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 the 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 that 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.
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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 dynamics analysis, it may be absorbed into the energy reference level. Also, the Hamiltonian operator has to be Hermitian, so that 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 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 the 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 that = 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).
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However, I would not like these the 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 effect has numerous practical applications including spectroscopies based on the electron paramagnetic resonance (EPR) and nuclear magnetic resonance (NMR), which 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 pulsing techniques, including the so-called Ramsey interferometry) in detail, and have to refer the reader to special literature.35
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 (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 that we can take Ann’2 out of the sum in Eq. (107) and then replace the sum with the corresponding integral:
35 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.
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4
2
2
A
nn
t
A
t
'
1
nn'
nn
n
t
2
4
'
1
2
W ( t)
sin
dn
sin
nn'
d(
t),
(6.108)
2
2
2
nn'
t
nn'
2
2
nn'
where n is the density of the states n on the energy axis:
dn
Density
.
(6.109)
n
of states
dEn
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,36 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.37 This is one of the most famous and useful results of quantum mechanics, its Golden Rule 38, 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,
36 See, e.g., MA Eq. (6.12).
37 In some texts, the density of states in Eq. (111) is replaced with a formal expression ( 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 that for most applications the more explicit form (111) is preferable.
38 Sometimes Eq. (111) is called “Fermi’s Golden Rule”. This is rather unfair, because this result was 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, which 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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Initial
W ( t) W (0) Γ t
e
,
(6.114)
n'
n'
occupancy’s
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 contributions to the integral (108), shrinks with time: En ~ / t. 39 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 that 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.
Finally, the physical picture of the initial state’s decay (which will also be the key to our discussion of quantum-mechanical “open” systems in the next chapter) 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. This 39 This is one more appearance of the “energy-time uncertainty relation”, which was discussed in Sec. 2.5.
Chapter 6
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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 this 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 essentially requires these states to be broadly extended in space – being either free, or essentially 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 which was discussed, for a particular 1D wave-mechanical system, in Sec. 2.5.)40
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 that the excitation rate is different from zero only for frequencies
2
E
n'
W
m
.
(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 that 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:
40 This situation is also 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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*
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, 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
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 that their density n is the same as that in completely free space, and we could use Eq. (1.100). 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 that Eq. (1.100) 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,41
2 1/2
sin k x ,
(6.128)
n
l
n
and hence in the matrix element (124):
F
2
2
0
1/2 l
1/ 2
x
F 0
l ( ik
l
) x
( ik
) x
A
sin
.
. (6.129)
nn'
k x e
xdx
n
e n
xdx
e n
xdx
2 l
i
2
l
l
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 that 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:
41 The normalization to infinite volume, using Eq. (4.263), is also possible, but physically less transparent.
Chapter 6
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QM: Quantum Mechanics
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):
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 that drops as -7/2 at ), and as ( – min)1/2
when approaches the “red boundary” (121) of the ionization effect, so that ( – min)1/2 0 in that limit as well.
A conceptually very similar, but a bit more involved analysis of such 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 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 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, if we make the following replacements:
ˆ
ˆ
A H ,
.
(6.136)
nn'
nn'
nn'
Chapter 6
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Let us use this equivalency to consider the results of coupling between a discrete-energy state n’, into 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 that 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): 2
2
H
,
(6.137)
nn'
n
where the states n and n’ now have virtually the same energy.42
(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, 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 differences between the eigenenergies of the particle in an unperturbed potential well. Since the condition of validity of Eq. (138) is << En’, we see that 42 The condition of validity of Eq. (137) is again given by Eq. (117), just with = 0 in the upper limit for .
Chapter 6
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~
.
.
(6.141)
En
This (sufficiently general43) perturbative result confirms the conclusion of a more particular analysis carried out in 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 discrete states separated with the same potential barrier – at least for the case when both are much lower than En’/, so that the perturbation theory is valid. A very handwaving interpretation of this result is that the particle oscillates 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.44
Let me conclude this section (and this chapter) with the application of Eq. (137) to a very important case, which 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 the 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
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 .
43 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.
44 See, e.g., P. Delsing et al., Phys. Rev. Lett. 63, 1180 (1989).
Chapter 6
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In such composite systems, the relatively weak interaction of its components may be usually represented as a bilinear 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: 45
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 that of 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 a 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 the system b:
E E E E E ,
(6.148)
nb
n b
'
n b
'
n' a
na
so that 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 two 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 the system b is inside a state group with a quasi-continuous energy spectrum (Fig. 12), the process has the exponential character (114)46 and may be interpreted as the effect of energy relaxation of the system a, with the released energy quantum absorbed by the system b. Note that since the quasi-continuous spectrum essentially requires a system of large spatial size, such a model is very convenient for description of the environment b of the quantum system a. (In physics, the “environment” typically means all the Universe – less the system under consideration.) 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
Golden
2
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 the system b at the relevant energy (147). In particular, Eq.
(149), with the dipole Hamiltonian (146), enables a very straightforward calculation of the natural linewidth of atomic electric-dipole transitions. However, such calculation has to be postponed until Chapter 9, in which we will discuss the electromagnetic field quantization – i.e., the exact nature of the 45 See, e.g., EM Sec. 3.1, in particular Eq. (3.16), in which letter p is used for the electric dipole moment.
46 Such process is spontaneous: it does not require any external agent, and starts as soon as either the interaction (145) has been turned on, or (if it is always on) as soon as the system a is placed into the excited state n’a.
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states nb and n’b for this problem, and hence will be able to calculate Bnn’ and n. 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.
6.8. Exercise problems
6.1. Use Eq. (14) to prove the following general form of the Hellmann-Feynman theorem (whose proof in the wave-mechanics domain was the task of Problem 1.5):
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 discussed in Problem 1.1.
6.3. A weak, time-independent force F is exerted on a 1D particle that was 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
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 the perturbation theory 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 rotator (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.
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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:
(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 manifestations 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 charge-screening effect of other 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 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 electron’s penetration into its nucleus, modeling it as a spinless, uniformly charged sphere of radius R << r B/ Z.
6.12. Prove that the kinetic-relativistic correction operator (48) indeed has only diagonal matrix elements in the basis of unperturbed Bohr atom states (3.200).
6.13. Calculate the lowest-order relativistic correction to the ground-state energy of a 1D
harmonic oscillator.
6.14. 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 am estimate of the magnitude of this orbital susceptibility.
6.15. How to calculate the energy level degeneracy lifting, by a time-independent perturbation, in the 2nd order of the perturbation in
ˆ 1
H , assuming that it is not lifted in the 1st order? Carry out such
calculation for a plane rotator of mass m and radius R, carrying electric charge q, and placed into a weak, uniform, constant electric field E.
6.16.* The Hamiltonian of a quantum system is slowly changed in time.
(i) Develop a theory of quantum transitions in the system, and spell out its result in the 1st order in the speed of the change.
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(ii) Use the 1st-order result 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.
6.17. 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 quantum transition frequencies nn’. Based on the result, calculate and estimate the absorption cross-section of each atom.
Hint: In the single-particle model, 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.
6.18. 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 3.16 and 5.15) to the single-particle model with an arbitrary energy spectrum.
6.19. Use the solution of the previous problem to calculate the potential energy of interaction of two hydrogen atoms, both in their ground state, separated by distance r >> r B.
6.20. 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
are known, and are all proportional to the external ac field’s amplitude, find E 1
the time necessary to populate the first excited level almost completely (with a
1
given precision << 1), using the Rabi oscillation effect, if at t = 0 the system E 0
is completely in its ground state.
6.21.* 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), 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
6.22. A weak external force pulse F( t), of a finite time duration, is applied to a 1D harmonic oscillator that initially was in its ground state.
(i) Calculate, in the lowest non-vanishing order of the perturbation theory, the probability that the pulse drives the oscillator into its lowest excited state.
(ii) Compare the result with the exact solution of the problem.
(iii) Spell out the perturbative result for a Gaussian-shaped waveform,
F t F exp t / ,
0
2 2
and analyze its dependence on the scale of the pulse duration.
6.23. A spatially-uniform, but time-dependent external electric field E( t) is applied, starting from t = 0, to a charged plane rotator, initially in its ground state.
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(i) Calculate, in the lowest non-vanishing order in the field’s strength, the probability that by time t > 0, the rotator is in its n th excited state.
(ii) Spell out and analyze your results for a constant-magnitude field rotating, with a constant angular velocity , within the rotator’s plane.
(iii) Do the same for a monochromatic field of frequency , with a fixed polarization.
6.24. A spin-½ with a gyromagnetic ratio is placed into a magnetic field including a time-independent component B0, and a perpendicular field of a constant magnitude Br, rotated with a constant angular velocity . Can this magnetic resonance problem be reduced to one already discussed in Chapter 6?
6.25. Develop 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)/ .
6.26. A heavy, relativistic particle, with electric charge q = Ze, passes 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 probabilities of the atom’s transition to its lowest excited states.
6.27. A particle of mass m is initially in the localized ground state, with energy E g < 0, of a very small, 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
2
/ m r c / r ,
e B
B
where r B is the Bohr radius. 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 the discussion at the end of the last chapter, in which one component system (in Fig. 6.13, system b) may be used as a model of the environment of its counterpart (system a), was still based on the assumption of a pure initial state (6.143) of the composite system. If the interaction of the two components of such a system is described by a certain Hamiltonian (the one given by Eq. (6.145) for example), and the energy spectrum of each component system is discrete, for state of the composite system at an arbitrary instant we may write
n n n ,
(7.1)
n
n
a
b
n
n
with a unique correspondence between the eigenstates na and nb.
However, in many important cases, our knowledge of a quantum system’s state is even 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 a weak3 but substantial contact with its environment e – here understood in the most general sense, say, as all the whole Universe less 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.
Much of this analysis will pertain also to another category of cases – when the system of our interest is isolated from its environment, at present, with acceptable precision, but our knowledge of its state is still incomplete for some other reason. Most typically, the system could be in contact with its 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 Indeed, a system, possibly apart from our Universe as a whole (who knows? – see below), is never exactly coherent, though in many cases, such as the ones discussed in the previous chapters, deviations from the coherence may be ignored with acceptable accuracy.
3 If the interaction between a system and its environment is very strong, their very partition is impossible.
© K. Likharev
QM: Quantum Mechanics
environment at earlier times, and its reduction to a pure state is impracticable. 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.
weak
The Universe
interaction
system of
environment ( e)
interest ( s)
Fig. 7.1. A quantum system and its environment
(VERY schematically :-).
In classical physics, the analysis of mixed statistical ensembles is based on the notion of the probability W (or the probability density w) of each detailed (“microscopic”) state of the system of interest.4 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,5 may be described as the following linear superposition:
Universe:
s e .
(7.2)
quantum
jk
j
k
j, k
state
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 exactly the quantum-mechanical description of a mixed state of the system s.
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), to find the expectation value of an arbitrary observable A in the state (2), we would need to calculate the long bracket
A A ˆ *
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 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
4 See, e.g., SM Sec. 2.1.
5 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.
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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 that, according to Eq. (4.96), this relation may be represented as Intrinsic
observable:
expectation
A A w Tr
(Aw),
(7.5)
jj '
j' j
value
jj '
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 suggested 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).
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so that 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 Statistical
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
value of
A Tr (Aw) A w
A w ,
(7.14)
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 form
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 coherent (pure) 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) by j; then
Density
w
*
,
(7.19) matrix:
jj ' in s
j
j'
pure state
Chapter 7
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so that N 2 elements of the whole N N matrix is 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) in the form
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 probability uniformly distributed over the trigonometric circle. Then the ensemble averaging is equivalent to the averaging over from 0 to 2,8
which kills the off-diagonal terms of the density matrix (22), so that 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 (coherent) state.
After this example, the reader should not be much shocked by the main claim9 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 the thermal equilibrium with a much larger environment of a fixed temperature T (such an environment is usually called a heat bath) the statistical physics gives a very simple expression, called the Gibbs distribution, for the probabilities Wn:10
Gibbs
1
E
E
distribution
W exp n ,
Z
with
exp
.
(7.24)
n
n
Z
k T
B
n
k T
B
8 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. In Sec. 3
below, we will study the dynamics of this dephasing process.
9 This fact follows from the basic postulate of statistical physics, called the microcanonical distribution – see, e.g., SM Sec. 2.2.
10 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.
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where En is the eigenenergy of the corresponding stationary state, and the normalization coefficient Z is called the statistical sum.
A detailed analysis of classical and quantum ensembles in thermodynamic equilibrium is a major focus of statistical physics courses (such as the SM 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,11 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 get12
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
k T
k T
0
B
1
/
, for
,
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 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.) Following Eq. (4.47), it is natural to define the following function of two arguments (traditionally, also called the density matrix):
11 See, e.g., SM Sec. 2.8.
12 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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QM: Quantum Mechanics
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),13 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:14
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 that ( x) = j’( x), and (
w x, x' ) ( x) *
( x' ) ( x) *
( x' ) .
(7.31)
j'
j'
We see that the density matrix carries the information not only about the modulus but also the phase of the wavefunction. (Of course one may argue rather convincingly that in this ultimate limit the density-matrix description is redundant because all this information is contained in the wavefunction itself.) How may be the density matrix interpreted? In the simple case (31), we can write
(
w x, x' ) 2 (
w x, x' ) w* ( x, x' ) ( x) *
( x) ( x' ) *
( x' ) (
w x) (
w x' ) ,
(7.32)
so that the modulus squared of the density matrix is just as the joint probability density to find the system at the point x and the point x’. For example, for a simple wave packet with a 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. The interpretation becomes more complex if we deal with an incoherent mixture of several wavefunctions, for example, 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 (1D) particle in the 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 that we need to replace the sum (33) with an integral, using for example the “delta-normalized” traveling-wave eigenfunctions (4.264): 13 For now, I will focus on a fixed time instant (say, t = 0), and hence write ( x) instead of ( x, t).
14 This fact is the origin of the density matrix’s name.
Chapter 7
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QM: Quantum Mechanics
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, 15
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
B
2
thermal
2
2
equilibrium
This is a very interesting result: the density matrix depends only on the difference of its arguments, dropping to zero fast as the distance between the points x and x’ exceeds the following characteristic scale (called the correlation length)
1/ 2
x x x' 2
(7.37) Correlation
c
length
mk T
B
.
1/ 2
This length may be interpreted in the following way. It is straightforward to use Eq. (24) to verify that the average energy E = p 2/2 m of a free particle in the thermal equilibrium, i.e. in the classical mixture (33), equals k B T/2. Hence the average magnitude of the particle’s momentum may be estimated as 1/ 2
2
p p
m E
mk T
(7.38)
c
2
1/2 B 1/2,
so that x c is of the order of the minimal length allowed by the Heisenberg-like “uncertainty relation”: x / p .
(7.39)
c
c
Note that with the growth of temperature, the correlation length (37) goes to zero, and the density matrix (36) tends to a delta function:
(