Quantum Mechanics by Konstantin K. Likharev - HTML preview
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coth
0 .
(7.151)
2
m 2
0
2 k T
2 m
2 k T
0
B
0
B
But this is exactly Eq. (48), which was derived in Sec. 2 from the Gibbs distribution, without any explicit account of the environment – though implying it by using the notion of the thermally-equilibrium ensemble.56
54 See, e.g., MA Eq. (6.5a).
55 Note that this calculation remains correct even if the dissipation’s dispersion law deviates from the Ohmic model (138), provided that the drag coefficient is replaced with its effective value Im(0)/0.
56 By the way, the simplest way to calculate SF(), i.e. to derive the FDT, is to require that Eqs. (48) and (150) give the same result for an oscillator with any eigenfrequency . This was exactly the approach used by H.
Nyquist (for the classical case) – see also SM Sec. 5.5.
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Notice that in the final form of Eq. (151), the coefficient , which characterizes the oscillator-to-environment interaction strength, has canceled! Does this mean that in Sec. 4 we toiled in vain? By no means. First of all, the result (150), augmented by the FDT (134), has an important conceptual value.
For example, let us consider the low-temperature limit k B T << 0 where Eq. (151) is reduced to 2
x
2
0
x
.
(7.152)
2 m
2
0
Let us ask a naïve question: what exactly is the origin of this coordinate’s uncertainty? From the point of view of the usual quantum mechanics of absolutely closed (Hamiltonian) systems, there is no doubt: this non-vanishing variance of the coordinate is the result of the final spatial extension of the ground-state wavefunction (2.275), reflecting Heisenberg’s uncertainty relation – which in turn results from the fact that the operators of coordinate and momentum do not commute. However, from the point of view of the Heisenberg-Langevin equation (145), the variance (152) is an inalienable part of the oscillator’s
~
response to the fluctuation force F t exerted by the environment at frequencies 0. Though it is impossible to refute the former, absolutely legitimate point of view, in many applications it is easier to subscribe to the latter standpoint and treat the coordinate’s uncertainty as the result of the so-called quantum noise of the environment, which, in equilibrium, obeys the FTD (134). This notion has received numerous confirmations in experiments that did not include any oscillators with their own frequencies 0 close to the noise measurement frequency .57
The second advantage of the Heisenberg-Langevin approach is that it is possible to use Eq.
(148) to calculate the (experimentally measurable!) distribution Sx(), i.e. decompose the fluctuations into their spectral components. This procedure is not restricted to the limit of small (i.e. of large Q); for any damping, we may just plug the FDT (134) into Eq. (148). For example, let us have a look at the so-called quantum diffusion. A free 1D particle, moving in a viscous medium providing it with the Ohmic damping (137), may be considered as a particular case of a 1D harmonic oscillator (145), but with 0 = 0, so combining Eqs. (134) and (149), we get
2
S
( ) d
1
F
x
2
2
coth
d
.
(7.153)
2 2
2
0 (
m
)
2 2
2
0 (
m
)
2
2 k T
B
This integral has two divergences. The first one, of the type d/2 at the lower limit, is just a classical effect: according to Eq. (85), the particle’s displacement variance grows with time, so it cannot have a finite time-independent value that Eq. (153) tries to calculate. However, we still can use that result to single out the quantum effects on diffusion – say, by comparing it with a similar but purely classical case. These effects are prominent at high frequencies, especially if the quantum noise overcomes the thermal noise before the dynamic cut-off, i.e. if
k T
B
.
(7.154)
m
In this case, there is a broad range of frequencies where the quantum noise gives a substantial contribution to the integral:
57 See, for example, R. Koch et al., Phys. Lev. B 26, 74 (1982).
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/ m
/ m
d
2
1
Quantum
x
2
diffusion
d
(7.155)
ln
~ .
2
Q
2
mk T
B
k T /
k T /
B
B
Formally, this contribution diverges at either m 0 or T 0, but this logarithmic (i.e. extremely weak) divergence is readily quenched by almost any change of the environment model at very high frequencies, where the “Ohmic” approximation (136) becomes unrealistic.
The Heisenberg-Langevin approach is very powerful because its straightforward generalizations enable analyses of fluctuations in virtually arbitrary linear systems, i.e. the systems described by linear differential (or integro-differential) equations of motion, including those with many degrees of freedom, and distributed systems ( continua), and such systems dominate many fields of physics. However, this approach also has a major limitation: if the equations of motion of the Heisenberg operators are not linear, then there is no linear relation, such as Eq. (146), between the Fourier images of the generalized forces and the generalized coordinates, and as the result, there is no simple relation, such as Eq. (148), between their spectral densities. In other words, if the Heisenberg equations of motion are nonlinear, there is no regular simple way to use them to calculate the statistical properties of the observables.
For example, let us return for a second to the dephasing problem described by Eqs. (68)-(70), and assume that the deterministic and fluctuating parts of the effective force – f exerted by the environment, are characterized by relations similar, respectively, to Eqs. (124) and (134). Now writing the Heisenberg equations of motion for the two remaining spin operators, and using the commutation relations between them, we get
1
1
2
2
ˆ~
ˆ
ˆ
ˆ
ˆ
ˆ
,
ˆ
,
ˆ
ˆ
ˆ
ˆ
,
(7.156)
x
H
x
x c f
z
z y c f
z
c
f
y
z
z
i
i
and a similar equation for ˆ . Such nonlinear equations cannot be used to calculate the statistical y
properties of the Pauli operators in this system exactly – at least analytically.
For some calculations, this problem may be circumvented by linearization: if we are only interested in small fluctuations of the observables, their nonlinear Heisenberg equations of motion, such as Eq. (156), may be linearized with respect to small deviations of the operators from their (generally, deterministic “values”, and then the resulting linear equations for the operator variations may be solved either as has been demonstrated above, or (if the deterministic “values” evolve in time) using their Fourier expansions. Sometimes this approach gives relatively simple and important results,58 but for many other problems, this approach is insufficient, leaving a lot of space for alternative methods.
7.6. Density matrix approach
The main alternative approach to the dynamics of open quantum systems, which is essentially a generalization of the one discussed in Sec. 2, is to extract the final results of interest from the dynamics of the density operator of our system s. Let us discuss this approach in detail.59
58 For example, the formula used for processing the experimental results by R. Koch et al. (mentioned above), had been derived in this way. (This derivation will be suggested to the reader as an exercise.) 59 As in Sec. 4, the reader not interested in the derivation of the basic equation (181) of the density matrix evolution may immediately jump to the discussion of this equation and its applications.
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We already know that the density matrix allows the calculation of the expectation value of any observable of the system s – see Eq. (5). However, our initial recipe (6) for the density matrix element calculation, which requires the knowledge of the exact state (2) of the whole Universe, is not too practicable, while the von Neumann equation (66) for the density matrix evolution is limited to cases in which probabilities Wj of the system states are fixed – thus excluding such important effects as the energy relaxation. However, such effects may be analyzed using a different assumption – that the system of interest interacts only with a local environment that is very close to its thermally-equilibrium state described, in the stationary-state basis, by a diagonal density matrix with the elements (24).
This calculation is facilitated by the following general observation. Let us number the basis states of the full local system (the system of our interest plus its local environment) by l, and use Eq. (5) to write
A
ˆ
Tr w
A
ˆ
ˆ
ˆ
,
(7.157)
l A w
ll'
l'l
l A l' l' w l
l
l l'
,
l,l'
where w îs the full density operator of this local system. At a weak interaction between the system s l
and the local environment e, their states reside in different Hilbert spaces, so we can write60
l s e ,
(7.158)
j
k
and if the observable A depends only on the coordinates of the system s of our interest, we may reduce Eq. (157) to the form similar to Eq. (5):
ˆ
A e s A s e e s ˆ w s e k
j
j'
k '
k '
j'
l
j
k
j,j' ; k,k'
(7.159)
ˆ
A s e ˆ w e s Tr ( ˆ)
w
A ,
jj'
j'
k
l
k
j
j
j, j'
k
where
w ˆ e w ˆ e Tr w ˆ ,
(7.160)
k
l
k
k
l
k
showing how the density operator w ôf the system s (sometimes called the reduced density operator) may be calculated from the full operator w ˆ .
l
Now comes the key physical assumption of this approach: since we may select the local environment e to be much larger than the system s of our interest, we may consider the composite system l as a Hamiltonian one, with time-independent probabilities of its stationary states, so for the description of the evolution in time of its full density operator w ˆ (again, in contrast to that, w ˆ , of the l
system of our interest) we may use the von Neumann equation (66). Partitioning its right-hand side in accordance with Eq. (68), we get:
i w ˆ
H ˆ , w ˆ H ˆ , w ˆ H ˆ , w ˆ .
(7.161)
l
s l e l int l
The next step is to use the perturbation theory to solve this equation in the lowest order in ˆ
H , that
int
would yield a non-vanishing contribution due to the interaction. For that, Eq. (161) is not very 60 Let me emphasize that this simple representation is valid only for the basis states of our local system but, generally, not for its quantum state! Indeed, the calculation we are performing is valid (and is most relevant) even when the local system is not in any definite quantum state, and may be only described by the density matrix wll’.
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convenient, because its right-hand side contains two other terms, of a much larger scale than the interaction Hamiltonian. To mitigate this technical difficulty, the interaction picture that was discussed at the end of Sec. 4.6, is very natural. (It is not necessary though, and I will use this picture mostly as an exercise of its application – unfortunately, the only example I can afford to give in this course.) As a reminder, in that picture (whose entities will be marked with index “I”, with the unmarked operators assumed to be in the Schrödinger picture), both the operators and the state vectors (and hence the density operator) depend on time. However, the time evolution of the operator of any observable A is described by an equation similar to Eq. (67), but with the unperturbed part of the Hamiltonian only – see Eq. (4.214). In the model (68), this means
ˆ
ˆ
ˆ
i A A , H
I
I 0.
(7.162)
where the unperturbed Hamiltonian consists of two parts defined in different Hilbert spaces: ˆ
ˆ
ˆ
H H H
0
s
e .
(7.163)
On the other hand, the state vector’s dynamics is governed by the interaction evolution operator ˆ u that I
obeys Eqs. (4.215). Since this equation, using the interaction-picture Hamiltonian (4.216), ˆ
H ˆ† ˆ
u H ˆ u ,
(7.164)
I
0
int 0
is absolutely similar to the ordinary Schrödinger equation using the full Hamiltonian, we may repeat all arguments given at the beginning of Sec. 3 to prove that the dynamics of the density operator in the interaction picture of a Hamiltonian system is governed by the following analog of the von Neumann equation (66):
i ˆ
ˆ
w H , ˆ w ,
(7.165)
I
I I
where the index l is dropped for the notation simplicity. Since this equation is similar in structure (with the opposite sign) to the Heisenberg equation (67), we may use the solution Eq. (4.190) of the latter equation to write its analog:
ˆ w t u t
w
u t
.
(7.166)
I
Î 0
, ˆ ( )
0 ˆ†
l
I 0
,
It is also straightforward (and hence is left for the reader) to verify that in this picture, the expectation value of any observable A may be found from an expression similar to the basic Eq. (5): A
ˆ
Tr A ˆ w
I
I ,
(7.167)
showing again that the interaction and Schrödinger pictures give the same final results.
In the case of the factorable interaction (90),61 Eq. (162) is simplified for both operators participating in that product – for each one in its own way. In particular, for A ˆ x ˆ , it yields i x ˆ
.
(7.168)
I
x ˆ H ˆ
,
I
0
x ˆ H ˆ
,
x ˆ H ˆ
,
I
s
I e
61 An analysis of a much more general case when the interaction with the environment is represented as a sum of several/many products of the type (90), may be found, for example, in the monograph by K. Blum, Density Matrix Theory and Applications, 3rd ed., Springer, 2012.
Chapter 7
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Since the coordinate operator is defined in the Hilbert space of our system s, it commutes with the Hamiltonian of the environment, so we get
i x ˆ
.
(7.169)
I
x ˆ H ˆ
,
I
s
On the other hand, if A ˆ F ˆ , this operator is defined in the Hilbert space of the environment, and commutes with the Hamiltonian of the unperturbed system s. As a result, we get i F ˆ
I
F ˆ H ˆ
,
I
e
.
(7.170)
This means that with our time-independent unperturbed Hamiltonians, H ând H ˆ , the time s
e
evolution of the interaction-picture operators is rather simple. In particular, the analogy between Eq.
(170) and Eq. (93) allows us to immediately write the following analog of Eq. (94):
i
i
F ˆ
,
(7.171)
I t
H êxp
t ˆ
ˆ
0 exp
e F
H te
so in the stationary-state basis n of the environment,
i
i
E E
n
n'
F ˆ
,
(7.172)
I
t
( ) exp
( )
0 exp
( )
0 exp
nn'
E t
n Fnn'
E t
n' Fnn'
i
t
and similarly (but in the basis of the stationary states of system s) for the operator x ˆ . As a result, the right-hand side of Eq. (164) may be also factored:
ˆ
†
i
i
H t u t
H u t
H H t F
x
H H t
I
ˆ0 ˆ
0
,
înt
0 0
, exp
ˆ ˆ
s
e
ˆ ˆexp ˆ ˆ
s
e
(7.173)
i ˆ
i
i
i
exp H t ˆ
ˆ
ˆ
ˆ
ˆ
x exp H t exp
H t F(0) exp H t ˆ x
s
s
e
e
I t ˆ
F t
I .
So, the transfer to the interaction picture has taken some time, but now it enables a smooth ride.62
Indeed, just as in Sec. 4, we may rewrite Eq. (165) in the integral form:
t
1
w ˆ
ˆ
, ˆ
;
(7.174)
I t
H I t' w I t' dt'
i
plugging this result into the right-hand side of Eq. (165), we get
t
t
1
1
w ˆ
,
(7.175)
I t
2 H Î t H ˆ
,
I t' , w
Î t' dt' 2 x ˆ t( F ˆ
) t
( ), x ˆ t'
( F ˆ
) t'
( ), w ˆ t'
( )
I
dt'
where, for the notation’s brevity, from this point on I will strip the operators x ând F ôf their index
“I ” . (I hope their time dependence indicates the interaction picture clearly enough.) So far, this equation is exact (and cannot be solved analytically), but this is a good time to notice that even if we approximate the density operator on its right-hand side by its unperturbed, factorable
“value”
62 If we used either the Schrödinger or the Heisenberg picture instead, the forthcoming Eq. (175) would pick up a rather annoying multitude of fast-oscillating exponents, of different time arguments, on its right-hand side.
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ˆ w t' ˆ w t' ˆ w ,
with e ˆ w e
W ,
(7.176)
I
e
n
e
n'
n nn'
corresponding to no interaction between the system s and its thermally-equilibrium environment e, where en are the stationary states of the environment and Wn are the Gibbs probabilities (24), Eq. (175) still describes a nontrivial time evolution of the density operator.63 This is exactly the first nonvanishing approximation (in the weak interaction) we have been looking for. Now using Eq. (160), we find the equation of evolution of the density operator of the system of our interest: 64
t
1
ˆ
w t
,
(7.177)
2 Tr
x ˆ t
( F ˆ
) t
( ), x ˆ t'
( F ˆ
) t'
( ), w ˆ t'
( ) w ˆ dt'
n
e
where the trace is over the stationary states of the environment. To spell out the right-hand side of Eq.
(177), note again that the coordinate and force operators commute with each other (but not with themselves at different time moments!) and hence may be swapped at will, so we may write Tr , , ˆ ˆ
ˆ
ˆ
ˆ
Tr
ˆ ˆ
ˆ
ˆ
ˆ
Tr
ˆ ˆ
n
x t x t'
w t' n F t F t' we x t
w t' x t' n F t w F
e
t'
x ˆ t' ˆ
w t' x ˆ t
ˆ
Tr
ˆ ˆ
ˆ
ˆ
ˆ Tr ˆ ˆ
ˆ
n F t' w F
e
t
w t' x t' x t n w F
e
t' F t
ˆ x tˆ x t' ˆ
w t' F t F t' W x t w t' x t'
F
t W F t'
nn' n'n
ˆ
n
ˆ ˆ nn' n' n'n
n, n'
n, n'
ˆ
(7.178)
x t' ˆ
w t' ˆ x t F t' W F t w t' x t' x t
W F
t' F
t
nn'
n'
n'n ˆ ˆ ˆ
n
nn' n'n .
n, n'
n, n'
Since the summation over both indices n and n’ in this expression is over the same energy level set (of all stationary states of the environment), we may swap these indices in any of the sums. Doing this only in the terms including the factors Wn’, we turn them into Wn, so this factor becomes common: Tr
n
, ,
Wn ˆ x tˆ x t' ˆ
w t' Fnn' t Fn'n t' ˆ x t ˆ
w t' ˆ x t' Fn'n t Fnn' t'
n, n'
(7.179)
ˆ x t' ˆ w ˆ x t F
n'n t' Fnn' t
ˆ w ˆ x t' ˆ x t Fnn' t' Fn'n t.
Now using Eq. (172), we get
~
~
iE t t'
iE t t'
x ˆ t x ˆ t' ˆ
w t'
exp
x ˆ t ˆ
w t' x ˆ t'
exp
Tr ..., ...,...
W F 2
n
n
nn'
~
~
n, n'
iE t t'
iE t t'
x ˆ t' ˆ
w t' x ˆ t
exp
ˆ
w t' x ˆ t' x ˆ t
exp
~
2
E t t'
~
2
E t t'
W F
cos
x t x t' w t'
i
W F
x t x t' w t'
n
nn'
ˆ ,ˆ , ˆ
sin
n
nn'
ˆ ,ˆ , ˆ . (7.180)
n, n'
n,n'
Comparing the two double sums participating in this expression with Eqs. (108) and (111), we see that they are nothing else than, respectively, the symmetrized correlation function and the temporal Green’s 63 This is exactly the moment of transition from the reversible quantum dynamics to irreversible one, in this approach.
64 For the notation simplicity, the fact that here (and in all following formulas) the density operator w ôf the system s of our interest is taken in the interaction picture, is just implied.
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function (multiplied by /2) of the time-difference argument = t – t’ 0. As a result, Eq. (177) takes a compact form:
t
t
Density
1
i
ˆ
w t
K t t' x ˆ t
( ), x ˆ t'
( ), w ˆ t'
( ) dt'
G t t' x ˆ t
( ), x ˆ t'
( ), w ˆ t'
( ) dt' . (7.181) matrix:
2
F
time
2
evolution
Let me hope that the readers (especially the ones who have braved this derivation) enjoy this beautiful result as much as I do. It gives an equation for the time evolution of the density operator of the system of our interest ( s), with the effects of its environment represented only by two real, c-number functions of τ: one ( KF) describing the fluctuation force exerted by the environment, and the other one ( G) representing its ensemble-averaged environment’s response to the system’s evolution. And most spectacularly, these are exactly the same functions that participate in the alternative, Heisenberg-Langevin approach to the problem, and hence related to each other by the fluctuation-dissipation theorem (134).
After a short celebration, let us acknowledge that Eq. (181) is still an integro-differential equation, and needs to be solved together with Eq. (169) for the system coordinate’s evolution. Such equations do not allow explicit analytical solutions, besides a few very simple (and not very interesting) cases. For most applications, further simplifications should be made. One of them is based on the fact (which was already discussed in Sec. 3) that both environmental functions participating in Eq. (181) tend to zero when their argument t – t’ becomes much larger than the environment’s correlation time
c, which is independent of the system-to-environment coupling strength. If the coupling is sufficiently weak, the time scales Tnn’ of the evolution of the density matrix elements, following from Eq. (181), are much longer than this correlation time, and also than the characteristic time scale of the coordinate operator’s evolution. In this limit, all arguments t’ of the density operator, giving substantial contributions to the right-hand side of Eq. (181), are so close to t that it does not matter whether its argument is t’ or just t. This simplification, w( t’) w( t), is known as the Markov approximation. 65
However, this approximation alone is still insufficient for finding the general solution of Eq.
(181). Substantial further progress is possible in two important cases. The most important of them is when the intrinsic Hamiltonian H ôf the system s of our of interest does not depend on time explicitly s
and has a discrete eigenenergy spectrum En,66 with well-separated levels: E E
.
(7.182)
n
n'
Tnn'
Let us see what this condition yields for Eq. (181) rewritten for the matrix elements in the stationary state basis, in the Markov approximation:
65 Named after A. A. Markov (1856-1922; in older Western literature, “Markoff”), a mathematician famous for his general theory of the so-called Markov processes, whose future development is completely determined by its present state, but not its pre-history.
66 Here, rather reluctantly, I will use this standard notation, En, for the eigenenergies of our system of interest ( s), in the hope that the reader would not confuse these discrete energy levels with the quasi-continuous energy levels of its environment ( e), participating in particular in Eqs. (108) and (111). As a reminder, by this stage of our calculations, the environment’s levels have disappeared from our formulas, leaving behind their functionals KF( ) and G().
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t
t
1
i
w
K t t' x ˆ t
( ), x ˆ t'
( ), w ˆ
dt'
G t t' x ˆ t
( ), x ˆ t'
( ), w ˆ
dt' .
(7.183)
nn'
2
F
nn'
nn'
2
After spelling out the commutators, the right-hand side of this expression includes four operator products, which differ “only” by the operator order. Let us first have a look at one of these products,
x ˆ t
( ) x ˆ t'
( ) ˆ
w
x
t
( ) x
t'
( ) w
,
(7.184)
nn'
nm mm'
m'n'
m,m'
where the indices m and m’ run over the same set of stationary states of the system s of our interest as the indices n and n’. According to Eq. (169) with a time-independent Hs, the matrix elements xnn’ (in the stationary state basis) oscillate in time as exp{ i nn’t}, so
x ˆ t
( ) x ˆ t'
( ) ˆ
w
x x
exp i t t' w
,
(7.185)
nn'
nm mm'
nm
mm' m'n'
m, m'
where on the right-hand side, the coordinate matrix elements are in the Schrödinger picture, and the usual notation (6.85) is used for the quantum transition frequencies:
E E .
(7.186)
nn'
n
n'
According to condition (182), frequencies nn’ with n n’ are much higher than the speed of evolution of the density matrix elements (in the interaction picture!) – on both the left-hand and right-hand sides of Eq. (183). Hence, on the right-hand side of Eq. (183), we may keep only the terms that do not oscillate with these frequencies nn’, because rapidly-oscillating terms would give negligible contributions to the density matrix dynamics.67 For that, in the double sum (185) we should save only the terms proportional to the difference ( t – t’) because they will give (after the integration over t’) a slowly changing contribution to the right-hand side.68 These terms should have nm + mm’ = 0, i.e. ( En –
Em) + ( Em – Em’) En – Em’ = 0. For a non-degenerate energy spectrum, this requirement means m’ = n; as a result, the double sum is reduced to a single one:
x ˆ t
( ) x ˆ t'
( ) w ˆ w
x x exp i
t t'
w
x
2 exp i t t' .
(7.187)
nn'
nn' nm mn
nm nn' nm
nm
m
m
Another product, w ˆ x ˆ t'
( ) x ˆ t
( ) , which appears on the right-hand side of Eq. (183) may be simplified
nn'
absolutely similarly, giving
ˆ x
w ˆ t'
( ) x ˆ t
( )
x
2 exp i t' t w .
(7.188)
nn'
n'm
n'm nn'
m
These expressions hold whether n and n’ are equal or not. The situation is different for two other products on the right-hand side of Eq. (183), with w sandwiched between x( t) and x( t’). For example,
x ˆ t
( ) w ˆ x ˆ t'
( )
x
t
( ) w
x
t'
( )
x w
x
exp i t t' .
(7.189)
nn'
nm
mm'
m'n'
nm mm' m'n'
nm
m'n'
m, m'
m, m'
67 This is essentially the same rotating-wave approximation (RWA) as was used in Sec. 6.5.
68 As was already discussed in Sec. 4, the lower-limit substitution ( t’ = –) in the integrals participating in Eq.
(183) gives zero, due to the finite-time “memory” of the system, expressed by the decay of the correlation and response functions at large values of the time delay = t – t’.
Chapter 7
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For this term, the same requirement of keeping an oscillating function of ( t – t’) only, yields a different condition: nm + m’n’ = 0, i.e.
E E
E
E
.
(7.190)
n
m
m'
n' 0
Here the double sum’s reduction is possible only if we make an additional assumption that all interlevel energy distances are unique, i.e. our system of interest has no equidistant levels (such as in the harmonic oscillator). For the diagonal elements ( n = n’), the RWA requirement is reduced to m = m’, giving sums over all diagonal elements of the density matrix:
x ˆ t() w ˆ x ˆ t' () x 2 exp i t t' w .
(7.191)
nn
nm
nm mm
m
(Another similar term, x ˆ t'
( ) w ˆ x ˆ t
( ) , is just a complex conjugate of this one.) However, for off-diagonal
nn
matrix elements ( n n’), the situation is different: Eq. (190) may be satisfied only if m = n and m’ = n’, so the double sum is reduced to just one, non-oscillating term:
x ˆ t
( ) w ˆ x ˆ t'
( ) x w x ,
n
for n' .
(7.192)
nn'
nn
nn'
n'n'
The second similar term, x ˆ t'
( ) w ˆ x ˆ t
( ) , is exactly the same, so in the first of the integrals in Eq. (183),
nn
these terms add up, while in the second one, they cancel.
This is why the final equations of evolution look differently for diagonal and off-diagonal elements of the density matrix. For the former case ( n = n’), Eq. (183) is reduced to the so-called master equation 69 relating diagonal elements wnn of the density matrix, i.e. the energy level occupancies Wn: 70
2
1
W x
K
n
nm
F W
W
n
m
exp
i
nm
exp
i
nm
2
m n
0
(7.193)
i
G W W
n
m
exp
i
nm
exp
i
nm
d ,
2
where t – t’. Changing the summation index notation from m to n’, we may rewrite the master equation in its canonical form
W
W
W ,
(7.194) Master
n
n' n
n'
n n'
n
equation
n' n
where the coefficients
2
2
1
Γ
x
K
G
dt'
(7.195) Interlevel
n' n
nn'
F cos
nn'
sin
,
2
nn'
transition
0
rates
are called the interlevel transition rates.71 Formula (194) has a very clear physical meaning of the level occupancy dynamics (i.e. the balance of the probability flows W) due to quantum transitions between 69 The master equations, which were first introduced to quantum mechanics in 1928 by W. Pauli, are sometimes called the “Pauli master equations”, or “kinetic equations”, or “rate equations”.
70 As Eq. (193) shows, the term with m = n would vanish and thus may be legitimately excluded from the sum.
71 As Eq. (193) shows, the result for n n’ is described by Eq. (195) as well, provided that the indices n and n’ are swapped in all components of its right-hand side, including the swap nn’ n’n = – nn’.
Chapter 7
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the energy levels (see Fig. 7), in our current case caused by the interaction between the system of our interest and its environment.
higher levels
E
W
n
n
Fig. 7.7. Probability flows in a discrete-
spectrum system. Solid arrows: the
n' n
n n'
exchange between the two energy levels, n
E
W
and n’, described by one term in the master
n'
n'
equation (194); dashed arrows: other
transitions to/from these two levels.
lower levels
The Fourier transforms (113) and (123) enable us to express the two integrals in Eq. (195) via, respectively, the symmetrized spectral density SF() of environment force fluctuations and the imaginary part ” () of the generalized susceptibility, both at frequency = nn’. After that we may use the fluctuation-dissipation theorem (134) to exclude the former function, getting finally72
Transition
1
2
"
nn'
2
2
(
)
rates via
x
"
x
nn'
.
(7.196)
n' n
nn'
nn' coth
1
” ()
2
k T
nn'
exp E E
k T
B
n
n' /
B
1
Note that since the imaginary part ” of the generalized susceptibility is an odd function of frequency, Eq. (196) is in compliance with the Gibbs distribution for arbitrary temperature. Indeed, according to this formula, the ratio of the “up” and “down” rates for each pair of levels equals
n'
"
n
nn'
" n'n
E E
exp n
n'
.
(7.197)
exp{( E E ) / k T} 1 exp{( E E ) / k T} 1
n
k T
n'
n
n'
B
n'
n
B
B
On the other hand, according to the Gibbs distribution (24), in thermal equilibrium the level populations should be in the same proportion. Hence, Eq. (196) complies with the so-called detailed balance equation,
Detailed
balance
W
W
,
n n n'
n' n' n
(7.198)
valid in the equilibrium for each pair { n, n’}, so all right-hand sides of all Eqs. (194), and hence the time derivatives of all Wn vanish – as they should. Thus, the stationary solution of the master equations indeed describes the thermal equilibrium correctly. As will be shown in the next section, the speed of reaching this equilibrium, within each pair of levels, is determined by the sum of these two rates: 72 It is straightforward (and highly recommended to the reader as an exercise) to show that at low temperatures ( k B T << En’ – En), Eq. (196) gives the same result as the Golden Rate formula (6.111), with A = x. (The low-temperature condition ensures that the initial occupancy of the excited level n is negligible, as was assumed at the derivation of Eq. (6.111).)
Chapter 7
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,
1
for k T ,
coth
B
nn '
nn'
nn'
n n'
n ' n
nn'
nn'
2 k T
B
2 k T /
, for k T
,
B
nn'
B
nn'
(7.199)
2
with
2
x
"
nn'
nn '
nn' .
In situations when the level specification is obvious, the rate is frequently denoted as the reciprocal energy relaxation time 1/ T 1.
The system of master equations (194), frequently complemented by additional right-hand-side terms describing interlevel transitions due to other factors (e.g., by an external ac force with a frequency close to one of nn’), is the key starting point for practical analyses of many applied quantum systems, notably including optical quantum amplifiers and generators (lasers). It is important to remember that they are strictly valid only in the rotating-wave approximation, i.e. if the condition (182) is satisfied for all n and n’ of substance.
The relaxation times T 1, characterizing the dynamics of the diagonal elements of the density matrix, should not be confused with the characteristic times T 2 of the off-diagonal element decay, i.e. of the dephasing, which was preliminary discussed in Sec. 3. Now let us see what Eqs. (183) have to say about the dephasing rates. Taking into account our intermediate results (187)-(192), and merging the non-oscillating components (with m = n and m = n’) of the sums Eq. (187) and (188) with the terms (192), which also do not oscillate in time, we get the following equation:73
1
w
K
nn'
F
2
xnm
exp i nm
2
xn'm
exp i n'm x
x
nn
n'n' 2
2
0
m n
m n'
(7.200)
i
G
2
x
nm
exp i nm
2
xn'm
exp i n'm d w
for
,
n n'.
2
nn'
m n
m n'
In contrast with Eq. (194), the right-hand side of this equation includes both a real and an imaginary part, and hence may be represented as
w 1/ T i
w ,
(7.201)
nn'
nn'
nn' nn'
where both factors 1/ Tnn’ and nn’ are real. As Eq. (201) shows, the second term in the right-hand side of this equation causes slow oscillations of the matrix elements wnn’, which, after returning to the Schrödinger picture, add just small corrections74 to the unperturbed frequencies (186) of their oscillations, and are not important for most applications. More important is the first term, proportional to 73 Sometimes Eq. (200) (in any of its numerous alternative forms) is called the Redfield equation, after the 1965
work by A. Redfield. Note, however, that in the mid-1960s, several other authors, notably including (in the alphabetical order) H. Haken, W. Lamb, M. Lax, W. Louisell, and M. Scully, also made major contributions to the very fast development of the density-matrix approach to open quantum systems.
74 Such corrections are sometimes called Lamb shifts, because they are the generic form of the particular effect more commonly called the Lamb shift: a minute (~1 GHz) difference between the frequencies of transitions 2 s
1 s and 2 p 1 s (with all states having j = ½) of hydrogen, due to the electric-dipole coupling to the free-space electromagnetic environment, first observed experimentally in 1947 by Willis Lamb and Robert Retherford.
(These frequencies have to be equal not only in the non-relativistic theory described in Sec. 3.6 but also in the relativistic quantum theory (see Secs. 6.3, 9.7), if the electromagnetic environment is ignored.) The explanation of Chapter 7
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1
1
2
2
2
K
x
x
x
x
F
cos
cos
2
nm
nm
n'm
n'm
nn
n'n'
Tnn'
0
m n
m n'
1
G
2
x
sin
2
x
sin d , for n n' ,
(7.202)
2
nm
nm
n'm
n'm
m n
m n'
because it describes the effect completely absent without the environment coupling: exponential decay of the off-diagonal matrix elements, i.e. the dephasing. Comparing the first two terms of Eq. (202) with Eq. (195), we see that the dephasing rates may be described by a very simple formula: 1
1
x x
S
.
(7.203a)
n m
n'
m
nn n'n' 2 0
T
2
2
F
nn'
m n
m n'
Moreover, since at low frequencies, the dissipation provided by most real environments is Ohmic (138), this expression may be further simplified:
Dephasing
1
1
k T
B
rate
(7.203b)
n m
n'
m
x
x
nn
n'n' 2 ,
for n n' .
T
2
2
nn'
m n
m n'
This result shows that two effects yield independent contributions to the dephasing. The first of them may be interpreted as a result of “virtual” transitions of the system from the levels n and n’ of our interest to other energy levels m; according to Eq. (196), this contribution is proportional to the strength of coupling to the environment at relatively high frequencies nm and n’m. On the contrary, the second contribution is due to low-frequency, essentially classical fluctuations of the environment, and hence to the low-frequency dissipative susceptibility. In the Ohmic dissipation case, when the ratio ” ()/
is frequency-independent, both contributions are of the same order, but their exact relation depends on the matrix elements xnn’ of a particular system. Note also that Eq. (203a), as well as the analysis carried out in Sec. 3, implies that low-frequency fluctuations of any other origin, not taken into account in our analysis (say, an unintentional noise from experimental equipment), may also contribute to dephasing.
Such “technical fluctuations” are indeed a very serious challenge for the experimental implementation of coherent qubit systems – see Sec. 8.5 below. On the other hand, in optical systems, the low-frequency contribution to dephasing is usually negligible.
7.7. Application to two-level systems
Let us see what these results mean for the particular but very important case of a two-level system, with just two relevant states. As was discussed in Sec. 3, it may be described by the Hamiltonian (7.68), with the interaction term in the form (7.70) which coincides with Eq. (7.90) used for the calculations in the previous section, at the replacement x ˆ ˆ . However, instead of the simple intrinsic z
Hamiltonian (7.69) excluding interstate transitions, which was discussed in Sec. 3, the theory described above is valid for the most general Hamiltonian (5.3) in the two-function Hilbert space. As was discussed in Sec. 5.1, after dropping just the trivial term b I, which may be always removed by selecting the proper energy reference, this Hamiltonian, in the usual z-representation, is described by the following matrix
the Lamb shift by H. Bethe, in the same 1947, essentially launched the whole field of quantum electrodynamics –
to be briefly discussed in Chapter 9.
Chapter 7
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c
c ic
z
x
y
cos
sin
i
e
H c ˆσ
c
,
(7.204)
s
c ic
c
i
x
y
z
sin e
cos
where and are the angles describing the direction of the c-number “field vector” c: c c n c n c n c n
n
n
.
(7.205)
x
x
y
y
z
z
sin cos
sin sin
cos
x
y
z
As was discussed in Sec. 5.1, this Hamiltonian may describe a large variety of two-level quantum systems, from spin-½ in a magnetic field, where it is just the Pauli Hamiltonian (4.163), to a particle placed into a system of two coupled quantum wells – see Figs. 2.19 and 2.21.
First, let us consider the case when the field vector c is time-independent. The two stationary states of the system (let us call them + and –), or rather the coefficients of their expansion in the z-basis, and their energies E may be easily found from the corresponding system of equations (4.102):75
c cos
E
i
c sin
e
,
0
(7.206)
c sin ei c cos E 0.
The results are E = c (so that E+ – E– = 2 c), and (up to an arbitrary common phase multiplier):
*
i
cos ,
sin e .
(7.207)
2
2
Now we may readily calculate the matrix elements that participate (after the replacement x ˆ ˆ ) in z
Eqs. (196) and (203):
z
*
*
1
0
ˆ
,
cos ,
z
0 1
(7.208)
z
*
*
1
0
ˆ
,
sin
i
e
.
z
0 1
With these substitutions, Eq. (199) with nn’ = 2 c/ gives
1
2
2
4
"
coth
sin
coth
sin 2 ,
(7.209)
T
k T
k T
1
B
B
where the last expression is valid only for the Ohmic dissipation (138), while Eq. (203) yields 1
1
1
4 k T
B
cos2 .
(7.210)
T
T
2
2
T
2
1
So, the relaxation ( T 1) and dephasing ( T 2) times generally depend on the angle between the z-
axis (i.e., with our assumption (70), the Bloch-sphere direction of the system’s dissipative coupling to the environment) and the dc “field” c that determines the position of the stationary states on the sphere –
75 Since particular cases of this procedure were repeatedly performed in Chapter 4 (see also the solutions of Problems 4.27-4.29 and 5.2-5.4), I hope that more detailed explanations are not needed.
Chapter 7
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see Fig. 5.2.76 In particular, at = 0, i.e. in the absence of interlevel transitions, Eq. (210) reduces to the result (142) obtained under the same assumption, by using the Heisenberg-Langevin approach.
Now let us have a brief look at what the master equations (194), now taking the form W W W ,
W W W ,
(7.211)
say about the environment’s effects on the two-level system’s dynamics. Since the total probability W+
+ W– of finding the system on some energy level has to equal 1, we can eliminate one of these probabilities from these (compatible) equations, and readily integrate the remaining linear equation, for arbitrary initial conditions, getting
t
t
W t W
W
(7.212)
0exp
1 exp ,
T
T
1
1
where
1
1
W
W
.
(7.213)
k T
k T
exp /
B
,
1
exp
/ B 1
If the system was initially incoherent (i.e. was a classical mixture of two states), then this result, describing the exponential transient of both probabilities from their initial values W(0) to the final (stationary) values, would be all we could say about its dynamics. However, if the initial state is pure, there is more to the story, and we also need to use the dephasing equation (201) for the off-diagonal matrix element w+ –. (According to Eq. (6), w– + = w+ –*.) Using the relation reciprocal to Eq. (166) to transfer from the interaction picture back to the Schrödinger one, and neglecting the small Lamb-shift correction due to the environment, we get a simple equation:
1
w
i
,
(7.214)
w
T 2
with the obvious solution
1
1
w t
w
0 exp
i
t ,
so w
t
w
0 exp
i
t .
(7.215)
T 2
T 2
For example, let the system be initially in a pure state . (For a spin-½, this would mean that it is z-polarized, while for the two-well implementation shown in Fig. 4, this is the state with the particle is definitely in the right well.) Then, according to Eqs. (18b) or (20),
W
2
2
0 w 0 cos2 ,
W 0 w 0 sin2 ,
2
2
(7.216)
w
*
i
0
sin cos
e ,
w 0
*
w 0 sin cos
i
e
,
2
2
2
2
where Eqs. (207) were used. Formally, Eqs. (212)-(216) give the full solution of the problem, but in order to comprehend its meaning, let us have a look at the probability W( t) to find our system in the initial state at an arbitrary moment t. Since t > 0, due to dephasing, the system’s state is no longer pure, in order to recalculate the density matrix elements (216) calculated in the stationary-state basis, 76 Since in most optical systems, where the low-frequency contribution to dephasing is small, Eq. (210) gives a very simple (and frequently used) relation T 2 2 T 1.
Chapter 7
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into W( t), i.e. one of the elements of the density matrix in the z-basis, we need to use the general Eq.
(4.93), which is valid for any operator, in particular for the density operator:
*
*
*
*
W t w t
w
t
w t
w
t
. (7.217)
Plugging in Eqs. (207), (212), (213), (215), and (216), we finally get
W t
t
t
sin 4 cos4 exp
W
1 exp
2
2
T
T
1
1
(7.218)
t
2sin 2 cos 2 cos t exp
,
2
2
T 2
where
W
2
2
1
1
cos
sin
1
cos
1
tanh
cos
. (7.219)
2
2
2
2
2
k T
B
So, the probability of the initial state of the system not only relaxes with the time constant T 1 to its final, thermal-equilibrium values W() and W() = 1 – W(), but also performs the quantum oscillations with frequency 2 c/ = ( E+ – E–)/ between these two states, decaying with the time constant T 2. (For spin-½ implementations of the two-level system, this means that the spin’s precession about the field’s direction decays, with the expectation values Sx,y tending to zero at t ~ T 2, and Sz tending to its thermally-equilibrium value [ W() – ½] at t ~ T 1.) In many experimental situations when Eqs. (211) and (214) are valid but the constants T 1 and T 2 cannot be reliably calculated,77 they may be measured by observation of these two relaxation effects.
More complex problems of this type, for example those described by time-dependent Hamiltonians, may evade such simple analytical solutions because the very notions of stationary state and energy levels cannot be used. However, if the time-dependent part of a Hamiltonian is small and may be considered a perturbation, the two-level system may be still described by Eqs. (211) and (214), with additional terms describing this perturbation. In some of these cases, the Bloch equation (5.22), also with additional terms on its right-hand side, may be very useful for analysis. In Sec. 10, one of such problems is offered to the reader as an exercise.
7.8. Damped harmonic oscillator
As was explained in Section 6, the performed calculations starting from Eq. (191) are not valid for systems with equidistant energy spectra – for example, the harmonic oscillator. For this particular but very important system, with its simple matrix elements xnn’ given by Eqs. (5.92), it is longish but straightforward to perform similar calculations, starting from (183), to obtain an equation similar in structure to Eq. (200), but with two other terms, proportional to wn1, n’1, on its right-hand side.
Neglecting the minor Lamb-shift term, the equation reads
w
n
n n'
n n n'
w
nn'
e
1
e
2 nn'
.
(7.220)
2 n
n
n'
w
n nn'
w
e
1
1
11/2
2
n ,
1 n' 1
e
1/2 n ,1 n' 1.
77 For example, in systems whose coupling to the environment cannot be expressed as a single product (46).
Chapter 7
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Here is the effective damping coefficient,78
2
x
Im
0
Im
,
(7.221)
0
0
2
2
m 0
equal to just /2 m for the Ohmic dissipation, and n e is the equilibrium number of oscillator’s excitations, given by Eq. (26b), with the environment’s temperature T. (I am using this new notation because in dynamics, the instant expectation value n may be time-dependent, and is generally different from its equilibrium value n e.)
Alternatively, the derivation of Eq. (220) may be started at a bit earlier point, from the Markov approximation applied to Eq. (181), by expressing the coordinate operator via the creation-annihilation operators (5.65). This procedure gives the result in the operator (i.e. basis-independent) form:79
†
†
†
†
w ˆ n
.
(7.222)
e
1 a ˆ a ˆ, w ˆ 2 ˆ w
a ˆ a ˆ n e ˆ a
a ˆ , w ˆ 2 a ˆ w ˆ a ˆ
In the Fock state basis, this equation immediately reduces to Eq. (220); however, Eq. (222) may be more convenient for some applications.
Returning to Eq. (220), we see that it inter-relates only the elements wnn’ located at the same distance ( n – n’) from the principal diagonal of the density matrix. This means, in particular, that the dynamics of the diagonal elements wnn of the matrix, i.e. the Fock state probabilities Wn, is independent of the off-diagonal elements, and may be represented in the form (194) truncated to the transitions between the adjacent energy levels only ( n’ = n 1):
W
W
W
W
W ,
(7.223)
n
n 1 n n 1 n n 1 n n 1 n n 1 n n 1 n
with the following rates:
2 n n
n n
n 1
n
1 e 1,
2
n n 1
1 ,
e
(7.224)
2 n n ,
2 n n
n 1
n
e
n n 1
e 1.
According to the definition of n e, given by Eq. (26b),
1
1
n
,
so
1
,
(7.225)
e
exp
/ k
1
1 exp
/
0
n
T
e
B
k T
0
B
78 This coefficient participates prominently in the classical theory of damped oscillations (see, e.g., CM Sec. 5.1), in particular defining the oscillator’s Q-factor as Q 0/2, and the decay time of the amplitude A and the energy E of free oscillations: A( t) = A(0)exp{- t}, E( t) = E(0)exp{-2 t}.
79 Sometimes Eq. (222) is called the Lindblad equation, but I believe this terminology is inappropriate. It is true that its structure falls into a general category of equations suggested by G. Lindblad in 1976 for the density operators in the Markov approximation, whose diagonalized form in the interaction picture is w ˆ
L ˆ
2 w ˆ L ˆ†
L ˆ L ˆ† , w ˆ .
j
j
j
j j
j
However, Eq. (222) was derived much earlier (by L. Landau in 1927 for zero temperature, and by M. Lax in 1960
for an arbitrary temperature), and in contrast to the general Lindblad equation, spells out the participating operators L ând coefficients
j
j for a particular physical system – the harmonic oscillator.
Chapter 7
Page 47 of 54
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so taking into account Eqs. (5.92), (186), and (221), and the asymmetry of the function ” (), we see that these rates are again described by Eq. (196), even though the last formula was derived for non-equidistant energy spectra.
Hence the only substantial new feature of the master equation for the harmonic oscillator is that the decay of the off-diagonal elements of its density matrix is scaled by the same parameter (2) as that of the decay of its diagonal elements, i.e. there is no radical difference between the dephasing and energy-relaxation times T 2 and T 1. This fact may be interpreted as the result of the independence of the energy level distances, 0, of the fluctuations F( t) exerted on the oscillator by the environment, so their low-frequency density, SF(0), does not contribute to dephasing. (This fact formally follows also from Eq. (203) as well, taking into account that for the oscillator, xnn = xn’n’ = 0.) The simple equidistant structure of the oscillator’s spectrum makes it possible to readily solve the system of Eqs. (223), with n = 0, 1, 2, …, for some important cases. In particular, if the initial state of the oscillator is a classical mixture, with no off-diagonal elements, its further relaxation proceeds as such a mixture: wnn’( t) = 0 for all n’ n.80 In particular, it is straightforward to use Eq. (208) to verify that if the initial classical mixture obeys the Gibbs distribution (25), but with a temperature T i different from that of the environment ( T e), then the relaxation process is reduced to a simple exponential transient of the effective temperature from T i to T e:
W
0
0
2
2
exp
1 exp
,
with
1
, (7.226)
n t
t
t
n k T
B ef t
k T
B ef t
T ef t T e
T
i
e
e
with the corresponding evolution of the expectation value of the full energy E – cf. Eq. (26b):
1
E t
0
n t ,
n t
n .
(7.227)
0
2
exp / k T t 1 t
0
B
e
ef
However, if the initial state of the oscillator is different (say, corresponds to some excited Fock state), the relaxation process described by Eqs. (223)-(224), is more complex – see, e.g., Fig. 8.
At low temperatures (Fig. 8a), it may be interpreted as a gradual “roll” of the probability distribution down the energy staircase, with a gradually decreasing velocity d n/ dt n. However, at substantial temperatures, with k B T ~0, i.e. n e ~ 1 (Fig. 8b), this “roll-down” is saturated when the level occupancies Wn( t) approach their equilibrium values (25).81
The analysis of this process may be simplified in the case when W( n, t) Wn( t) is a smooth function of the energy level number n, limited to high levels: n >> 1. In this limit, we may use the Taylor expansion of this function (written for the points n = 1), truncated to three leading terms: 2
W
,
n t
1 W ,
n t
W
t W n ,
1 t W ,
n t
.
(7.228)
n 1
2
n
2
n
80 Note, however, that this is not true for many applications, in which a damped oscillator is also under the effect of an external time-dependent field, which may be described by additional, typically off-diagonal terms on the right-hand side of Eqs. (220).
81 The reader may like to have a look at the results of very nice measurements of such functions Wn( t) in microwave oscillators, performed using their coupling with Josephson-junction circuits: H. Wang et al., Phys.
Rev. Lett. 101, 240401 (2008), and with Rydberg atoms: M. Brune et al., Phys. Rev. Lett. 101, 240402 (2008).
Chapter 7
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1
1
n 0
n 0
0
T 0
k T
B
2
W
1
n 0
1
n 0.157
e
e
W
n
n
0.1
0.1
2
2
3
3
5 4
6 5
4
0.01
0.01
0
1
2
3
4
5
6
0
1
2
3
4
5
6
2 t
2 t
Fig. 7.8. Relaxation of a harmonic oscillator, initially in its 5th Fock state, at: (a) T = 0, and (b) T > 0. Note that in the latter case, even the energy levels with n > 5 get populated, due to their thermal excitation.
Plugging this expression into Eqs. (223)-(224), we get for the function W( n, t) a partial differential equation, which may be recast in the following form:
W
2
f ( n) W
d n W,
with f n 2 n n
. (7.229)
2
e
, d n 2 n ½
e
n
t
n
n
Since the energy E of an oscillator with n >> 1 is close to 0 n, this energy diffusion equation essentially describes the time evolution of the continuous probability density w( E, t) that in this case, may be defined as w( E, t) W( E/0, t)/0. In the classical limit n e >> 1, Eq. (229) is analytically solvable for arbitrary initial conditions.82 Note, however, that the most important properties of the damped harmonic oscillator (including its relaxation dynamics) may be analyzed much simpler by using the Heisenberg-Langevin approach which was discussed in Section 5.
7.9. Continuous-spectrum systems
The continuous approximation explored at the end of the last section naturally reminds us of the need to discuss dissipative systems with continuous spectra. Unfortunately, for such systems the few (relatively :-) simple results that may be obtained from the basic Eq. (181), are essentially classical in nature and are discussed in detail in the SM part of this series. Here, I will give only a simple illustration.
Let us consider a 1D particle that interacts weakly with a thermally-equilibrium environment, but otherwise is free to move along the x- axis. As we know from Chapters 2 and 5, in this case, the most convenient basis is that of the momentum eigenstates p. In the momentum representation, the density matrix is just the c-number function w( p, p’) defined by Eq. (54), which was already discussed in brief 82 See, e.g., the paper by B. Zeldovich et al., Sov. Phys. JETP 28, 308 (1969), which also gives some more intricate solutions of Eqs. (223)-(224).
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in Sec. 2. On the other hand, the coordinate operator, which participates on the right-hand side of Eq.
(181), has the form given by the first of Eqs. (4.269),
x ˆ i
,
(7.230)
p
dual to the coordinate-representation formula (4.268). As we already know, such operators are local –
see, e.g., Eq. (4.244). Due to this locality, the whole right-hand side of Eq. (181) is local as well, and hence (within the framework of our perturbative treatment) the interaction with the environment affects only the diagonal values w( p, p) of the density matrix, i.e. the momentum’s probability density w( p).
Let us find the equation governing the evolution of this function in time in the Markov approximation, when the time scale of the density matrix evolution is much longer than the correlation time c of the environment, i.e. the time scale of the functions KF() and G(). In this approximation, we may take the matrix elements out of the first integral of Eq. (181),
1 t
K
F t t' dt' ˆ x( t),ˆ x( t' ), ˆ(
w t'
1
)
K
F d ˆ x,ˆ x, ˆ
w
2
2
0
(7.231)
S
F
k T
0ˆ x,ˆ x, ˆ
w
B
ˆ, xˆ x, ˆ
w ,
2
2
and calculate the last double commutator in the Schrödinger picture. This may be done either using an explicit expression for the matrix elements of the coordinate operator or in a simpler way – using the same trick as at the derivation of the Ehrenfest theorem in Sec. 5.2. Namely, expanding an arbitrary function f( p) into the Taylor series in p,
k
1 f
k
f ( p)
p ,
(7.232)
k
0
!
k
k p
and using Eq. (240), we can write
1 k
f
k
1 k f
k
k 1
1
1
f
k
f
ˆ x, f
(7.233)
k ˆ
x, p
k i kp
1
i
p
i
k
k 0 k!
p
k 0 k!
p
k 1 k
.
1 !
1
p
p
p
Now applying this result sequentially, first to w and then to the resulting commutator, we get 2
w
w
w
ˆ x,ˆ x,
w
2
ˆ x, i
i
i
.
(7.234)
2
p
p
p
p
It may look like the second integral in Eq. (181) might be simplified similarly. However, it vanishes at p’ p, and t’ t, so in order to calculate the first non-vanishing contribution from that integral for p = p’, we have to take into account the small difference t – t’ ~ c between the arguments of the coordinate operators under that integral. This may be done using Eq. (169) with the free particle’s Hamiltonian consisting of the kinetic-energy contribution alone: 1
1
p ˆ2
p ˆ
x ˆ t' x ˆ t
x ˆ
x ˆ H ˆ
,
ˆ,
,
(7.235)
s
x
i
i
2 m
m
where the exact argument of the operator on the right-hand side is already unimportant and may be taken for t. As a result, we may use the last of Eqs. (136) to reduce the second term on the right-hand side of Eq. (181) to
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i t
i
p ˆ
p ˆ
G t t' x ˆ t
( ), x ˆ t'
( ), w ˆ t'
(
) dt'
G
d x ˆ, , ˆ
w
x ˆ, , w ˆ
. (7.236)
2
2
m
i
2
m
0
In the momentum representation, the momentum operator and the density matrix w are just c-numbers and commute, so by applying Eq. (233) to the product pw, we get
p ˆ
p
p
x ˆ, , ˆ
w
x ˆ,2 w
i
2
w
,
(7.237)
m
m
p m
and may finally reduce the integro-differential equation Eq. (181) to a partial differential equation: Fokker –
Planck
w
2 w
p
equation:
Fw k T
,
with F
.
(7.238)
free
t
p
B
p
2
m
1D particle
This is the 1D form of the famous Fokker-Planck equation describing the classical statistics of motion of a particle (in our particular case, of a free particle) in an environment providing a linear drag characterized by the coefficient ; it belongs to the same drift-diffusion type as Eq. (229). The first, drift term on its right-hand side describes the particle’s deceleration due to the drag force (137), F = – p/ m =
– v, provided by the environment. The second, diffusion term on the right-hand side of Eq. (238) describes the effect of fluctuations: the particle momentum’s random walk around its average (drift-affected, and hence time-dependent) value. The walk obeys the law similar to Eq. (85), but with the momentum-space diffusion coefficient
D k T .
(7.239)
p
B
This is the reciprocal-space version of the fundamental Einstein relation between the dissipation (friction) and fluctuations, in this classical limit represented by their thermal energy scale k B T.83
The Fokker-Planck equation (238) may be readily generalized to the 3D motion of a particle under the effect of an additional external force,84 and in this more general form is the basis for many important applications; however, due to its classical character, its discussion is also left for the SM part of this series.85
To summarize our discussion of the two alternative approaches to the analysis of quantum systems interacting with a thermally-equilibrium environment, described in the last five sections, let me emphasize again that they give different descriptions of the same phenomena, and are characterized by the same two functions G( τ) and KF( τ). Namely, in the Heisenberg-Langevin approach, we describe the 83 Note that Eq. (224), as well as the original Einstein’s relation between the diffusion coefficient D in the direct
space and temperature, may be derived much simpler by other means – for example, from the Nyquist formula (139). These issues are discussed in detail in SM Chapter 5.
84 Moreover, Eq. (223) may be generalized to the motion of a quantum particle in an additional periodic potential U(r). In this case, due to the band structure of the energy spectrum (which was discussed in Secs. 2.7 and 3.4), the coupling to the environment produces not only a continuous drift-diffusion of the probability density in the space of the quasimomentum q but also quantum transitions between different energy bands at the same q –
see, e.g., K. Likharev and A. Zorin, J. Low Temp. Phys. 59, 347 (1985).
85 See SM Secs. 5.6-5.7. Some examples of quantum effects in dissipative systems with continuous spectra, mostly for particular models of the environment, may be found, e.g., in the monographs by U. Weiss, Quantum Dissipative Systems, 2nd ed., World Scientific, 1999, and H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford U. Press, 2007, and in references therein.
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system by operators that change (fluctuate) in time, even in thermal equilibrium, while in the density-matrix approach, the system is described by deterministic probability functions, such as Wn( t) or w( p, t), which are stationary in equilibrium. In all cases when a problem may be solved analytically to the end by both methods (for example, for a harmonic oscillator), they give identical results.
7.10. Exercise problems
7.1. Calculate the density matrix of a two-level system whose Hamiltonian is described, in a certain basis, by the following matrix:
H c σ c σ c σ c σ ,
x
x
y
y
z
z
where k are the Pauli matrices and cj are c-numbers, in thermal equilibrium at temperature T.
7.2. In the usual z-basis, spell out the density matrix of a spin-½ with gyromagnetic ratio : (i) in a pure state with the spin definitely directed along the z-axis, (ii) in a pure state with the spin definitely directed along the x-axis, (iii) in thermal equilibrium at temperature T, in a magnetic field directed along the z-axis, and (iv) in thermal equilibrium at temperature T, in a magnetic field directed along the x-axis.
7.3. Calculate the Wigner function of a harmonic oscillator, with mass m and frequency 0, in thermodynamic equilibrium at temperature T. Discuss the relation between the result and the Gibbs distribution.
7.4. Calculate the Wigner function of a harmonic oscillator, with mass m and frequency 0: (i) in the ground state,
(ii) in the first excited stationary state ( n = 1),
(iii) in the Glauber state with an arbitrary dimensionless complex amplitude , and (iv) in the so-called cat state:86 a linear superposition of two Glauber states with equal and opposite values of .
In the last case, explore and interpret the behavior of the function near the origin at >>1.
7.5.* A harmonic oscillator is weakly coupled to an Ohmic environment that is in thermal equilibrium at temperature T.
(i) Use the rotating-wave approximation to write the reduced equations of motion for the Heisenberg operators of the complex amplitude of oscillations.
(ii) Calculate the expectation values of the correlators of the fluctuation force operators participating in these equations, and express them via the average number n e of thermally-induced excitations in equilibrium, given by the second of Eqs. (26b).
7.6. Calculate the average potential energy of the long-range electrostatic interaction between two similar isotropic 3D harmonic oscillators, each with the electric dipole moment d = qs, where s is the oscillator’s displacement from its equilibrium position, at arbitrary temperature T.
86 This state is frequently used to discuss the well-known Schrödinger cat paradox – see Sec. 10.1 below.
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7.7. A semi-infinite string with mass per unit length is attached to a wall and stretched with a constant force (tension) T. Calculate the spectral density of the transverse force exerted on the wall, in thermal equilibrium at temperature T.
7.8.* Calculate the low-frequency spectral density of small fluctuations of the voltage V across a Josephson junction shunted with an Ohmic conductor and biased with a dc external current I > I c.
Hint: You may use Eqs. (1.73)-(1.74) to describe the junction’s dynamics, and assume that the shunting conductor remains in thermal equilibrium.
7.9. Prove that in the interaction picture of quantum dynamics, the expectation value of an arbitrary observable A may be indeed calculated using Eq. (167).
7.10. Show that the quantum-mechanical Golden Rule (6.149) and the master equation (196) give the same results for the rate of spontaneous quantum transitions n’ n in a system with a discrete energy spectrum, which is weakly coupled to a low-temperature heat bath (with k B T << nn’).
Hint: You may establish the relation between the function ” ( nn’) that participates in Eq. (196) and the density of states n that participates in the Golden Rule, by considering the particular case of sinusoidal classical oscillations in the system of interest.
7.11. A spin-½ with gyromagnetic ratio had been placed into a constant magnetic field with magnitude B >> k B T/, and let relax into its ground state. Then the direction of the field was suddenly changed by /2 and kept constant after that. Taking into account the spin’s weak coupling to a dissipative environment:
(i) calculate the time evolution of the spin’s density matrix (in any basis you like), and (ii) calculate the time evolution of the spin vector’s expectation value S and sketch its trajectory.
~
7.12. A spin-½ with gyromagnetic ratio is placed into the magnetic field B t B B ( t) 0
with an arbitrary but relatively small time-dependent component, and is also weakly coupled to a dissipative environment in thermal equilibrium at temperature T. Derive the differential equations describing the time evolution of the expectation values of the spin’s Cartesian components.
7.13. Use the Bloch equations derived in the previous problem to analyze the magnetic resonance87 in a spin-½ which is weakly connected to a dissipative environment in thermal equilibrium.
Use the result for a semi-quantitative discussion of the environmental broadening of arbitrary quantum transitions in systems with discrete energy spectra.
Hint: You may use the same rotating field model as in Problem 5.5.
7.14. Use the Bloch equations (see the solution of Problem 12) to analyze the dynamics of spin-½ with gyromagnetic ratio under the effect of an external ac magnetic field with a relatively low 87 See the discussion in Sec. 5.2 and the solution of Problem 5.5.
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frequency and/or large amplitude Bmax (so that Bmax >> , 1/ T 1,2), assuming that the constants T 1,2
are field-independent.
7.15. Derive Eq. (220) from Eq. (222).
7.16. For a harmonic oscillator with weak Ohmic dissipation, use Eq. (220) to find the time evolution of the expectation value E of oscillator’s energy for an arbitrary initial state, and compare the result with that following from the Heisenberg-Langevin approach.
7.17. Derive Eq. (234) in an alternative way – by using an expression dual to Eq. (4.244).
Chapter 7
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Chapter 8. Multiparticle Systems
This chapter provides a brief introduction to the quantum mechanics of systems of similar particles, with special attention to the case when they are indistinguishable. For such systems, theory predicts (and experiment confirms) very specific effects even in the case of negligible explicit (“direct”) interactions between the particles, in particular, the Bose-Einstein condensation of bosons and the exchange interaction of fermions. In contrast, the last section of the chapter is devoted to quite a different topic – quantum entanglement of distinguishable systems, and attempts to use this effect for high-performance processing of information.
8.1. Distinguishable and indistinguishable particles
The importance of quantum systems of many similar particles is probably self-evident; just the very fact that most atoms include several/many electrons is sufficient to attract our attention. There are also important systems where the total number of electrons is much higher than in one atom; for example, a cubic centimeter of typical metal houses ~1023 conduction electrons that cannot be attributed to particular atoms, and have to be considered common parts of the system as the whole. Though quantum mechanics offers virtually no exact analytical results for systems of substantially interacting particles,1 it reveals very important new quantum effects even in the simplest cases when particles do not interact, and least explicitly ( directly).
If non-interacting particles are either different from each other by their nature, or physically similar but still distinguishable because of other reasons, everything is simple – at least, conceptually.
Then, as was already discussed in Sec. 6.7, a system of two particles, 1 and 2, each in a pure quantum state, may be described by a state vector
' ,
(8.1a)
1
2
Distinguish- which is a direct product of single-particle vectors, describing their states and ’ defined in different able Hilbert spaces. (Below, I will frequently use, for this direct product, the following convenient shorthand: particles
' ,
(8.1b)
in which the particle’s number is coded by the state symbol’s position.) Hence the permuted state P ˆ ' ' ,
(8.2)
where P îs the permutation operator (which is defined by Eq. (2) itself), is different from the initial one.
The permutation operator may also be used for states of systems of identical particles. In physics, the last term may be used to describe:
1 As was already noted in Sec. 7.3, for such systems of similar particles, the powerful methods discussed in the last chapter do not work well, because of the absence of a clear difference between some “system of interest” and elementary parts of its “environment”.
© K. Likharev
QM: Quantum Mechanics
(i) the “really elementary” particles like electrons, which (at least at this stage of development of physics) are considered as structure-less entities, and hence are all identical; (ii) any objects (e.g., hadrons or mesons) that may be considered as a system of “more elementary” particles (e.g., quarks and gluons), but are placed in the same internal quantum state – most simply, though not necessarily, in the ground state.2
It is important to note that identical particles still may be distinguishable – say by their clear spatial separation. Such systems of similar but distinguishable particles (or subsystems) are broadly discussed nowadays in the context of quantum computing and encryption – see Sec. 5 below. This is why it is insufficient to use the term “identical particles” if we want to say that they are genuinely indistinguishable, so I below I will use the latter term, despite it being rather unpleasant grammatically.
It turns out that for a quantitative description of systems of indistinguishable particles, we need to use, instead of direct products of the type (1), linear combinations of such direct products; in the above example, of ’ and ’.3 To see that, let us discuss the properties of the permutation operator defined by Eq. (2). Consider an observable A, and a system of eigenstates/eigenvalues of its operator: A ˆ a A a .
(8.3)
j
j
j
If the particles are indistinguishable, the observable’s expectation value should not be affected by their permutation. Hence the operators A ând P ˆ have to commute and share their eigenstates. This is why the eigenstates of the operator P âre so important: in particular, they include the eigenstates of the Hamiltonian, i.e. the stationary states of the system. Let us have a look at the action, on an elementary direct product, of the permutation operator squared:
P ˆ 2 ' P ˆ P ˆ ' P ˆ ' ' , (8.4)
i.e. 2
ˆ
P brings the state back to its original form. Since any pure state of a two-particle system may be represented as a linear combination of such products, this result does not depend on the state, and may be represented as the following operator relation:
ˆ 2
P I.ˆ
(8.5)
Now let us find the possible eigenvalues Pj of the permutation operator. Acting by both sides of Eq. (5) on any of the eigenstates j of the permutation operator, we get a very simple equation for its eigenvalues:
2 Note that from this point of view, even complex atoms or molecules, in the same internal quantum state, may be considered on the same footing as the “really elementary” particles. For example, the already mentioned recent spectacular interference experiments by R. Lopes et al., which require particle identity, were carried out with couples of 4He atoms in the same internal quantum state.
3 A very legitimate question is why, in this situation, we need to introduce the particles’ numbers to start with. A partial answer is that in this approach, it is much simpler to derive (or guess) the system’s Hamiltonians from the correspondence principle – see, e.g., Eq. (27) below. Later in this chapter, we will discuss an alternative approach (the so-called “second quantization”), in which particle numbering is avoided. While that approach is more logical, writing adequate Hamiltonians (which, in particular, would avoid spurious self-interaction of the particles) within it is more challenging – see Sec. 3 below.
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2
P 1,
(8.6)
j
with two possible solutions:
P 1.
(8.7)
j
Let us find the eigenstates of the permutation operator in the simplest case when each of the two particles can be only in one of two single-particle states – say, and ’. Evidently, none of the simple products ’ and ’, taken alone, qualifies for such an eigenstate – unless the states and ’ are identical. This is why let us try their linear combination
a ' b ' ,
(8.8)
j
giving
P ˆ P a ' b ' .
(8.9)
j
j
j
For the case Pj = +1 we have to require the states (8) and (9) to be the same, so a = b, giving the so-called symmetric eigenstate 4
1
Symmetric
,
(8.10)
' '
entangled
2
eigenstate
where the front coefficient guarantees the orthonormality of the two-particle state vectors, provided that the single-particle vectors are orthonormal. Similarly, for Pj = –1 we get a = – b, i.e. an antisymmetric eigenstate
Anti-
1
symmetric
.
(8.11)
' '
entangled
2
eigenstate
These are the simplest (two-particle, two-state) examples of entangled states, defined as multiparticle system states whose vectors cannot be factored into direct products of single-particle vectors.
So far, our math does not preclude either sign of Pj, in particular the possibility that the sign would depend on the state (i.e. on the index j). Here, however, comes a crucial fact: all indistinguishable particles fall into two groups:5
(i) bosons, particles with integer spin s, for whose states Pj = +1, and (ii) fermions, particles with half-integer spin, with Pj = –1.
This fundamental connection between the particle’s spin and parity (“statistics”) can be proved using quantum field theory.6 In non-relativistic quantum mechanics we are discussing now, it is usually considered experimental; however, our discussion of spin in Chapter 5 enables its following interpretation. In free space, the permutation of particles 1 and 2 may be viewed as a result of their pair’s common rotation by angle = about an arbitrary z-axis. As we have seen in Sec. 5.7, at the 4 As in many situations we have met before, the kets given by Eqs. (10) and (11) may be multiplied by the common factor exp{ i} with an arbitrary real phase . However, until we discuss coherent superpositions of various states , there is no good motivation for taking different from 0; that would only clutter the notation.
5 This fact is often described as two different “statistics”: the Bose-Einstein statistics of bosons and Fermi-Dirac statistics of fermions because their statistical distributions in thermal equilibrium are indeed different – see, e.g., SM Sec. 2.8. However, this difference is actually bigger: we are dealing with two different quantum mechanics.
6 Such proofs were first offered by M. Fierz in 1939 and W. Pauli in 1940, and later refined by others.
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rotation by this angle, the state vector of a particle with a definite quantum number ms acquires an extra factor exp{ ims}, where the quantum number ms may be either an integer or a half-integer. As a result, for bosons, i.e. the particles with integer s, ms can take only integer values, so exp{ ims} = 1, and the product of two such factors in the state vector ’ is equal to +1. On the contrary, for the fermions with their half-integer s, all ms are half-integer as well, so exp{ ims} = i, and the product of two such factors in the state vector ’ is equal to ( i)2 = –1.7
The most impressive corollaries of Eqs. (10) and (11) are for the case when the partial states of the two particles are the same: = ’. The corresponding Bose state + defined by Eq. (10) is possible; in particular, at sufficiently low temperatures, a set of many non-interacting Bose particles may be in the same ground state – the so-called Bose-Einstein condensate (“BEC”).8 The most fascinating feature of the condensates is that their dynamics is governed by quantum mechanical laws, which may show up in the behavior of their observables with virtually no quantum uncertainties9 – see, e.g., Eqs. (1.73)-(1.74).
On the other hand, if we take = ’ in Eq. (11), we see that the Fermi state – becomes the null state, i.e. cannot exist at all. This is the mathematical expression of Pauli’s exclusion principle:10 two indistinguishable fermions cannot be placed into the same quantum state. (As will be discussed below, this is true for systems with more than two fermions as well.) Perhaps, the key importance of this principle is obvious: if it were not valid for electrons (that are fermions), all electrons of each atom would condense in their ground (1 s- like) state, and all the usual chemistry (and biochemistry, and biology, including dear us!) would not exist. Thus, the Pauli principle makes fermions indirectly interact even if they do not interact directly, in the usual sense of the word “interaction”.
8.2. Singlets, triplets, and the exchange interaction
Now let us discuss possible approaches to quantitative analyses of identical particles, starting from a simple case of two spin-½ particles (say, electrons), whose explicit interaction with each other and the external world does not involve spin. The description of such a system may be based on factorable states with ket-vectors
o s
,
(8.12)
12
12
with the orbital state vector o 12 and the spin vector s 12 belonging to different Hilbert spaces. It is frequently convenient to use the coordinate representation of such a state, sometimes called the spinor: r , r
r , r o s
(r , r ) s .
(8.13) 2-particle
1
2
1
2
12
12
1
2
12
spinor
Unfortunately, the simple generalization of these arguments to an arbitrary quantum state runs into problems, so they cannot serve as a strict proof of the universal relation between s and Pj.
8 For a quantitative discussion of the Bose-Einstein condensation, see, e.g., SM Sec. 3.4. Examples of such condensates include superfluids like helium, Cooper-pair condensates in superconductors, and BECs of weakly interacting atoms.
9 For example, for a coherent condensate of N >> 1 particles of mass m, Heisenberg’s uncertainty relation takes the form x p = x( Nmv) /2, so its coordinate x and velocity v may be measured simultaneously with much higher precision than those of a single particle.
10 It was first formulated for electrons by Wolfgang Pauli in 1925, on the background of less general rules suggested by Gilbert Lewis (1916), Irving Langmuir (1919), Niels Bohr (1922), and Edmund Stoner (1924) for the explanation of experimental spectroscopic data.
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Since the spin-½ particles are fermions, the particle permutation has to change the spinor’s sign: ˆ
P (r , r ) s
(r , r ) s
(r , r ) s ,
(8.14)
1
2
12
2
1
21
1
2
12
i.e. to change the sign of either its orbital factor or the spin factor.
In particular, in the case of symmetric orbital factor,
(r , r ) (r , r ),
(8.15)
2
1
1
2
the spin factor has to obey the relation
s
s .
(8.16)
21
12
Let us use the ordinary z-basis (where z, in the absence of an external magnetic field, is an arbitrary spatial axis) for both spins. In this basis, the ket-vector of any two spins-½ may be represented as a linear combination of the following four basis vectors:
,
, ,
and .
(8.17)
The first two kets evidently do not satisfy Eq. (16), and cannot participate in the state. Applying to the remaining kets the same argumentation as has resulted in Eq. (11), we get
1
Singlet
s
s
(8.18)
12
.
state
2
Such an orbital-symmetric and spin-antisymmetric state is called the singlet.
The origin of this term becomes clear from the analysis of the opposite (orbital-antisymmetric and spin-symmetric) case:
(r , r ) (r , r ),
s
s .
(8.19)
2
1
1
2
12
21
For the composition of such a symmetric spin state, the first two kets of Eq. (17) are completely acceptable (with arbitrary weights), and so is an entangled spin state that is the symmetric combination of the two last kets, similar to Eq. (10):
1
s
,
(8.20)
2
so the general spin state is a triplet:
1
Triplet
s
c c c
(8.21)
12
0
.
state
2
Note that any such state (with any values of the coefficients c satisfying the normalization condition), corresponds to the same orbital wavefunction and hence the same energy. However, each of these three states has a specific value of the z-component of the net spin – evidently equal to, respectively, +, –, and 0. Because of this, even a small external magnetic field lifts their degeneracy, splitting the energy level in three; hence the term “triplet ” .
In the particular case when the particles do not interact directly, for example ˆ 2
p
ˆ
ˆ
ˆ
ˆ
H h h ,
h k ˆ u(r ),
with k ,
1 2 ,
(8.22)
1
2
k
2 m
k
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the two-particle Schrödinger equation for the symmetrical orbital wavefunction (15) is obviously satisfied by the direct products,
(r , r ) (r ) (r ),
(8.23)
1
2
n
1
n'
2
of single-particle eigenfunctions, with arbitrary sets n, n’ of quantum numbers. For the particular but very important case n = n’, this means that the eigenenergy of the (only acceptable) singlet state, 1
(r ) (r ),
(8.24)
2
n
1
n
2
is just 2 n, where n is the single-particle energy level.11 In particular, for the ground state of the system, such singlet spin state gives the lowest energy E g = 2g, while any triplet spin state (19) would require one of the particles to be in a different orbital state, i.e. in a state of higher energy, so the total energy of the system would be also higher.
Now moving to the systems in which two indistinguishable spin-½ particles do interact, let us consider, as the simplest but important12 example, the lower energy states of a neutral atom13 of helium
– more exactly, 4He. Such an atom consists of a nucleus with two protons and two neutrons, with the total electric charge q = +2 e, and two electrons “rotating” about the nucleus. Neglecting the small relativistic effects that were discussed in Sec. 6.3, the Hamiltonian describing the electron motion may be expressed as
2
2
2
ˆ pk
2 e
e
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
H h h U ,
h
,
U
.
(8.25)
1
2
int
k
int
2 m
4 r
4 r r
0 k
0
1
2
As with most problems of multiparticle quantum mechanics, the eigenvalue/eigenstate problem for this Hamiltonian does not have an exact analytical solution, so let us carry out its approximate analysis considering the electron-electron interaction U int as a perturbation. As was discussed in Chapter 6, we have to start with the “0th-order” approximation in which the perturbation is ignored, so the Hamiltonian is reduced to the sum (22). In this approximation, the ground state of the atom is the singlet (24), with the orbital factor
(r , r ) (r ) (r ) ,
(8.26)
g
1
2
100
1
100
2
and energy E g = 2g. Here each factor 100(r) is the single-particle wavefunction of the ground (1 s) state of the hydrogen-like atom with Z = 2, with quantum numbers n = 1, l = 0, and m = 0 – hence the wavefunctions’ indices. According to Eqs. (3.174) and (3.208),
r r
r
0
1
2
(r)
Y ( ,) R ( r)
/
r
0
e
, with
B
B
r
,
(8.27)
100
0
,
1 0
4
3 / 2
0
r
Z
2
0
and according to Eqs. (3.191) and (3.201), in this approximation the total ground state energy is 11 In this chapter, I try to use lowercase letters for all single-particle observables (in particular, for their energies), in order to distinguish them as clearly as possible from the system’s observables (including the total energy E of the system), which are typeset in uppercase (capital) letters.
12 Indeed, helium makes up more than 20% of all “ordinary” matter of our Universe.
13 Note that the positive ion He+1 of this atom, with just one electron, is fully described by the hydrogen-like atom theory with Z = 2, whose ground-state energy, according to Eq. (3.191), is – Z 2 E H/2 = –2 EH –55.4 eV.
Chapter 8