.7. Classical images of (a)
L
B
the orbital angular momentum’s
S
quantization in a magnetic field,
(S )
J z
and (b) the fine-structure level
S J
B L
L S
const
const
splitting.
0
0
20 In this terminology, the normal Zeeman effect, observed in atoms with zero net spin, is the one with no spin splitting, i.e. without the second terms in the parentheses of Eqs. (66), (67), and (69).
21 This formula is frequently used with capital letters J, S, and L, which denote the quantum numbers of the atom as a whole.
Chapter 6
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This physical picture allows us to conjecture that what is important for the slow precession rate are only the vectors L and S averaged over the period of their much faster precession around vector J –
in other words, only their components L J and S J along the vector J. Classically, these components may be calculated as
L J
S J
L
J,
and S
J.
J
(6.71)
2
2
J
J
J
The scalar products participating in these expressions may be readily expressed via the squared lengths of the vectors, using the following geometric formulas:
2
S (J L)2
2
2
J L 2L J,
2
L (J S)2
2
2
J S 2J .
S
(6.72)
As a result, we get the following time average:
L J
S J
L
J
2 S L
S
J
J
L J
S J
z
z
2
J
J
z
2
z 2
2
2
2
J
J
J
z
(6.73)
( 2
2
2
J L S ) 2( 2
2
2
J S L )
2
2
2
J S L
J
J 1
.
z
2 2
z
J
2 2
J
The last move is to smuggle in some quantum mechanics by using, instead of the vector lengths squared and the z-component of Jz, their eigenvalues given by Eqs. (5.169), (5.175), and (5.177). As a result, we immediately arrive at the exact Eqs. (70). This coincidence encourages thinking about quantum mechanics of angular momenta in the classical terms of torque-induced precession, which turns out to be very fruitful in some more complex problems of atomic and molecular physics.
The high-field limit and low-field limits of the Zeeman effect, described respectively by Eqs.
(66) and (69), are separated by a medium field range, in which the Zeeman splitting is of the order of the fine-structure splitting analyzed in Sec. 3. There is no time in this course for a quantitative analysis of this (conceptually simple) crossover, which involves rather cumbersome algebra.22
6.5. Time-dependent perturbations
Now let us proceed to the case when the perturbation
)
1
(
ˆ
H in Eq. (1) is a function of time, while
(0)
ˆ
H is time-independent. The adequate perturbative approach to this problem, and its results, depend critically on the relation between the characteristic frequency of the perturbation and the distance between the initial system’s energy levels:
E E .
(6.74)
n
n'
In the case when all essential frequencies of a perturbation are very small in the sense of Eq.
(74), we are dealing with the so-called adiabatic change of parameters, that may be treated essentially as a time-independent perturbation – see the previous sections of this chapter). The most interesting observation here is that the adiabatic perturbation does not allow any significant transfer of the system’s 22 For a more complete discussion of the Stark, Zeeman, and fine-structure effects in atoms, I can recommend, for example, either the monograph by G. Woolgate cited above, or the one by I. Sobelman, Theory of Atomic Spectra, Alpha Science, 2006.
Chapter 6
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probability from one eigenstate to another. For example, in the WKB limit of the orbital motion, the Bohr quantization rule and its Wilson-Sommerfeld modification (2.110) guarantee that the integral
p dr ,
(6.75)
C
taken along the particle’s classical trajectory, is an adiabatic invariant, i.e. does not change at a slow change of system’s parameters. (It is curious that classical mechanics also guarantees the invariance of the integral (75), but its proof there23 is much harder than the quantum-mechanical derivation of this fact, carried out in Sec. 2.4.) This is why even if the perturbation becomes large with time (while changing sufficiently slowly), we can expect the classification of eigenstates and eigenvalues to persist.
Let us proceed to the harder case when both sides of Eq. (74) are comparable, using for this discussion the Schrödinger picture of quantum dynamics, given by Eq. (4.158). Combining it with Eq.
(1), we get the Schrödinger equation in the form
i
( t ˆ
)
(0)
)
1
(
H
H ( t) ( t) .
(6.76)
t
Very much in the spirit of our treatment of the time-independent case in Sec. 1, let us represent the time-dependent ket-vector of the system with its expansion,
( t) n n ( t) ,
(6.77)
n
over the full and orthonormal set of the unperturbed, stationary ket-vectors defined by the equation H (0)
ˆ
n E n .
(6.78)
n
(Note that these kets n are exactly what was called n(0) in Sec. 1; we may afford a less bulky notation in this section because only the lowest orders of the perturbation theory will be discussed.) Plugging the expansion (77), with n replaced with n’, into both sides of Eq. (76), and then inner-multiplying both its sides by the bra-vector n of another unperturbed (and hence time-independent) state of the system, we get the following set of linear, ordinary differential equations for the expansion coefficients: d
i
n ( t) E n ( t)
)
1
(
H
t n' t
(6.79)
n
( )
( ) ,
nn'
dt
n'
where the matrix elements of the perturbation, in the unperturbed state basis, defined similarly to Eq.
(8), are now functions of time:
H )1
(
t
( n H ˆ
)
)
1
(
t
( ) n' .
(6.80)
nn'
The set of differential equations (79), which are still exact, may be useful for numerical calculations.24 However, it has a certain technical inconvenience, which becomes clear if we consider its (evident) solution in the absence of perturbation:25
23 See, e.g., CM Sec. 10.2.
24 Even if the problem under analysis may be described by the wave-mechanics Schrödinger equation (1.25), direct numerical integration of that partial differential equation is typically less convenient than that of the ordinary differential equations (79).
25 This is of course just a more general form of Eq. (1.62) of the wave mechanics of time-independent systems.
Chapter 6
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En
n t
( ) n ( )
0 exp i
t.
(6.81)
We see that these solutions oscillate very fast, and their numerical modeling may represent a challenge for even the fastest computers. These spurious oscillations (whose frequency, in particular, depends on the energy reference level) may be partly tamed by looking for the general solution of Eqs. (79) in a form inspired by Eq. (81):
En
n t
( ) a t
( ) exp
.
(6.82)
n
i
t
Here an( t) are new functions of time (essentially, the stationary states’ probability amplitudes), which may be used, in particular, to calculate the time-dependent level occupancies, i.e. the probabilities Wn to find the perturbed system on the corresponding energy levels of the unperturbed system: 2
W ( t) n ( t)
a
.
(6.83)
n
n t 2
Plugging Eq. (82) into Eq. (79), for these functions, we readily get a slightly modified system of equations:
Probability
amplitudes:
i a
)
1
(
i
t
a H
t
( ) e
,
(6.84)
n
nn'
n'
nn'
evolution
n'
where the factors nn’, defined by the relation
Quantum
transition
frequencies
E E ,
(6.85)
nn'
n
n'
have the physical sense of frequencies of potential quantum transitions between the n th and n’ th energy levels of the unperturbed system. (The conditions when such transitions indeed take place will be clear soon.) The advantages of Eq. (84) over Eq. (79), for both analytical and numerical calculations, are their independence of the energy reference, and lower frequencies of oscillations of the right-hand side terms, especially when the energy levels of interest are close to each other.26
In order to continue our analytical treatment, let us focus on a particular but very important problem of a sinusoidal perturbation turned on at some moment – which may be taken for t = 0: Turning on
,
0
for
t ,
0
sinusoidal
ˆ )1
(
H ( t)
(6.86)
perturbation
ˆ i t
ˆ†
e
A
A e i t for
,
t ,
0
where the perturbation amplitude operators A ând ˆ †
A , 27 and hence their matrix elements,
26 Note that the relation of Eq. (84) to the initial Eq. (79) is very close to the relation of the interaction picture of quantum dynamics, discussed at the end of Sec. 4.6, to its Schrödinger picture, with the perturbation Hamiltonian playing the role of the interaction one – compare Eqs. (1) and Eq. (4.206). Indeed, Eq. (84) could be readily obtained from the interaction picture, and I did not do this just to avoid using this heavy bra-ket artillery for our current (relatively) simple problem, and hence to keep its physics more transparent.
27 The notation of the amplitude operators in Eq. (86) is justified by the fact that the perturbation Hamiltonian has to be self-adjoint (Hermitian), and hence each term on the right-hand side of that relation has to be a Hermitian conjugate of its counterpart, which is evidently true only if the amplitude operators are also the Hermitian conjugates of each other. Note, however, that each of these amplitude operators is generally not Hermitian.
Chapter 6
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ˆ
ˆ
n A n' A ,
†
*
n A n' A ,
(6.87)
nn'
n'n
are time-independent after the turn-on moment. In this case, Eq. (84) yields
i t
i
(
)
t
nn
*
(
)
'
nn'
i a
.
(6.88)
n
a A e
A e
,
for t 0
n'
nn'
n'
n
n'
This is, generally, still a nontrivial system of coupled differential equations; however, it allows simple and explicit solutions in two very important limits. First, let us assume that our system initially was definitely in one eigenstate n’ (usually, though not necessarily, in the ground state), and that the occupancies Wn of all other levels stay very low all the time. (We will find the condition when the second assumption is valid a posteriori – from the solution.) With these assumptions, a ;
1
a ,
1 for n n',
(6.89)
n'
n
Eq. (88) may be readily integrated, giving
*
A
nn'
i
t
A
(
)
nn
n'n
i(
)
a
t
(6.90)
n
e
e
nn
n n'
nn'
'
1
nn'
'
1 ,
for
.
This expression describes what is colloquially called the ac excitation of (other) energy levels.
Qualitatively, it shows that the probability Wn (83) of finding the system in each state (“on each energy level”) of the system does not tend to any constant value but rather oscillates in time. It also shows that the ac-field-induced transfer of the system from one state to the other one has a clearly resonant character: the maximum occupancy Wn of a level with number n n’ grows infinitely when the corresponding detuning 28
,
(6.91)
nn'
nn'
tends to zero. This conclusion is clearly unrealistic, and is an artifact of our initial assumption (89); according to Eq. (90), it is satisfied only if29
A
,
(6.92)
'
nn
nn'
and hence which does not allow a deeper analysis of the resonant excitation.
In order to overcome this limitation, we may perform the following trick – very similar to the one we used for the transfer to the degenerate case in Sec. 1. Let us assume that for a certain level n,
, ,
,
n"
all
for
n, n'
(6.93)
nn'
n"n
n"n'
– the condition illustrated in Fig. 8. Then, according to Eq. (90), we may ignore the occupancy of all but two levels, n and n’, and also the second, non-resonant term with frequency nn’ + 2 >> nn’ in Eqs. (88),30 now written for two probability amplitudes, an and an’.
28 The notion of detuning is also very useful in the classical theory of oscillations (see, e.g., CM Chapter 5), where the role of nn’ is played by the own frequency 0 of the oscillator.
29 Strictly speaking, one more condition is that the number of “resonance” levels is also not too high – see Sec. 6.
30 The second assumption, i.e. the omission of non-resonant terms in the equations for amplitudes is called the Rotating Wave Approximation (RWA); the same idea in the classical theory of oscillations is the basis of what is usually called the van der Pol method, and its result, the reduced equations – see, e.g., CM Secs. 5.3-5.5.
Chapter 6
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En
0
nn '
E
Fig. 6.8. The resonant excitation of
n"
an energy level.
E
n'
The result is the following system of two linear equations:
i t
i a a Ae
,
* i t
i a a A e ,
(6.94)
n
n'
n'
n
which uses the shorthand notation A Ann’ and nn’. (I will use this simplified notation for a while –
until other energy levels become involved, at the beginning of the next section). This system may be readily reduced to a form without explicit time dependence of the right-hand parts – for example, by introducing the following new probability amplitudes, with the same moduli:
i t / 2
i t
/ 2
b a e
,
b a e
,
(6.95)
n
n
n'
n'
so
i t
/ 2
i / 2
a b e
,
t
a b e
.
(6.96)
n
n
n'
n'
Plugging these relations into Eq. (94), we get two usual linear first-order differential equations:
i b
b Ab ,
i b A b
*
b .
(6.97)
n
n
n'
n'
n
n'
2
2
As the reader knows very well by now, the general solution of such a system is a linear combination of two exponential functions, exp{ t}, with the exponents that may be found by plugging any of these functions into Eq. (97), and requiring the consistency of the two resulting linear algebraic equations. In our case, the consistency condition (i.e. the characteristic equation of the system) is
/ 2 i
A
0
*
,
(6.98)
A
/ 2 i
and has two solutions = i, where
1/ 2
1/ 2
2
2
2
Rabi
Δ
A
A
oscillations:
2
Ω
,
2
i.e. 4
.
(6.99)
2
2
frequency
4
The coefficients at the exponents are determined by initial conditions. If, as was assumed before, the system was on the level n’ initially (at t = 0), i.e. if an’ (0) = 1, an(0) = 0, so bn’ (0) = 1, bn(0) = 0 as well, then Eqs. (97) yield, in particular:
A
b ( t) i
sin t
,
(6.100)
n
so the n th level occupancy is
Chapter 6
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A 2
2
A 2
W b
2
sin
t
2
sin .
(6.101) Rabi
n
n
2
2
2
formula
/ 2
t
A
2
This is the famous Rabi oscillation formula.31 It shows that if the detuning is large in comparison with A /, though still small in the sense of Eq. (93), the frequency 2 of the Rabi oscillations is completely determined by the detuning, and their amplitude is small:
2
A
t
W ( t) 4
sin
,
1
for 2
2
A (
(6.102)
n
)2 ,
2
2
2
– the result which could be obtained directly from Eq. (90), just neglecting the second term on its right-hand side. However, now we may also analyze the results of an increase of the perturbation amplitude: it leads not only to an increase of the amplitude of the probability oscillations but also of their frequency –
see Fig. 9. Ultimately, at A >> (for example, at the exact resonance, = 0., i.e. nn’ = , so En =
En’ + ), Eqs. (101)-(102) give = A / and ( Wn)max = 1, i.e. describe a periodic, full “repumping” of the system from one level to another and back, with a frequency proportional to the perturbation amplitude.32
A 3
0.8
0.6
W
1
n 0.4
0.3
Fig. 6.9. The Rabi oscillations
0.2
for several values of the
0.1
normalized amplitude of ac
0
perturbation.
0
0.2
0.4
0.6
0.8
t /(2 / )
This effect is a close analog of the quantum oscillations in two-level systems with time-independent Hamiltonians, which were discussed in Secs. 2.6 and 5.1. Indeed, let us revisit, for a moment, their discussion started at the end of Sec.1 of this chapter, now paying more attention to the time evolution of the system under a perturbation. As was argued in that section, the most general perturbation Hamiltonian lifting the two-fold degeneracy of an energy level, in an arbitrary basis, has the matrix (28). Let us describe the system’s dynamics using, again, the Schrödinger picture, representing the ket-vector of an arbitrary state of the system in the form (5.1), where and are the 31 It was derived in 1952 by Isaac Rabi, in the context of his group’s pioneering experiments with the ac (practically, microwave) excitation of quantum states, using molecular beams in vacuum.
32 As Eqs. (82), (96), and (99) show, the lowest frequency in the system is l = n’ – /2 + , so at A 0, l
n’ + 2 A2/. This effective shift of the lowest energy level (which may be measured by another “probe” field of a different frequency) is a particular case of the ac Stark effect, which was already mentioned in Sec. 2.
Chapter 6
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time-independent states of the basis in that Eq. (28) is written (now without any obligation to associate these states with the z-basis of any spin-½.) Then, the Schrödinger equation (4.158) yields
H
H
H H
1
i
H
11
12
11
12
.
(6.103)
H
H
H
H
21
22
21
22
As we know (for example, from the discussion in Sec. 5.1), the average of the diagonal elements of the matrix gives just a common shift of the system’s energy; for the purpose of the analysis, it may be absorbed into the energy reference level. Also, the Hamiltonian operator has to be Hermitian, so the off-diagonal elements of its matrix have to be complex-conjugate. With this, Eqs. (103) are reduced to the form,
i H ,
*
i H ,
with H H ,
(6.104)
12
12
22
11
2
2
which is absolutely similar to Eqs. (97). In particular, these equations describe the quantum oscillations of the probabilities W = 2 and W = 2 with the frequency33
1/ 2
2
H
12
2
2 4
.
(6.105)
2
The similarity of Eqs. (97) and (104), and hence of Eqs. (99) and (105), shows that the “usual”
quantum oscillations and the Rabi oscillations have essentially the same physical nature, besides that in the latter case the external ac signal quantum bridges the separated energy levels, effectively reducing their difference ( En – En’) to a much smaller difference – ( En – En’) – . Also, since the Hamiltonian (28) is similar to that given by Eq. (5.2), the dynamics of such a system with two ac-coupled energy levels, within the limits (93) of the perturbation theory, is completely similar to that of a time-independent two-level system. In particular, its state may be similarly represented by a point on the Bloch sphere shown in Fig. 5.3, with its dynamics described, in the Heisenberg picture, by Eq. (5.19).
This fact is very convenient for the experimental implementation of quantum information processing systems (to be discussed in more detail in Sec. 8.5), because it enables qubit manipulations in a broad variety of physical systems with well-separated energy levels, using external ac (usually either microwave or optical) sources.
Note, however, that according to Eq. (90), if a system has energy levels other than n and n’, they also become occupied to some extent. Since the sum of all occupancies equals 1, this means that ( Wn)max may approach 1 only if the other excitation amplitude is very small, and hence the state manipulation time scale T = 2/ = 2/ A is very long. The ultimate limit in this sense is provided by the harmonic oscillator where all energy levels are equidistant, and the probability repumping between all of them occurs at comparable rates. In particular, in this system, the implementation of the full Rabi oscillations is impossible even at the exact resonance.34
33 By the way, Eq. (105) gives a natural generalization of the relations obtained for the frequency of such oscillations in Sec. 2.6, where the coupled potential wells were assumed to be exactly similar, so = 0. Moreover, Eqs. (104) gives a long-promised proof of Eqs. (2.201), and hence a better justification of Eqs. (2.203).
34 From Sec. 5.5, we already know what happens to the ground state of an oscillator at its external sinusoidal (or any other) excitation: it turns into a Glauber state, i.e. a superposition of all Fock states – see Eq. (5.134).
Chapter 6
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However, I would not like these quantitative details to obscure from the reader the most important qualitative (OK, maybe semi-quantitative :-) conclusion of this section’s analysis: a resonant increase of the interlevel transition intensity at nn’. As will be shown later in the course, in a quantum system coupled to its environment at least slightly (hence in reality, in any quantum system), such increase is accompanied by a sharp increase of the external field’s absorption, which may be measured. This increase is used in numerous applications, notably including the magnetic resonance techniques already mentioned in Sec. 5.1.
6.6. Quantum-mechanical Golden Rule
One of the results of the past section, Eq. (102), may be used to derive one of the most important and nontrivial results of quantum mechanics. For that, let us consider the case when the perturbation causes quantum transitions from a discrete energy level En’ into a group of eigenstates with a very dense (essentially continuous) spectrum En – see Fig. 10a.
(a)
(b)
E
n
0.2
Fig. 6.10. Deriving the Golden
0.1
Rule: (a) the energy level
E
scheme, and (b) the function
n'
0
under the integral in Eq. (108).
15
0
15
t
nn '
If, for all states n of the group, the following conditions are satisfied 2
A
,
(6.106)
nn'
nn 2
'
nn 2'
then Eq. (102) coincides with the result that would follow from Eq. (90). This means that we may apply Eq. (102), with the indices n and n’ duly restored, to any level n of our tight group. As a result, the total probability of having our system transferred from the initial level n’ to that group is 2
4
A
t
W ( t)
nn
nn
W t
(6.107)
( )
'
n
sin 2
' .
2
2
n
n
2
nn'
Now comes the main, absolutely beautiful trick: let us assume that the summation over n is limited to a tight group of very similar states whose matrix elements Ann’ are virtually similar (we will check the validity of this assumption later on), so we can take Ann’2 out of the sum in Eq. (107) and then replace the sum with the corresponding integral:
4
2
2
A
nn
t
A
t
'
1
nn'
nn
n
t
2
4
'
1
2
W ( t)
nn'
(6.108)
dn
d
t
2
sin
2
sin
(
),
2
nn'
t
nn'
2
2
nn'
where n is the density of the states n on the energy axis:
dn
.
(6.109) Density
n
dE
of states
n
Chapter 6
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This density and the matrix element Ann’ have to be evaluated at nn’ = 0, i.e. at energy En = En’ + , and are assumed to be constant within the final state group. At fixed En’, the function under integral (108) is even and decreases fast at nn’t >> 1 – see Fig. 10b. Hence we may introduce a dimensionless integration variable nn’t, and extend the integration over it formally from – to +. Then the integral in Eq. (108) is reduced to a table one,35 and yields
4 A 2 t
2
1
4 A
t
W t
nn'
n
( )
sin2 d
nn'
n
t ,
(6.110)
2
2
2
where the constant
2
Golden
A 2
(6.111)
nn'
n
Rule
is called the transition rate.36 This is one of the most famous and useful results of quantum mechanics, its Golden Rule 37, which deserves much discussion.
First of all, let us reproduce the reasoning already used in Sec. 2.5 to show that the meaning of the rate is much deeper than Eq. (110) seems to imply. Indeed, due to the conservation of the total probability, Wn’ + W = 1, we can rewrite that equation as
W
.
(6.112)
n' t 0
Evidently, this result cannot be true for all times, otherwise the probability Wn’ would become negative.
The reason for this apparent contradiction is that Eq. (110) was obtained in the assumption that initially, the system was completely on level n’: Wn’(0) = 1. Now, if at the initial moment the value of Wn’ is different, the result (110) has to be multiplied by that number, due to the linear relation (88) between dan/ dt and an’. Hence, instead of Eq. (112), we get a differential equation similar to Eq. (2.159), W
W ,
(6.113)
n' t0
n'
which, for a time-independent , has the evident solution,
Initial
occupancy’s
W ( t) W (0) Γ t
e
,
(6.114)
n'
n'
decay
describing the exponential decay of the initial state’s occupancy, with the time constant = 1/.
I am inviting the reader to review this fascinating result again: by the summation of periodic oscillations (102) over many levels n, we have got an exponential decay (114) of the probability. This trick becomes possible because the effective range En of the state energies En giving substantial 35 See, e.g., MA Eq. (6.12).
36 In some texts, the density of states in Eq. (111) is replaced with a formal expression n( En – En’ – ). Indeed, applied to a finite energy interval En with n >> 1 levels, it gives the same result: n ( dn/ dEn) En n En.
Such replacement may be technically useful in some cases, but is incorrect for n ~ 1, and hence should be used with the utmost care, so for most applications, the more explicit form (111) is preferable.
37 Sometimes Eq. (111) is called “Fermi’s Golden Rule”. This is rather unfair, because this result had been developed mostly by the same P. A. M. Dirac in 1927, and Enrico Fermi’s role was not much more than advertising it, under the name of “Golden Rule No. 2”, in his influential lecture notes on nuclear physics that were published much later, in 1950. (To be fair to Fermi, he has never tried to pose as the Golden Rule’s author.) Chapter 6
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contributions to the integral (108), shrinks with time: En ~ / t. 38 However, since most of the decay (114) takes place within the time interval of the order of 1/, the range of the participating final energies may be estimated as
E ~
.
(6.115)
n
This estimate is very instrumental for the formulation of conditions of the Golden Rule’s validity. First, we have assumed that the matrix elements of the perturbation and the density of states are independent of the energy within the interval (115). This gives the following requirement
E
~ E E
,
(6.116)
n
~
n
n'
Second, for the transfer from the sum (107) to the integral (108), we need the number of states within that energy interval, Nn = n En, to be much larger than 1. Merging Eq. (116) with Eq. (92) for all the energy levels n” n, n’ not participating in the resonant transition, we may summarize all conditions of the Golden Rule validity as
Golden
1 Γ
.
(6.117) Rule’s
n
n'n"
validity
(The reader may ask whether I have neglected the condition expressed by the first of Eqs. (106).
However, for nn’ ~ En/ ~ , this condition is just Ann’2 << ()2, so plugging it into Eq. (111),
2
2 ,
(6.118)
n
and canceling one and one , we see that it coincides with the first relation in Eq. (117) above.) Let us have a look at whether these conditions may be satisfied in practice, at least in some cases. For example, let us consider the optical ionization of an atom, with the released electron confined in a volume of the order of 1 cm3 10-6 m3. According to Eq. (1.90), with E of the order of the atomic ionization energy En – En’ = ~ 1 eV , the density of electron states in that volume is of the order of 1021 1/eV, while the right-hand side of Eq. (117) is of the order of En ~ 1 eV. Thus the conditions (117) provide an approximately 20-orders-of-magnitude range for acceptable values of . This illustration should give the reader a taste of why the Golden Rule is applicable to so many situations.
The physical picture of the initial state’s decay is also very important. According to Eq. (114), the external excitation transfers the system into the continuous spectrum of levels n, and it never comes back to the initial level n’. However, it was derived from the quantum mechanics of Hamiltonian systems, whose equations are invariant with respect to time reversal.39 This paradox is a result of our generalization (113) of the exact result (112) This trick, breaking the time-reversal symmetry, is absolutely adequate for the physics under study. Indeed, some gut feeling of the physical sense of the resulting irreversibility may be obtained from the following observation. As Eq. (1.86) illustrates, the distance between the adjacent orbital energy levels tends to zero only if the system’s size goes to infinity. This means that our assumption of the continuous energy spectrum of the finial states n 38 This is one more appearance of the “energy-time uncertainty relation”, which was discussed in Sec. 2.5.
39 This situation is similar to the irreversible increase of entropy of macroscopic systems, despite the fact that their microscopic components obey reversible laws of motion, which is postulated in thermodynamics and explained in statistical physics – see, e.g., SM Secs. 1.2 and 2.2.
Chapter 6
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essentially requires these states to be broadly extended in space – being either fully free or virtually free de Broglie waves. Thus the Golden Rule corresponds to the (physically justified) assumption that in an infinitely large system, the traveling de Broglie waves excited by a local source and propagating outward from it, would never come back, and even if they did, unpredictable phase shifts introduced by minor uncontrollable perturbations on their way would never allow them to sum up in the coherent way necessary to bring the system back into the initial state n’. (This is essentially the same situation that was discussed, for a particular 1D wave-mechanical system, in Sec. 2.5.)
To get a feeling of the Golden Rule at work, let us apply it to the following simple problem –
which is a toy model of the photoelectric effect, briefly discussed in Sec. 1.1(ii). A 1D particle is initially trapped in the ground state of a narrow potential well described by Eq. (2.158): U ( x) W ( x),
with W 0.
(6.119)
Let us calculate the rate of the particle’s “ionization” (i.e. its excitation into a group of extended, delocalized states) by a weak classical sinusoidal force of amplitude F 0 and frequency , suddenly turned on at some instant, say t = 0.
As a reminder, the initial localized state (in our current notation, n’) of such a particle was already found in Sec. 2.6:
2
2
2
m
m
( x)
1/ 2
W
W
x
E
(6.120)
n'
exp
,
with
,
.
2
n'
2 m
2 2
The final, extended states n, with a continuous spectrum, for this problem exist only at energies En > 0, so the excitation rate is different from zero only for frequencies
2
En'
m
W
.
(6.121)
min
3
2
The weak sinusoidal force may be described by the following perturbation Hamiltonian, F
ˆ )1
(
H
F( t) ˆ x F ˆ x cos
0
t
ˆ
x ei t
e i t
,
for t 0 ,
(6.122)
0
2
so according to Eq. (86), which serves as the amplitude operator’s definition, in this case ˆ
ˆ †
F 0
A A
.
ˆ x
(6.123)
2
The matrix elements Ann’ that participate in Eq. (111) may be readily calculated in the coordinate representation:
*
F
A ( x A ˆ
) ( x) ( x) dx
0
*
( x) x ( x) dx .
(6.124)
nn'
n
n'
n
n
2
'
Since, according to Eq. (120), the initial n’ is a symmetric function of x, any non-vanishing contributions to this integral are given only by antisymmetric functions n( x), proportional to sin knx, with the wave number kn related to the final energy by the well-familiar equality (1.89): 2
k 2
n E .
(6.125)
n
2 m
Chapter 6
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As we know from Sec. 2.6 (see in particular Eq. (2.167) and its discussion), such antisymmetric functions, with n(0) = 0, are not affected by the zero-centered delta-functional potential (119), so their density n is the same as that in completely free space, and we could use Eq. (1.93). However, since that relation was derived for traveling waves, it is more prudent to repeat its derivation for standing waves, confining them to an artificial segment [- l/2, + l/2] – long in the sense k l, l
1,
(6.126)
n
so it does not affect the initial localized state and the excitation process. Then the confinement requirement n( l/2) = 0 immediately yields the condition knl/2 = n, so Eq. (1.93) is indeed valid, but only for positive values of kn, because sin knx with kn – kn does not describe an independent standing-wave eigenstate. Hence the final state density is
dn
dn
dE
l
2
k
lm
n
n
.
(6.127)
n
dE
dk
dk
2
m
2
2
k
n
n
n
n
It may look troubling that the density of states depends on the artificial segment’s length l, but the same l also participates in the final wavefunctions’ normalization factor,40
2 1/2
sin k x ,
(6.128)
n
l
n
and hence in the matrix element (124):
F l
x
F l
ik
l
x
ik x
0
2
1/ 2
/ 2
0
2
1/ 2
/ 2
/ 2
A
(6.129)
nn
sin k x e
.
(
)
(
)
xdx
n
e n
xdx
e
n
xdx .
'
2 l
2
i
l
l
/ 2
0
0
These two integrals may be readily worked out by parts. Taking into account that due to the condition (126), their upper limits may be extended to , the result is
1/ 2
2
2 k
A
n
F
.
(6.130)
nn'
0
l
2 k
n
2
2
Note that the matrix element is a smooth function of kn (and hence of En), so an important condition of the Golden Rule, the virtual constancy of Ann’ on the interval En ~ << En, is satisfied. So, the general Eq. (111) is reduced, for our problem, to the following expression:
2
1/ 2
2
3
2 2
2 k
lm
8 F mk
n
0
F
n
,
(6.131)
0
2
2 2
2
3
2
2 4
l
( k )
k
k
n
2
(
)
n
n
which is independent of the artificially introduced l – thus justifying its use.
Note that due to the above definitions of kn and , the expression in the parentheses in the denominator of the last expression does not depend on the potential well’s “weight” W, and is a function of only the excitation frequency (and the particle’s mass):
40 The normalization to infinite volume, by using Eq. (4.263), is also possible, but physically less transparent.
Chapter 6
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2
2
2
k
n
E E
(6.132)
n
n'
.
2 m
As a result, Eq. (131) may be recast simply as
2
3
F W k
0
n
(6.133)
.
2
4
What is hidden here is that kn, defined by Eq. (125) with En = En’ + , is a function of the external force’s frequency, changing as 1/2 at >> min (so drops as -7/2 at ), and as ( – min)1/2
when approaches the “red boundary” (121) of the ionization effect, so ( – min)1/2 0 in that limit as well.
A conceptually very similar but a bit more involved analysis of this effect in a more realistic 3D
case, namely the hydrogen atom’s ionization by an optical wave, is left for the reader’s exercise.
6.7. Golden Rule for step-like perturbations
Now let us reuse some of our results for a perturbation being turned on at t = 0, but after that time-independent:
for
,
0
t ,
0
Step-like
ˆ )1
(
H ( t)
(6.134)
perturbation
ˆ
H const, for t .
0
A superficial comparison of this equality and the former Eq. (86) seems to indicate that we may use all our previous results, taking = 0 and replacing
†
ˆ
ˆ
A A with
1
ˆ
H . However, that conclusion (which
would give us a wrong factor of 2 in the result) does not take into account the fact that when analyzing both the two-level approximation in Sec. 5 and the Golden Rule in Sec. 6, we have dropped the second (non-resonant) term in Eq. (90). In our current case (134), with = 0, there is no such difference between these terms. This is why it is more prudent to use the general Eq. (84),
i
t
nn'
i a
a H e
(6.135)
n
,
n'
nn'
n'
in which the matrix element of the perturbation is now time-independent at t > 0. We see that it is formally equivalent to Eq. (88) with only the first (resonant) term kept, provided that we make the following replacements:
ˆ
ˆ
A H ,
.
(6.136)
nn'
nn'
nn'
Let us use this equivalency to consider the results of coupling between a discrete-energy state n’, to which the particle is initially placed, and a dense group of states with a quasi-continuum spectrum, in the same energy range. Figure 11a shows an example of such a system: a particle is initially (say, at t =
0) placed into a potential well separated by a penetrable potential barrier from a formally infinite region with a continuous energy spectrum. Let me hope that the physical discussion in the last section makes the outcome of such an experiment evident: the particle will gradually and irreversibly tunnel out of the well, so the probability Wn’( t) of its still residing in the well will decay in accordance with Eq. (114).
The rate of this decay may be found by making the replacements (136) in Eq. (111): Chapter 6
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2
2
H
,
(6.137)
nn'
n
where the states n and n’ now have virtually the same energy.41
(a)
(b)
l
l l
l
l
con
con
con
con
n'
n
n'
n
2
Fig. 6.11. Tunneling from a discrete-energy state n’: (a) to a state continuum, and (b) to another discrete-energy state n.
It is very informative to compare this result, semi-quantitatively, with Eq. (105) for a symmetric ( En = En’) system of two potential wells separated by a similar potential barrier – see Fig. 11b. For the symmetric case, i.e. = 0, Eq. (105) is reduced to simply
1
H
.
(6.138)
nn' con
Here I have used the index “con” (from “confinement”) to emphasize that this matrix element is somewhat different from the one participating in Eq. (137), even if the potential barriers are similar.
Indeed, in the latter case, the matrix element,
H
n H ˆ n' * ˆ
,
(6.139)
nn'
H dx
n'
n
has to be calculated for two wavefunctions n and n’ confined to spatial intervals of the same scale l con, while in Eq. (137), the wavefunctions n are extended over a much larger distance l >> l con – see Fig.
11. As Eq. (128) tells us, in the 1D model this means an additional small factor of the order of ( l con/ l)1/2.
Now using Eq. (128) as a crude but suitable model for the final-state wavefunctions, we arrive at the following estimate, which is independent of the artificially introduced length l: 2
H
2
l
2
l
lm
nn'
con
con
con
2
~ 2 H
~ 2 H
~
,
(6.140)
nn'
n
nn'
2
con
con
l
l 2
k
E
E
n
n'
n'
where En’ ~ 2/ ml 2con is the scale of the distances between the adjacent eigenenergies of the particle in an unperturbed potential well. Since the condition of validity of Eq. (138) is << En’, we see that
~
.
.
(6.141)
En
This (sufficiently general42) perturbative result confirms the conclusion of a more particular analysis carried out at the end of Sec. 2.6: the rate of the (irreversible) quantum tunneling into a state continuum is always much lower than the frequency of (reversible) quantum oscillations between 41 The condition of validity of Eq. (137) is again given by Eq. (117), just with = 0 in the upper limit for .
42 It is straightforward to verify that the estimate (141) is valid for similar problems of any spatial dimensionality, not just for the 1D case we have analyzed.
Chapter 6
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discrete states separated with the same potential barrier – at least for the case when both are much lower than En’/, so the perturbation theory is valid. A very handwaving interpretation of this result is that the particle performs the quantum oscillations between the confined state in the well and the space-extended states behind the barrier many times before finally “deciding to perform” an irreversible transition into the unconfined continuum. This qualitative picture is consistent with experimentally observable effects of dispersive electromagnetic environments on electron tunneling.43
Let me conclude this section (and this chapter) with the application of Eq. (137) to a very important case that will provide a smooth transition to the next chapter’s topic. Consider a composite system consisting of two component systems, a and b, with the energy spectra sketched in Fig. 12.
system a
system b
n'
n
a
b
interaction
Fig. 6.12. Energy relaxation in
system a due to its weak coupling
ˆ )1
(
ˆ
ˆ
H
A( a) B( b)
to system b (which serves as the
environment of a).
n
n'
a
b
Let the systems be completely independent initially. The independence means that in the absence of their coupling, the total Hamiltonian of the system may be represented as a sum of two operators: ˆ (0)
ˆ
ˆ
H
H ( a) H ( b),
(6.142)
a
b
where arguments a and b symbolize the non-overlapping sets of the degrees of freedom of the two systems. Such operators, belonging to their individual, different Hilbert spaces, naturally commute.
Similarly, the eigenkets of the system may be naturally factored as
Direct
product
n n n .
(6.143)
a
b
The direct product sign is used here (and below) to denote the formation of a joint ket-vector from the kets of the independent systems, belonging to different Hilbert spaces. Evidently, the order of operands in such a product may be changed at will. As a result, its eigenenergies separate into a sum, just as the Hamiltonian (142) does:
ˆ (0)
H
n ˆ
ˆ
H H
(6.144)
a
b n
n
a
b
ˆ H n
a
a
nb
ˆ H n
b
b
na
E
E
na
nb n .
In such composite systems, the relatively weak interaction of its components may be usually represented as a product of two Hermitian operators, each depending only on the degrees of freedom of one component system:
ˆ )1
(
ˆ
ˆ
H
(
A a) B( b) .
(6.145)
A very common example of such an interaction is the electric-dipole interaction between an atomic-scale system (with a linear size of the order of the Bohr radius r B ~ 10-10 m) and the electromagnetic field at optical frequencies ~ 1016 s-1, with the wavelength = 2 c/ ~ 10-6 m >> r B: 43 See, e.g., P. Delsing et al., Phys. Rev. Lett. 63, 1180 (1989).
Chapter 6
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H ˆ )1
(
dˆ E , ˆ
dˆ
with q rˆ ,
(6.146)
k k
k
where the dipole electric moment d depends only on the positions r k of the charged particles (numbered with index k) of the atomic system, while the electric field E is a function of only the electromagnetic field’s degrees of freedom – to be discussed in Chapter 9 below.
Returning to the general situation shown in Fig. 12, if the component system a was initially in an excited state n’a, the interaction (145), turned on at some moment of time, may bring it into another discrete state na of lower energy – for example, the ground state. In the process of this transition, the released energy, in the form of an energy quantum
E E ,
(6.147)
n' a
na
is picked up by system b:
E E E E E ,
(6.148)
nb
n b
'
n b
'
n' a
na
so the total energy E = Ea + Eb of the system does not change. (If the states na and n’b are the ground states of the component systems, as they are in most applications of this analysis, and we take the ground state energy E g = Ena + En’b of the composite system for the reference, then Eq. (148) gives merely Enb = En’a.) If the final state nb of system b is inside a state group with a quasi-continuous energy spectrum (Fig. 12), the process has the exponential character (114)44 and may be interpreted as the effect of energy relaxation of system a, with the released energy quantum absorbed by system b.
If the relaxation rate is sufficiently low, it may be described by the Golden Rule (137). Since the perturbation (145) does not depend on time explicitly, and the total energy E does not change, this relation, with the account of Eqs. (143) and (145), takes the form
2
Golden
A 2 B 2 ,
A n A ˆ
where
n' ,
B
n B ˆ
and
n' ,
(6.149) Rule
nn'
nn'
n
nn'
a
a
nn'
b
b
for coupled
systems
where n is the density of the final states of system b at the relevant energy (147).45 In particular, Eq.
(149), with the dipole Hamiltonian (146), will enable us to readily calculate, in Chapter 9, the natural linewidth of atomic electric-dipole transitions.
Instead, I will now proceed to a general discussion of the effects of quantum systems’ interaction with their environment, toward which the situation shown in Fig. 12 provides a clear conceptual path.
Indeed, in this case the transition from the Hamiltonian (and hence reversible) quantum dynamics of the whole composite system a + b to the Golden-Rule-governed (and hence irreversible) dynamics of system a has been achieved essentially by following this component system alone, i.e. ignoring the details of the exact state of system b. (As was argued in the previous section, the quasi-continuous spectrum of the latter system essentially requires it to have a large spatial size, so it may be legitimately called the environment of the “open” system a.) This is exactly the approach that will be pursued in the next chapter.
44 This process is spontaneous: it starts as soon as either the interaction (145) has been turned on or (if it had been already on) as soon as the system a is placed into the excited state n’a.
45 Note that these partial matrix elements may be calculated in the Heisenberg picture as well, because due to the general Eq. (4.149) and the energy balance (147), the additional time dependences of these elements would be proportional to exp{ i t}, and cancel at their multiplication.
Chapter 6
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6.8. Exercise problems
6.1. Use Eq. (6.14) of the lecture notes to prove the following general form of the Hellmann-Feynman theorem:46
E
ˆ
n
H
n
n ,
where is an arbitrary c-number parameter.
6.2. Establish a relation between Eq. (16) and the result of the classical theory of weakly anharmonic (“nonlinear”) oscillations at negligible damping.
Hint: You may like to use N. Bohr’s reasoning that was discussed in Problem 1.1.
6.3. An additional weak time-independent force F is exerted on a 1D particle that had been placed into a hard-wall potential well
U x
0
for
,
0
x a,
,
otherwise.
Calculate, sketch, and discuss the 1st-order perturbation of its ground-state wavefunction.
6.4. A time-independent force F = (n xy+n yx), where is a small constant, is applied to a 3D
isotropic harmonic oscillator of mass m and frequency 0, located at the origin. Calculate, in the first order of the perturbation theory, the effect of the force upon the ground-state energy of the oscillator and its lowest excited energy level. How small should the constant be for your results to be quantitatively correct?
6.5. A 1D particle of mass m is localized at a narrow potential well that may be approximated with a delta function:
U x W x,
with W 0.
Calculate the change of its ground state energy by an additional weak time-independent force F, in the first non-vanishing approximation of the perturbation theory. Discuss the limits of validity of this result, taking into account that at F 0, the localized state of the particle is metastable.
6.6. Use Eq. (16) to calculate the eigenvalues of the operator 2
ˆ L , in the limit m l >> 1, by
purely wave-mechanical means.
Hint: Try the following substitution: () = f()/sin1/2.
6.7. In the lowest non-vanishing order of the perturbation theory, calculate the shift of the ground-state energy of an electrically charged spherical rotor (i.e. a particle of mass m, free to move over a spherical surface of radius R) due to a weak uniform time-independent electric field E.
6.8. Use the perturbation theory to evaluate the effect of a time-independent uniform electric field E on the ground state energy E g of a hydrogen atom. In particular: 46 As a reminder, proof of its wave-mechanics form was the task of Problem 1.7.
Chapter 6
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(i) calculate the 2nd-order shift of E g, neglecting the extended unperturbed states with E > 0, and bring the result to the simplest analytical form you can,
(ii) find the lower and the upper bounds on the shift, and
(iii) discuss the simplest experimental manifestation of this quadratic Stark effect.
6.9. A particle of mass m, with electric charge q, is in its ground s-state with a given energy E g < 0, being localized by a very-short-range, spherically symmetric potential well. Calculate its static electric polarizability.
6.10. In some atoms, the effect of nuclear charge screening by electrons on the motion of each of them may be reasonably well approximated by the replacement of the Coulomb potential (3.190), U = –
C/ r, with the so-called Hulthén potential
C / a
1/ r,
for
r a,
U
C
exp r /
a 1
exp r /
a / a,
for a r.
Assuming that the effective screening radius a is much larger than r 0 2/m C, use the perturbation theory to calculate the energy spectrum of a single particle of mass m, moving in this potential, in the lowest order needed to lift the l-degeneracy of the energy levels.
6.11. In the lowest non-vanishing order of the perturbation theory, calculate the correction to energies of the ground state and all lowest excited states of a hydrogen-like atom/ion, due to the electron’s penetration into the nucleus, by modeling it the latter a spinless, uniformly charged sphere of radius R << r B/ Z.
6.12. A particle of mass m is placed inside a hard-wall ellipsoid whose surface is described by the equation
2
2
2
x y z ,1
with b 1 a,
.
1
2
2
a
b
Calculate its ground-state energy in the 1st order in the small parameter , and interpret the result.
6.13. Prove that the relativistic correction operator (48) indeed has only diagonal matrix elements in the basis of unperturbed Bohr atom states (3.200).
6.14. Calculate the lowest-order relativistic correction to the ground-state energy of a 1D
harmonic oscillator.
6.15. Use the perturbation theory to calculate the contribution to the magnetic susceptibility m of a dilute gas, that is due to the orbital motion of a single electron inside each gas particle. Spell out your result for a spherically symmetric ground state of the electron, and give an estimate of the magnitude of this orbital susceptibility.
6.16. A certain energy level degeneracy is not lifted in the 1st order of the stationary perturbation theory. Calculate its lifting in the 2nd order of the theory. Apply the result to a planar rotor of mass m and radius R, with electric charge q, placed into a weak, uniform, time-independent electric field E.
Chapter 6
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6.17.* The Hamiltonian of a quantum system is slowly changed over time.
(i) Develop a theory of quantum transitions in the system, and spell out its result in the 1st approximation in the speed of the change.
(ii) Use this approximation to calculate the probability that a finite-time pulse of a slowly changing force F( t) drives a 1D harmonic oscillator, initially in its ground state, into an excited state.
(iii) Compare the last result with the exact one.
6.18. Use the single-particle model to calculate the complex electric permittivity () of a dilute gas of similar atoms, due to their induced electric polarization by a weak external ac field, for a field frequency very close to one of the quantum transition frequencies nn’. Based on the result, calculate and estimate the absorption cross-section of each atom.
Hint: In the single-particle model, the atom’s properties are determined by Z similar, non-interacting electrons, each moving in a similar static attracting potential, generally different from the Coulomb one, because it is contributed not only by the nucleus but also by other electrons.
6.19. Use the solution of the previous problem to generalize the expression for the London dispersion force between two atoms (whose calculation in the harmonic oscillator model was the subject of Problems 3.20 and 5.20) to the single-particle model with an arbitrary energy spectrum.
6.20. Use the solution of the previous problem to calculate the potential energy of the interaction of two hydrogen atoms, both in their ground state, separated by distance r >> r B.
6.21. In a certain quantum system, distances between the three lowest E 2
energy levels are slightly different – see the figure on the right ( << 1,2).
2
1
Assuming that the involved matrix elements of the perturbation Hamiltonian E
are known and are all proportional to the external ac field’s amplitude, find
1
the time necessary to populate the first excited level almost completely (with
1
a given precision << 1), by using the Rabi oscillation effect, if at t = 0, the E 0
system is in its ground state. Spell out your result for a weakly anharmonic
1D oscillator.
6.22.* Analyze the possibility of a slow transfer of a system from one of E 2
its energy levels to another one (in the figure on the right, from level 1 to level A A
3), by using the scheme shown in that figure, in which the monochromatic
external excitation amplitudes A
E 3
+ and A– may be slowly changed at will.
E
1
6.23. A weak external force pulse F( t), of a finite time duration, is applied to the particle in a system with a discrete energy spectrum, which initially was in its ground state.
(i) Derive, in the lowest non-vanishing order of the perturbation theory, a formula for the probability that the pulse drives the particle into its lowest excited state.
(ii) Specify this formula for a 1D harmonic oscillator and compare the result with the exact solution of the problem.
(iii) Spell out the perturbative result for the Gaussian-shaped waveform F( t) = F 0exp{– t 2/2} and analyze its dependence on the scale of the pulse duration.
Chapter 6
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6.24. A spatially uniform but time-dependent external electric field E( t) is applied, starting from t
= 0, to a charged planar rotor, initially in its ground state.
(i) Calculate, in the lowest non-vanishing order in the field’s strength, the probability that by a certain time t > 0, the rotor is in its m th excited state.
(ii) Spell out and analyze your results for a constant-magnitude field rotating, with a constant angular velocity , within the rotor’s plane.
(iii) Do the same for a monochromatic field of frequency , with a fixed direction.
6.25. A heavy relativistic particle, with electric charge q = Ze, flies by a hydrogen atom, initially in its ground state, with an impact parameter b within the range r B << b << r B/, where 1/137 is the fine structure constant. Calculate the total probability of the atom’s transition to one of its lowest excited states.
6.26. Develop a general theory of quantum excitations of the higher levels of a discrete-spectrum system, initially in the ground state, by a weak time-dependent perturbation, up to the 2nd order. Spell out and discuss the result for the case of monochromatic excitation, with a nearly perfect tuning of its frequency to the half of a certain quantum transition frequency n 0 ( En – E 0)/ .
6.27. A particle of mass m is initially in a localized ground state, with energy E g < 0, of a very-short-range, spherically symmetric potential well. Calculate the rate of its delocalization by an applied classical force F( t) = n F 0cos t with a time-independent direction n.
6.28.* Calculate the rate of ionization of a hydrogen atom, initially in its ground state, by a classical, linearly polarized electromagnetic wave with an electric field’s amplitude E0, and a frequency
within the range
c
.
2
m r
r
e B
B
Recast your result in terms of the cross-section of electromagnetic wave absorption. Discuss briefly what changes of the theory would be necessary if either of the above conditions had been violated.
6.29.* Use the quantum-mechanical Golden Rule to derive the general expression for the electric current I through a weak tunnel junction between two conductors, biased with dc voltage V, treating the conductors as degenerate Fermi gases of electrons with negligible direct interaction. Simplify the result in the low-voltage limit.
Hint: The electric current flowing through a weak tunnel junction is so low that it does not substantially perturb the electron states inside each conductor.
6.30.* Generalize the result of the previous problem to the case when a weak tunnel junction is biased with voltage V( t) = V 0 + A cos t, with generally comparable with eV 0 and eA.
6.31.* Use the quantum-mechanical Golden Rule to derive the Landau-Zener formula (2.257).
Chapter 6
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Chapter 7. Open Quantum Systems
This chapter discusses the effects of a weak interaction of a quantum system with its environment. Some part of this material is on the fine line between quantum mechanics and (quantum) statistical physics.
Here I will only cover those aspects of the latter field 1 that are of key importance for the major goals of this course, including the discussion of quantum measurements in Chapter 10.
7.1. Open systems, and the density matrix
All the way until the last part of the previous chapter, we have discussed quantum systems isolated from their environment. Indeed, from the very beginning, we have assumed that we are dealing with the statistical ensembles of systems as similar to each other as only allowed by the laws of quantum mechanics. Each member of such an ensemble, called pure (or coherent), may be described by the same state vector – in the wave mechanics case, by the same wavefunction . Even if we deal with a composite system, say, a two-components system like those discussed at the end of the previous chapter, we still can consider its coherent states of the type
n n n ,
(7.1)
n
n
a
b
n
n
with a unique correspondence between the pure states na and nb of the two subsystems.
However, in many important cases, our knowledge of a quantum system’s state is less complete.2
These cases fall into two categories. The first case is when a relatively simple quantum system s of our interest (say, an electron or an atom) is in substantial contact with its environment e – here understood in the most general sense, say, as all the whole Universe less the system s – see Fig. 1. Then there is virtually no chance of making two or more experiments with exactly the same composite system because that would imply a repeated preparation of the whole environment (including the experimenter
:-) in a certain quantum state – a rather challenging task, to put it mildly. Then it makes much more sense to consider a statistical ensemble of another kind – a mixed ensemble, with random states of the environment, though possibly with its macroscopic parameters (e.g., temperature, pressure, etc.) known with high precision. Such ensembles will be the focus of the analysis in this chapter.
weak
The Universe
interaction
system of
environment ( e)
interest ( s)
Fig. 7.1. A quantum system and its
environment (VERY schematically :-).
1 A broader discussion of statistical mechanics and physical kinetics, including those of quantum systems, may be found in the SM part of this series.
2 Actually, no system, possibly apart from our Universe as a whole (see below), is ever exactly coherent, though in many cases, deviations from the coherence may be ignored with acceptable accuracy.
© K. Likharev
QM: Quantum Mechanics
Much of this analysis will also pertain to another category of cases – when the system of our interest is, at present, isolated from its environment with acceptable precision, but our knowledge of its state is still incomplete for some reason. Most typically, the system could be in contact with its environment at earlier times. So, this second category of cases may be considered as a particular case of the first one, and may be described by the results of its analysis, with certain simplifications – which will be spelled out in appropriate places of my narrative.
In classical physics, the analysis of mixed statistical ensembles is based on the notion of the probability W of each detailed (“microscopic”) state of the system of interest.3 Let us see how such an ensemble may be described in quantum mechanics. In the case when the coupling between the system of our interest and its environment is so weak that they may be clearly separated, we can still use state vectors of their states, defined in completely different Hilbert spaces. Then the most general quantum state of the whole Universe, still assumed to be pure,4 may be described as the following linear superposition:
Universe:
s e .
(7.2)
quantum
jk
j
k
state
j, k
The “only” difference of such a state from the superposition described by Eq. (1), is that there is no one-to-one correspondence between the states of our system and its environment. In other words, a certain quantum state sj of the system of interest may coexist with different states ek of its environment.
This is the quantum-mechanical description of a mixed state of system s.5
Of course, the huge size of the Hilbert space of the environment, i.e. of the number of the ek
factors in the superposition (2), strips us of any practical opportunity to make direct calculations using that sum. For example, according to the basic Eq. (4.125), in order to find the expectation value of an arbitrary observable A in the state (2), we would need to calculate all long brackets in the sum A A *
e s A ˆ s
e .
(7.3)
jk
j'k'
k
j
j'
k'
j,j' ; k,k'
Even if we assume that each of the sets { s} and { e} is full and orthonormal, Eq. (3) still includes a double sum over the enormous basis state set of the environment!
However, let us consider a limited, but the most important subset of all operators – those of intrinsic observables, which depend only on the degrees of freedom of the system of our interest ( s).
These operators do not act upon the environment’s degrees of freedom, and hence in Eq. (3), we may move the environment’s bra-vectors ek over all the way to the ket-vectors ek’. Assuming, again, that the set of environmental eigenstates is full and orthonormal, Eq. (3) is now reduced to 3 In systems with a continuum of states, we have to discuss the probability density w instead – see below.
4 Whether this assumption is true is an interesting issue, still being debated (more by philosophers than by physicists), but it is widely believed that its solution is not critical for the validity of the results of this approach to all systems available for our experimentation.
5 Note also that in this definition, the notion of a mixed state includes pure states as particular cases, while in some texts, this term is limited to states that are not exactly pure. Due to the already discussed prevalence of the mixed states (in any definition!) in the world, with the exactly pure states serving only as (sometimes useful and acceptable) abstractions, this is not much of a difference.
Chapter 7
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*
ˆ
A α α s A s e e
A
α* α
(7.4)
jk
j'k'
j
j'
k
k'
jj'
.
jk
j'k
j,j' ; k,k'
jj'
k
This is already a big relief because we have “only” a single sum over k, but the main trick is still ahead. After the summation over k, the second sum in the last form of Eq. (4) is some function w of the indices j and j’, so according to Eq. (4.96), this relation may be represented as Intrinsic
observable:
A
expectation
A w Tr
(Aw),
(7.5)
jj '
j' j
jj '
value where the matrix w, with the elements
Density
matrix:
w
*
,
w
*
i.e.
,
(7.6)
j'j
jk
j'k
jj'
jk j'k
definition
k
k
is called the density matrix of the system.6 Most importantly, Eq. (5) shows that the knowledge of this matrix allows the calculation of the expectation value of any intrinsic observable A (and, according to the general Eqs. (1.33)-(1.34), its r.m.s. fluctuation as well, if needed), even for the very general state (2). This is why let us have a good look at the density matrix.
First of all, we know from the general discussion in Chapter 4, fully applicable to the pure state (2), the expansion coefficients in superpositions of this type may be always expressed as short brackets of the type (4.40); in our current case, we may write
e s
.
(7.7)
jk
k
j
Plugging this expression into Eq. (6), we get
*
w s e e s s ˆ w s .
(7.8)
jj'
jk
j'k
j
k
k
j'
j
j'
k
k
We see that from the point of our system (i.e. in its Hilbert space whose basis states may be numbered by the index j only), the density matrix is indeed just the matrix of some construct,7
Density
operator:
w ˆ e e ,
(7.9)
k
k
definition
k
which is called the density (or “statistical”) operator. As it follows from the definition (9), in contrast to the density matrix, this operator does not depend on the choice of a particular basis sj – just as all linear operators considered earlier in this course. However, in contrast to them, the density operator does depend on the composite system’s state , including the state of the system s as well. Still, in the j-
space, it is mathematically just an operator whose matrix elements obey all relations of the bra-ket formalism.
In particular, due to its definition (6), the density operator is Hermitian:
w*
*
*
w ,
(7.10)
jj'
jk
j'k
j'k
jk
j'j
k
k
6 This notion was introduced in 1927 by John von Neumann.
7 Note that the “short brackets” in this expression are not c-numbers, because the state is defined in a larger Hilbert space (of the environment plus the system of interest) than the basis states ek (of the environment only).
Chapter 7
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so according to the general analysis of Sec. 4.3, in the Hilbert space of the system s, there should be a certain basis { w} in that the matrix of this operator is diagonal:
w
w .
(7.11)
jj' in w
j
jj'
Since any operator, in any basis, may be represented in the form (4.59), in the basis { w} we may write Density
w ˆ w w w .
(7.12) operator in
j
j
j
w-basis
j
This expression reminds but is not equivalent to Eq. (4.44) for the identity operator, that has been used so many times in this course, and in the basis wj has the form
I ˆ w w .
(7.13)
j
j
j
In order to comprehend the meaning of the coefficients wj participating in Eq. (12), let us use Eq.
(5) to calculate the expectation value of any observable A whose eigenstates coincide with those of the special basis { w}, and whose matrix is, therefore, diagonal in this basis: Expectation
A Tr (Aw) A w
A w ,
(7.14)
value of
jj'
j
jj'
j j
wj-compatible
jj'
j
variable
where Aj is just the expectation value of the observable A in the state wj. Hence, to comply with the general Eq. (1.37), the real c-number wj must have the physical sense of the probability Wj of finding the system in the state j. As the result, we may rewrite Eq. (12) in the form8
w ˆ w W w .
(7.15)
j
j
j
j
In the ultimate case when only one of the probabilities (say, Wj” ) is different from zero, W ,
(7.16)
j
jj"
the system is in a pure (coherent) state wj” . Indeed, it is fully described by one ket-vector wj” , and we can use the general rule (4.86) to represent it in another (arbitrary) basis { s} as a coherent superposition w
U† s U* s ,
(7.17)
j"
j"j'
j'
j'j"
j'
j'
j'
where U is the unitary matrix of transform from the basis { w} to the basis { s}. According to Eqs. (11) and (16), in such a pure state, the density matrix is diagonal in the { w} basis, w
,
(7.18a)
jj'
w
in
j, j"
j' , j"
but not in an arbitrary basis. Indeed, using the general rule (4.92), we get
w
U† w
U U † U
U* U .
(7.18b)
jj' in s
jl
ll' in w
l'j'
jj"
j"j'
j"j
j"j'
l, l '
To make this result more transparent, let us denote the matrix elements Uj”j wj” sj (which, for a fixed j” , depend on just one index j) as j; then
8 In some textbooks, this relation is taken for the theory’s starting point, leaving the physical sense of the density operator obscure.
Chapter 7
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Density
matrix:
w
*
,
(7.19)
jj ' in s
j
j'
pure state
so N 2 elements of the whole N N matrix are determined by just one string of N c-numbers j. For example, for a two-level system ( N = 2),
*
*
w
1 1
2
1
.
(7.20)
in s
*
*
1
2
2
2
We see that the off-diagonal terms are, colloquially, “as large as the diagonal ones”, in the following sense:
w w w w .
(7.21)
12
21
11
22
Since the diagonal terms have the sense of the probabilities W 1,2 to find the system in the corresponding state, we may represent Eq. (20) as
W
( W W )1/ 2 i
e
w
1
1
2
.
(7.22)
state
pure
( W W )1/2 i
e
W
1
2
2
The physical sense of the (real) constant is the phase shift between the coefficients in the linear superposition (17), which represents the pure state wj” in the basis { s 1,2}.
Now let us consider a different statistical ensemble of two-level systems, that includes the member states identical in all aspects (including similar probabilities W 1,2 in the same basis s 1,2), besides that the phase shifts are random, with the phase’s probability uniformly distributed over the trigonometric circle. Then the ensemble averaging is equivalent to the averaging over from 0 to 2,9
which kills the off-diagonal terms of the density matrix (22), so the matrix becomes diagonal:
W
0
w
1
.
(7.23)
mixture
classical
0
W 2
The mixed statistical ensemble with the density matrix diagonal in the stationary state basis is called the classical mixture and represents the limit opposite to the pure state.
After this example, the reader should not be much shocked by the main claim10 of statistical mechanics that any large ensemble of similar systems in thermodynamic (or “thermal”) equilibrium is exactly such a classical mixture. Moreover, for systems in thermal equilibrium with a much larger environment of a fixed temperature T (such an environment is frequently called either a “heat bath” or a
“thermostat”), statistical physics gives a very simple expression, called the Gibbs distribution, for the probabilities Wn:
Gibbs
1
E
E
distribution
W
exp n ,
Z
with
exp
.
(7.24)
n
n
Z
k T
B
n
k T
B
9 For a system with a time-independent Hamiltonian, such averaging is especially plausible in the basis of the stationary states n of the system, in which the phase is just the difference of integration constants in Eq. (4.158), and its randomness may be naturally produced by minor fluctuations of the energy difference E 1 – E 2.
10 This fact follows from the basic postulate of statistical physics, called the microcanonical distribution – see, e.g., SM Sec. 2.2.
Chapter 7
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where En is the eigenenergy of the corresponding stationary state, and the normalization coefficient Z is called the statistical sum.11
A detailed analysis of classical and quantum ensembles in thermodynamic equilibrium is a major focus of statistical physics courses (such as the SM part of this series) rather than this course of quantum mechanics. However, I would still like to attract the reader’s attention to the key fact that, in contrast with the similarly-looking Boltzmann distribution for single particles,12 the Gibbs distribution is general, not limited to classical statistics. In particular, for a quantum gas of indistinguishable particles, it is absolutely compatible with the quantum statistics (such as the Bose-Einstein or Fermi-Dirac distributions) of the component particles. For example, if we use Eq. (24) to calculate the average energy of a 1D harmonic oscillator of frequency 0 in thermal equilibrium, we easily get13
W exp n
0
0
0
0
1 exp
,
exp
1 exp
, (7.25)
n
Z
k T
2
B
k T
B
k T
B
k T
B
0
E
W E
coth
0
0
0
.
(7.26a)
n
n
n
k T
k T
0
2
2
2
B
exp
/
0
B
1
The final form of the last result,
k T
0
1
,
0
for
,
E
n , with n
B
0 (7.26b)
2
0
exp
/ k T
0
B
1
k T /
, for
k T,
B
0
0
B
may be interpreted as an addition, to the ground-state energy 0/2, of the average number n of thermally-induced excitations, with the energy 0 each. In the harmonic oscillator, whose energy levels are equidistant, such a language is completely appropriate, because the transfer of the system from any level to the one just above it adds the same amount of energy, 0. Note that the above expression for
n is actually the Bose-Einstein distribution (for the particular case of zero chemical potential); we see that it does not contradict the Gibbs distribution (24) of the total energy of the system, but rather immediately follows from it.
Because of the fundamental importance of Eq. (26) for virtually all fields of physics, let me draw the reader’s attention to its two main properties. At low temperatures, k B T << 0, there are virtually no excitations, n 0, and the average energy of the oscillator is dominated by that of its ground state. In the opposite limit of high temperatures, n k B T /0 >> 1, and E approaches the classical value k B T.
7.2. Coordinate representation and the Wigner function
For many applications of the density operator, its coordinate representation is convenient. (I will only discuss it for the 1D case; the generalization to multi-dimensional cases is straightforward.) 11 See. e.g., SM Sec. 2.4. The Boltzmann constant k B is only needed if the temperature is measured in non-energy units – say in kelvins.
12 See, e.g., SM Sec. 2.8.
13 See, e.g., SM Sec. 2.5 – but mind a different energy reference level, E 0 = 0/2, used for example in SM Eqs.
(2.68)-(2.69), affecting the expression for Z. Actually, the calculation, using Eqs. (24) and (5.86), is so straightforward that it is highly recommended to the reader as a simple exercise.
Chapter 7
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Following Eq. (4.47), it is natural to define the following function of two arguments – traditionally, but a bit misleadingly, also called the density matrix:
Density
matrix:
w( x, x' ) x w ˆ x' .
(7.27)
coordinate
representation Inserting, into the right-hand side of this definition, two closure conditions (4.44) for an arbitrary (but full and orthonormal) basis { s}, and then using Eq. (4.233),14 we get
(
w x, x' ) x s s ˆ w s
s x'
x w
x' .
(7.28)
j
j
j'
j'
( )
* ( )
j
jj' in s
j'
j, j'
j, j'
In the special basis { w}, in which the density matrix is diagonal, this expression is reduced to w( x, x' ) ( x W
)
*
( x' ) .
(7.29)
j
j
j
j
Let us discuss the properties of this function. At coinciding arguments, x’ = x, this is just the probability density:15
w( x, x) ( x W
)
*
( x)
w ( x W
)
w( x) .
(7.30)
j
j
j
j
j
j
j
However, the density matrix gives more information about the system than just the probability density.
As the simplest example, let us consider a pure quantum state, with Wj = j,j’, so ( x) = j’( x), and (
w x, x' ) ( x) *
( x' ) ( x) *
( x' ) ,
(7.31)
j'
j'
so
(
w x, x' ) 2 (
w x, x' ) w* ( x, x' ) ( x) *
( x) ( x' ) *
( x' ) (
w x) (
w x' ) .
(7.32)
For example, for a simple wave packet with a small spatial extent x, w( x, x’) has an appreciable magnitude only if both points are not farther than ~ x from the packet center, and hence from each other. Note that the density matrix carries information not only about the modulus but also the phase of the wavefunction. However, in the ultimate limit of a pure state, the density-matrix description is redundant because all this information is contained in the wavefunction itself.
The density matrix becomes really invaluable when we deal with an incoherent mixture of several wavefunctions, for example, with the classical mixture describing the thermodynamic equilibrium. In this case, we can use Eq. (24) to rewrite Eq. (29) as follows:
*
1
E
w( x, x' ) ( x W
) ( x' )
( x)exp
*
( x' ) .
(7.33)
n
n
n
n
n
n
n
Z n
k T
B
As the simplest example, let us see what is the density matrix of a free particle in thermal equilibrium. As we know very well by now, in this case, the set of energies Ep = p 2/2 m of stationary states (monochromatic waves) forms a continuum, so we need to replace the sum (33) with an integral, using, for example, the “delta-normalized” traveling-wave eigenfunctions (4.264): 14 For now, I will focus on a fixed time instant (say, t = 0), and hence write ( x) instead of ( x, t).
15 This fact is the origin of the density matrix’s name.
Chapter 7
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1
ipx
p 2
ipx'
w( x, x' )
exp exp
exp
dp .
(7.34)
2
Z
2 mk T
B
This is a usual Gaussian integral and may be worked out, as we have done repeatedly in Chapter 2 and beyond, by complementing the exponent to the full square of the momentum p plus a constant. The statistical sum Z may be also readily calculated, 16
Z 2 mk
T
(7.35)
B
1/2,
However, for what follows it is more useful to write the result for the product wZ (the so-called un-normalized density matrix):
Free
mk T
mk T ( x x' )
B
1/ 2
2
(
w x, x' ) Z
exp
.
(7.36) particle:
2