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 it 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 corrections67 to the unperturbed frequencies (186) of their oscillations, and are not important for most applications. More important is the first term, proportional to
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'
66 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.
67 Such corrections are sometimes called Lamb shifts, due to their conceptual similarity to the genuine Lamb shift
– the effect first observed experimentally in 1947 by Willis Lamb and Robert Retherford: a minor difference between energy levels of the 2 s and 2 p states of hydrogen, due to the electric-dipole coupling of hydrogen atoms to the free-space electromagnetic environment. (These energies are equal not only in the non-relativistic theory described in Sec. 3.6 but also in the relativistic theory (see Secs. 6.3, 9.7), if the electromagnetic environment is ignored.) The explanation of 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.
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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
n m
n' m
x
x
nn
n'n' 2 S F 0
Dephasing
T
2
2
rate
nn'
m n
m n'
(7.203)
1
k T
B
n m
n' m
x
x
nn
n'n' 2 ,
for n n' ,
2
2
m n
m n'
where the low-frequency drag coefficient is again defined as lim0 ” ()/ – see Eq. (138).
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. (195), this contribution is proportional to the strength of coupling to the environment at relatively high frequencies nm and n’m. (If the energy quanta of these frequencies are much larger than the thermal fluctuation scale k B T, then only the lower levels, with Em < max[ En, En’] are important.) 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.
For example, returning for a minute to the two-level system discussed in Sec. 3, described by our current theory with the replacement x ˆ ˆ , the high-frequency contributions to dephasing vanish z
because of the absence of transitions between energy levels, while the low-frequency contribution yields 1
1
k T
2
k T
2
4 k T
B
x x
(7.204)
nn
n'n'
B
σ z σ z
2
2
11
22
B
,
2
T
T
2
12
thus exactly reproducing the result (142) of the Heisenberg-Langevin approach.68 Note also that the expression for T 2 is very close in structure to Eq. (199) for T 1 (in the high-temperature limit). However, for the simple interaction model (70) that was explored in Sec. 3, the off-diagonal elements of the operator x ˆ în the stationary-state z-basis vanish, so that T
z
1 , while T 2 says finite. The physics
of this result is very clear, for example, for the two-well implementation of the model (see Fig. 4 and its discussion): it is suitable for the case of a very high energy barrier between the wells, which inhibits tunneling, and hence any change of the well occupancies. However, T 1 may become finite, and comparable with T 2, if tunneling between the wells is substantial.69
68 The first form of Eq. (203), as well as the analysis of Sec. 3, implies that low-frequency fluctuations of any other origin, not taken into account in own current 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.
69 As was discussed in Sec. 5.1, the tunneling may be described by using, instead of Eq. (70), the full two-level Hamiltonian (5.3). Let me leave for the reader’s exercise to spell out the equations for the time evolution of the density matrix elements of this system, and of the expectation values of the Pauli operators, for this case.
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Because of the reason explained above, the derivation of Eqs. (200)-(204) is 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 repeat the above 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
n 1 n n'
n n n' 2 w
e
e
nn'
w
.
(7.205)
nn'
2 n 1 n 1 n' 1 w
2 n nn'
w
e
1/2
1/ 2
n ,
1 n' 1
e
n ,
1 n' 1
Here is the effective damping coefficient,70
2
x
Im
0
Im
,
(7.206)
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.)
As a remark: the derivation of Eq. (205) might be started at a bit earlier point, from the Markov approximation applied to Eq. (181), expressing the coordinate operator via the creation-annihilation operators (5.65). This procedure gives the result in the operator (i.e. basis-independent) form:71
†
†
†
†
w ˆ n
.
(7.207)
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. (205); however, Eq. (207) may be more convenient for some applications.
Returning to Eq. (205), we see that it 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):
70 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}.
71 Sometimes Eq. (207) 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. (207) 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.
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W
W
W
W
W ,
(7.208)
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
(7.209)
2 n n ,
2 n n
n 1
n
e
n n 1
e 1.
Since according to the definition of n e, given by Eq. (26b),
1
exp / k T
0
B
1
n
n
, (7.210)
e
exp / k T
k T
k T
0
,
that
so
1
1
e
B
exp
/
0
B
1
exp
/
0
B
1
taking into account Eqs. (5.92), (186), (206), 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 that their low-frequency density, SF(0), does not contribute to the 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. (208), 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.72 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
, (7.211)
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 .
(7.212)
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 upper Fock state), the relaxation process, described by Eqs. (208)-(209), 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 dn/ dt n. However, at substantial temperatures, 72 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. (205).
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with k B T ~0 (Fig. 8b), this “roll-down” is saturated when the level occupancies Wn( t) approach their equilibrium values (25).73
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.
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
.
(7.213)
n 1
2
n
2
n
Plugging this expression into Eqs. (208)-(209), 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
. (7.214)
2
e
, d n 2 n ½
e
n
t
n
n
Since at n >> 1, the oscillator’s energy E is close to 0 n, this energy diffusion equation (sometimes incorrectly called the Fokker-Planck equation – see below) essentially describes the time evolution of the continuous probability density w( E, t), which may be defined as w( E, t) W( E/0, t)/0.74
73 The reader may like to have a look at the results of 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).
74 In the classical limit n e >> 1, Eq. (214) is analytically solvable for any initial conditions – 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.
(208)-(209). Note, however, that the most important properties of the damped harmonic oscillator (including its relaxation dynamics) may be analyzed simpler by using the Heisenberg-Langevin approach discussed in the previous section.
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This continuous approximation naturally reminds us of the need to discuss dissipative systems with a continuous spectrum. 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 in Sec. 2. On the other hand, the coordinate operator, which participates in the right-hand side of Eq. (181), has the form given by the first of Eqs. (4.269),
x ˆ i
,
(7.215)
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 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
(7.216)
S
F
k T
0ˆ x,ˆ x, ˆ
w
B
ˆ x,ˆ 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 ,
(7.217)
k
0 !
k
k p
and using Eq. (215), we can prove the following simple commutation relation:
1 k
f
k
1 k f
k
k 1
1
1
f
k
f
ˆ x, f
(7.218)
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
.
(7.219)
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 that 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 Chapter 7
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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,
,
(7.220)
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
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
. (7.221)
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 that, applying Eq. (218) to the product pw, we get
p ˆ
p
p
x ˆ, , ˆ
w
x ˆ,2 w
i
2
w
,
(7.222)
m
m
p m
and may finally reduce the integro-differential equation Eq. (181) to a partial differential equation: Fokker –
w
2 w
p
Planck
Fw k T
,
with F
.
(7.223) equation:
t
p
B
p
2
m
free
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. (214). 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. (223) describes the effect of fluctuations: the particle’s momentum’ 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 .
(7.224)
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.75
Just for the reader’s reference, let me note that the Fokker-Planck equation (223) may be readily generalized to the 3D motion of a particle under the effect of an additional external force,76 and in this 75 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.
76 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).
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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.77
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 three 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 system by operators that change (fluctuate) in time, even in the thermal equilibrium, while in the density-matrix approach, the system is described by non-fluctuating probability functions, such as Wn( t) or w( p, t), which are stationary in equilibrium. In the cases when a problem may be solved analytically to the end by both methods (for example, for a harmonic oscillator), they give identical results.
7.8. 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 thermodynamic 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 the pure state with the spin definitely directed along the z-axis,
(ii) in the 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 in:
(i) in thermodynamic equilibrium at temperature T,
(ii) in the ground state, and
(ii) in the Glauber state with dimensionless complex amplitude .
Discuss the relation between the first of the results and the Gibbs distribution.
7.4. Calculate the Wigner function of a harmonic oscillator, with mass m and frequency 0, in its first excited stationary state ( n = 1).
7.5.* A harmonic oscillator is weakly coupled to an Ohmic environment.
(i) Use the rotating-wave approximation to write the reduced equations of motion for the Heisenberg operators of the complex amplitude of oscillations.
77 See SM Secs. 5.6-5.7. For a more detailed discussion of quantum effects in dissipative systems with continuous spectra see, e. g., either U. Weiss, Quantum Dissipative Systems, 2nd ed., World Scientific, 1999, or H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford U. Press, 2007.
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(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 of thermally-induced excitations in equilibrium, given by the second of Eqs. (26b).
7.6. Calculate the average potential energy of 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.
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, weakly coupled to a low-temperature heat bath (with k B T << nn’).
Hint: You may start by establishing a relation between the function ” ( nn’), which participates in Eq. (196), and the density of states n, which participates in the Golden Rule formula, using the particular case of sinusoidal classical oscillations in the system of interest.
7.11. For a harmonic oscillator with weak Ohmic dissipation, use Eqs. (208)-(209) 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.
7.12. Derive Eq. (219) in an alternative way, using an expression dual to Eq. (4.244).
7.13. A particle in a system of two coupled potential wells (see, e.g., Fig. 7.4 in the lecture notes) is weakly coupled to an Ohmic environment.
(i) Derive equations describing the time evolution of the density matrix elements.
(ii) Solve these equations in the low-temperature limit, when the energy level splitting is much larger than k B T, to calculate the time evolution of the probability of finding the particle in one of the wells, after it had been placed there at t = 0.
~
7.14.* 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. Derive differential equations describing the time evolution of the expectation values of spin’s Cartesian components, at arbitrary temperature.
Chapter 7
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Chapter 8. Multiparticle Systems
This chapter provides a brief introduction to 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. These effects notably include the Bose-Einstein condensation of bosons and the exchange interaction of fermions.
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 a typical metal houses ~1023 conduction electrons that cannot be attributed to particular atoms, and have to be considered as 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 which is a direct product,
' ,
(8.1a)
1
2
Distinguish- of single-particle vectors, describing their states and ’ defined in different Hilbert spaces. (Below, I able will frequently use, for such 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 ˆ
P ' ' ' ,
(8.2)
1
2
where P îs the permutation operator defined by Eq. (2), is clearly different from the initial one.
This operator may be also used for states of systems of identical particles. In physics, the last term may be used to describe:
(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;
1 As was emphasized in Sec. 7.3, for such systems of similar particles the powerful methods discussed in the last chapter, based on the separation of the whole Universe into a “system of our interest” and its “environment”, typically do not work well – mostly because the quantum state of the “particle of interest” may be substantially correlated (in particular, entangled) with those of similar particles forming its “environment” – see below.
© K. Likharev
QM: Quantum Mechanics
(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 products, for example of
’ and ’.3 To see this, let us discuss the properties of the permutation operator defined by Eq. (2).
Consider an observable A, and a system of eigenstates 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 a system of indistinguishable particles.
Let us have a look at the action of the permutation operator squared, on an elementary ket-vector product:
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 eigenstates j of the permutation operator, we get a very simple equation for its eigenvalues: 2
P 1,
(8.6)
j
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 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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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 component particles can be only in one of two single-particle states – say, and ’. Evidently, none of the simple products ’ and ’, taken alone, does qualify for the eigenstate – unless the states and
’ are identical. This is why let us try their linear combination
a ' b ' ,
(8.8)
j
so that
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 that 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 a direct product (1) 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 in 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.
In the non-relativistic theory we are discussing now, this key fact should be considered as an experimental one. (The relativistic quantum theory, whose elements will be discussed in Chapter 9, offers proof that the half-integer-spin particles cannot be bosons and the integer-spin ones cannot be fermions.) However, our discussion of spin in Sec. 5.7 enables the following handwaving interpretation of the difference between these two particle species. In the free space, the permutation of particles 1 and 2 may be viewed as a result of their pair’s common rotation by angle = about a properly selected z-
4 As in many situations we have met earlier, the kets given by Eqs. (10) and (11) may be multiplied by exp{ i}
with an arbitrary real phase . However, until we discuss coherent superpositions of various states , there is no good motivation for taking the phase different from 0; that would only clutter the notation.
5 Sometimes this fact is described as having 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 deeper: we are dealing with two different quantum mechanics.
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axis. As we have seen in Sec. 5.7, at the rotation by this angle, the state vector of a particle with a definite quantum number ms acquires an extra factor exp{ ims}. As we know, the quantum number ms ranges from – s to + s, in unit steps. As a result, for bosons, with integer s, ms can take only integer values, so that exp{ ims} = 1, so that the product of two such factors in the state product ’ is equal to +1. On the contrary, for the fermions with their half-integer s, all ms are half-integer as well, so that exp{ ims} = i so that the product of two such factors in vector ’ is equal to ( i)2 = –1.
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 non-interacting Bose particles condenses on the ground state – the so-called Bose-Einstein condensate (“BEC”).6 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 uncertainties7 – see, e.g., Eqs. (1.73)-(1.74).
On the other hand, if we take = ’ in Eq. (11), we see that state - becomes the null-state, i.e.
cannot exist at all. This is the mathematical expression of the Pauli exclusion principle:8 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.) Probably, the key importance of this principle is self-evident: if it was not valid for electrons (that are fermions), all electrons of each atom would condense on in their ground (1 s- like) state, and all the usual chemistry (and biochemistry, and biology, including dear us!) would not exist. The Pauli principle makes fermions implicitly interacting even if they do not interact directly, i.e. in the usual sense of this word.
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
Since the spin-½ particles are fermions, the particle permutation has to change the sign: 6 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.
7 For example, for a coherent condensate of N >> 1 particles, Heisenberg’s uncertainty relation takes the form
x p = x( Nmv) /2, so that its coordinate x and velocity v may be measured simultaneously with much higher precision than those of a single particle.
8 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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ˆ
P (r , r ) s
(r , r ) s
(r , r ) s ,
(8.14)
1
2
12
2
1
21
1
2
12
of either the 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 that 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 at all, 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.9 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 that 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 their simplest but important10 example, the lower energy states of a neutral atom11 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 that 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 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 9 In this chapter, I try to use lower-case 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 capital letters.
10 Indeed, helium makes up more than 20% of all “ordinary” matter of our Universe.
11 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.
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2
Z E
(0)
E
2 (0)
2
0
2
H
4
E
eV.
109
(8.28)
g
g
2 2
n
2
H
n ,
1 Z 2
Z2
This is still somewhat far (though not terribly far!) from the experimental value E g –78.8 eV – see the bottom level in Fig. 1a.
(a)
(b)
Δ E
singlet state
( s )
(eV)
(“parahelium”)
“parahelium” “orthohelium”
E ex
0
3 p 3 d
3 s
3 s 3 p 3 d
2 p
2 p
E
2 s
ex
triplet state
2 s
-5
( s )
(“orthohelium”)
-20
E dir
B 0
1 s (ground state)
-25
l
l
100
nlm
Fig. 8.1. The lower energy levels of a helium atom: (a) experimental data and (b) a schematic structure of an excited state in the first order of the perturbation theory. On panel (a), all energies are referred to that (-2 E H –55.4 eV) of the ground state of the positive ion He+1, so that their magnitudes are the (readily measurable) energies of the atom’s single ionization starting from the corresponding state of the neutral atom. Note that the “spin direction” nomenclature on panel (b) is rather crude: it does not reflect the difference between the entangled states s+ and s-.
Making a minor (but very useful) detour from our main topic, let us note that we can get a much better agreement with experiment by calculating the electron interaction energy in the 1st order of the perturbation theory. Indeed, in application to our system, Eq. (6.14) reads
(1)
ˆ
E
g U g
3
3
*
d r d r (r , r ) U (r , r ) (r , r ).
(8.29)
g
int
1 2 g 1 2 int 1 2 g 1 2
Plugging in Eqs. (25)-(27), we get
2
e
r r
(1)
1 4
2
3
3
2(
)
E
d r d r
exp
1
2
.
(8.30)
g
4 3
1
2
r
4 r
r
r
0
0 1
2
0
As may be readily evaluated analytically (this exercise is left for the reader), this expression equals (5/4) E H, so that the corrected ground state energy,
(0)
(1)
E E
E
E
,
(8.31)
g
g
g
4 5/ 4
74. eV
8
H
is much closer to experiment.
There is still room here for a ready improvement, using the variational method discussed in Sec.
2.9. For our particular case of the 4He atom, we may try to use, as the trial state, the orbital wavefunction given by Eqs. (26)-(27), but with the atomic number Z considered as an adjustable Chapter 8
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parameter Z ef < Z = 2 rather than a fixed number. The physics behind this approach is that the electric charge density (r) = – e(r)2 of each electron forms a negatively charged “cloud” that reduces the effective charge of the nucleus, as seen by the other electron, to Z ef e, with some Z ef < 2. As a result, the single-particle wavefunction spreads further in space (with the scale r 0 = r B/ Z ef > r B/ Z), while keeping its functional form (27) nearly intact. Since the kinetic energy T in the system’s Hamiltonian (25) is proportional to r -2
2
-1
1
0 Z ef , while the potential energy is proportional to r 0 Z ef , we can write 2
Z
Z
ef
ef
E ( Z )
T
U
.
(8.32)
g
ef
g
g
Z 2
Z 2
2
2
Now we can use the fact that according to Eq. (3.212), for any stationary state of a hydrogen-like atom (just as for the classical circular motion in the Coulomb potential), U = 2 E, and hence T = E –
U = – E. Using Eq. (30), and adding the correction (31) to the potential energy, we get 2
Z
Z
ef
5
E ( Z ) 4
8
ef
E .
(8.33)
g
ef
2
4 2
H
This expression allows an elementary calculation of the optimal value of Z ef, and the corresponding minimum of the function E g( Z ef):
5
( Z )
21
,
6875
.
1
E
E
.
(8.34)
ef opt
g
85
.
2
eV
5
.
77
32
H
min
Given the trial state’s crudeness, this number is in surprisingly good agreement with the experimental value cited above, with a difference of the order of 1%.
Now let us return to the main topic of this section – the effects of particle (in this case, electron) indistinguishability. As we have just seen, the ground-level energy of the helium atom is not affected directly by this fact, but the situation is different for its excited states – even the lowest ones. The reasonably good precision of the perturbation theory, which we have seen for the ground state, tells us that we can base our analysis of wavefunctions (e) of the lowest excited state orbitals, on products like
100(r k) nlm(r k’), with n > 1. To satisfy the fermion permutation rule, Pj = –1, we have to take the orbital factor of the state in either the symmetric or the antisymmetric form:
Orthohelium
1
and
(r , r )
r
r
r
r
,
(8.35) parahelium:
e
1
2
( )
( )
( )
( )
100
1
nlm
2
nlm
1
100
2
2
orbital
wavefunctions
with the proper total permutation asymmetry provided by the corresponding spin factor (18) or (21), so that the upper/lower sign in Eq. (35) corresponds to the singlet/triplet spin state. Let us calculate the expectation values of the total energy of the system in the first order of the perturbation theory. Plugging Eq. (35) into the 0th-order expression
(0)
3
3
*
ˆ
ˆ
E
d r d r r , r h
h r , r ,
(8.36)
e
1
2
e 1
2 1
2
e 1
2
we get two groups of similar terms that differ only by the particle index. We can merge the terms of each pair by changing the notation as (r1 r, r2 r ’ ) in one of them, and (r1 r ’, r2 r) in the counterpart term. Using Eq. (25), and the mutual orthogonality of the wavefunctions 100(r) and nlm(r), we get the following result:
Chapter 8
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2
2
(0)
*
2 2
2
2
e
e
r
3
*
2 2
E
(r)
(r) d r ( '
r )
'
r
( '
r ) 3
d r'
e
100
2 m
4
100
nlm
r
2 m
4
r'
nlm
(8.37)
0
0
,
with n .
1
100
nlm
It may be interpreted as the sum of eigenenergies of two separate single particles, one in the ground state 100, and another in the excited state nlm – although actually the electron states are entangled. Thus, in the 0th order of the perturbation theory, the electron entanglement does not affect their energy.
However, the potential energy of the system also includes the interaction term U int, which does not allow such separation. Indeed, in the 1st approximation of the perturbation theory, the total energy E e of the system may be expressed as
(1)
100 + nlm + E int
, with
)
1
(
3
3
*
E
U
d r d r (r , r ) U (r , r ) (r , r ) , (8.38)
int
int
1 2 e 1 2 int 1 2 e 1 2
Plugging Eq. (35) into this result, using the symmetry of the function U int with respect to the particle number permutation, and the same particle coordinate re-numbering as above, we get
)
1
(
E
E E ,
(8.39)
int
dir
ex
with the following, deceivingly similar expressions for the two components of this sum/difference: Direct
interaction
3
3
*
E
d r d r' (r) *
( '
r ) U (r, '
r )
(r)
( '
r ),
(8.40)
dir
energy
100
nlm
int
100
nlm
Exchange
interaction
3
3
*
E d r d r'
(r) *
( '
r ) U (r, '
r )
(r)
( '
r ).
(8.41)
ex
energy
100
nlm
int
nlm
100
Since the single-particle orbitals can be always made real, both components are positive – or at least non-negative. However, their physics and magnitude are different. The integral (40), called the direct interaction energy, allows a simple semi-classical interpretation as the Coulomb energy of interacting electrons, each distributed in space with the electric charge density (r) = – e*(r)(r):12
r
'
r
3
3
( )
( )
100
E d r d r'
nlm
(r) (r) 3
d r (r) (r) 3
d r,
(8.42)
dir
4
100
nlm
nlm
100
r '
r
0
where (r) are the electrostatic potentials created by the electron “charge clouds”:13
1
3
( '
r )
1
100
3
( '
r )
r
( )
d r'
,
r
( )
d r' nlm
.
(8.43)
100
4
r '
nlm
r
4
r '
r
0
0
However, the integral (41), called the exchange interaction energy, evades a classical interpretation, and (as it is clear from its derivation) is the direct corollary of electrons’
indistinguishability. The magnitude of E ex is also very much different from E dir because the function under the integral (41) disappears in the regions where the single-particle wavefunctions 100(r) and
nlm(r) do not overlap. This is in full agreement with the discussion in Sec. 1: if two particles are identical but well separated, i.e. their wavefunctions do not overlap, the exchange interaction disappears, 12 See, e.g., EM Sec. 1.3, in particular Eq. (1.54).
13 Note that the result for E dir correctly reflects the basic fact that a charged particle does not interact with itself, even if its wavefunction is quantum-mechanically spread over a finite space volume. Unfortunately, this is not true for some popular approximate theories of multiparticle systems – see Sec. 4 below.
Chapter 8
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i.e. measurable effects of particle indistinguishability vanish. (In contrast, the integral (40) decreases with the growing electron separation only slowly, due to the long-range Coulomb interaction.) Figure 1b shows the structure of an excited energy level, with certain quantum numbers n > 1, l, and m, given by Eqs. (39)-(41). The upper, so-called parahelium 14 level, with the energy E
E E
(8.44)
para
100 nlm
,
dir
ex
100
nlm
corresponds to the symmetric orbital state and hence to the singlet spin state (18), while the lower, orthohelium level, with
E
E E E
(8.45)
orth
100 nlm
,
dir
ex
para
corresponds to the degenerate triplet spin state (21).
This degeneracy may be lifted by an external magnetic field, whose effect on the electron spins15
is described by the following evident generalization of the Pauli Hamiltonian (4.163),
e
ˆ
H
ˆs B ˆ
ˆ
B
s B
S B ,
with
2
,
(8.46)
field
1
2
e
m e
where
ˆS ˆs ˆs ,
(8.47)
1
2
is the operator of the (vector) sum of the system of two spins.16 To analyze this effect, we need first to make one more detour, to address the general issue of spin addition. The main rule17 here is that in a full analogy with the net spin of a single particle, defined by Eq. (5.170), the net spin operator (47) of any system of two spins, and its component S âlong the (arbitrarily selected) z-axis, obey the same z
commutation relations (5.168) as the component operators, and hence have the properties similar to those expressed by Eqs. (5.169) and (5.175):
S ˆ 2 S, M
2
S
ˆ
1 ,
,
,
,
,
with
, (8.48)
S
S S M
S S M
M S M
S
M
S
S
z
S
S
S
S
where the ket vectors correspond to the coupled basis of joint eigenstates of the operators of S 2 and Sz (but not necessarily all component operators – see again the Venn shown in Fig. 5.12 and its discussion, with the replacements S, L s1,2 and J S). Repeating the discussion of Sec. 5.7 with these replacements, we see that in both coupled and uncoupled bases, the net magnetic number MS is simply expressed via those of the components
14 This terminology reflects the historic fact that the observation of two different hydrogen-like spectra, corresponding to the opposite signs in Eq. (39), was first taken as evidence for two different species of 4He, which were called, respectively, the “orthohelium” and the “parahelium”.
15 As we know from Sec. 6.4, the field also affects the orbital motion of the electrons, so that the simple analysis based on Eq. (46) is strictly valid only for the s excited state ( l = 0, and hence m = 0). However, the orbital effects of a weak magnetic field do not affect the triplet level splitting we are analyzing now.
16 Note that similarly to Eqs. (22) and (25), here the uppercase notation of the component spins is replaced with their lowercase notation, to avoid any possibility of their confusion with the total spin of the system.
17 Since we already know that the spin of a particle is physically nothing more than a (specific) part of its angular momentum, the similarity of the properties (48) of the sum (47) of spins of different particles to those of the sum (5.170) of different spin components of the same particle it very natural, but still has to be considered as a new fact – confirmed by a vast body of experimental data.
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M m
m
.
(8.49)
S
s
1
s 2
However, the net spin quantum number S (in contrast to the Nature-given spins s 1,2 of its elementary components) is not universally definite, and we may immediately say only that it has to obey the following analog of the relation l – s j ( l + s) discussed in Sec. 5.7: s s S s s .
(8.50)
1
2
1
2
What exactly S is (within these limits), depends on the spin state of the system.
For the simplest case of two spin-½ components, each with s = ½ and ms = ½, Eq. (49) gives three possible values of MS, equal to 0 and 1, while Eq. (50) limits the possible values of S to just either 0 or 1. Using the last of Eqs. (48), we see that the possible combinations of the quantum numbers are
S ,
0
S ,
1
and
(8.51)
M ,
0
M
S
,
0 1.
S
It is virtually evident that the singlet spin state s– belongs to the first class, while the simple (separable) triplet states and belong to the second class, with MS = +1 and MS = –1, respectively. However, for the entangled triplet state s+, evidently with MS = 0, the value of S is less obvious. Perhaps the easiest way to recover it18 to use the “rectangular diagram”, similar to that shown in Fig. 5.14, but redrawn for our case of two spins, i.e., with the replacements ml ( ms)1 = ½, ms ( ms)2 = ½ – see Fig. 2.
ms
2
½
S 1
Fig. 8.2. The “rectangular diagram”
s
showing the relation between the
S 0, 1
uncoupled-representation states (dots)
½
0
½ ms 1
and the coupled-representation states
(straight lines) of a system of two spins-
½ – cf. Fig. 5.14.
½
S 1
Just as at the addition of various angular momenta of a single particle, the top-right and bottom-left corners of this diagram correspond to the factorable triplet states and , which participate in both the uncoupled-representation and coupled-representation bases, and have the largest value of S, i.e.
1. However, the entangled states s, which are linear combinations of the uncoupled-representation states and , cannot have the same value of S, so that for the triplet state s+, S has to take the value different from that (0) of the singlet state, i.e. 1. With that, the first of Eqs. (48) gives the following expectation values for the square of the net spin operator:
2 2
,
state,
et
each tripl
for
2
S
(8.52)
0,
state.
singlet
for the
18 Another, a bit longer but perhaps a more prudent way is to directly calculate the expectation values of 2
ˆ S for
the states s, and then find S by comparing the results with the first of Eqs. (48); it is highly recommended to the reader as a useful exercise.
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Note that for the entangled triplet state s+, whose ket-vector (20) is a linear superposition of two kets of states with opposite spins, this result is highly counter-intuitive, and shows how careful we should be interpreting entangled quantum states. (As will be discussed in Chapter 10, the entanglement brings even more surprises for quantum measurements.)
Now we may return to the particular issue of the magnetic field effect on the triplet state of the 4He atom. Directing the z-axis along the field, we may reduce Eq. (46) to
S ˆ
ˆ
ˆ
z
H
S
.
(8.53)
field
e
B
2
z
BB
Since all three triplet states (21) are eigenstates, in particular, of the operator S ˆ , and hence of the z
Hamiltonian (53), we may use the second of Eqs. (48) to calculate their energy change simply as
,
1
state
triplet
factorable
for the
,
E
2
M
B
(8.54)
S
2 B
s
field
B
B
,
0
state
triplet
entangled
for the
,
,1
state
triplet
factorable
for the
.
This splitting of the “orthohelium” level is schematically shown in Fig. 1b.19
8.3. Multiparticle systems
Leaving several other problems on two-particle systems for the reader’s exercise, let me proceed to the discussion of systems with N > 2 indistinguishable particles, whose list notably includes atoms, molecules, and condensed-matter systems. In this case, Eq. (7) for fermions is generalized as ˆ
P ,
k
all
for
, k' ,
1 ,...,
2
N ,
(8.55)
kk '
where the operator ˆ
P permutes particles with numbers k and k’. As a result, for systems with non-kk '
directly-interacting fermions, the Pauli principle forbids any state in which any two particles have similar single-particle wavefunctions. Nevertheless, it permits two fermions to have similar orbital wavefunctions, provided that their spins are in the singlet state (18), because this satisfies the permutation requirement (55). This fact is of paramount importance for the ground state of the systems whose Hamiltonians do not depend on spin because it allows the fermions to be in their orbital single-particle ground states, with two electrons of the spin singlet sharing the same orbital state. Hence, for the limited (but very important!) goal of finding ground-state energies of multi-fermion systems with negligible direct interaction, we may ignore the actual singlet spin structure, and reduce the Pauli 19 It is interesting that another very important two-electron system, the hydrogen (H2) molecule, which was briefly discussed in Sec. 2.6, also has two similarly named forms, parahydrogen and orthohydrogen. However, their difference is due to two possible (respectively, singlet and triplet) states of the system of two spins of the two hydrogen nuclei – protons, which are also spin-½ particles. The resulting energy of the parahydrogen is lower than that of the orthohydrogen by only ~45 meV per molecule – the difference comparable with k B T at room temperature (~26 meV). As a result, at the ambient conditions, the equilibrium ratio of these two spin isomers is close to 3:1. Curiously, the theoretical prediction of this minor effect by W. Heisenberg (together with F. Hund) in 1927 was cited in his 1932 Nobel Prize award as the most noteworthy application of quantum theory.
Chapter 8
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exclusion principle to the simple picture of single-particle orbital energy levels, each “occupied with two fermions”.
As a very simple example, let us find the ground energy of five fermions, confined in a hard-wall, cubic-shaped 3D volume of side a, ignoring their direct interaction. From Sec. 1.7, we know the single-particle energy spectrum of the system:
n n n
n n n
(8.56)
n , n , n
0 2
2
2
x
y
z
2
2
,
with
,
and
,
,
,
1 ,...
2
0
x
y
z
2
2
x
y
z
ma
so that the lowest-energy states are:
- one ground state with { nx,ny,nz} = {1,1,1}, and energy 111= (12+12+12)0 = 30, and