temperature kT/hν

Figure 3.3 Free energy A and energy U of a 1D harmonic oscillator evaluated as

an imaginary-time path integral approximated by n nodes. The curves are labeled

with n; the broken line represents the classical approximation (n = 1); n = ∞

represents the exact quantum solution. Energies are expressed in units of hν = ω.

the energy of a 1D harmonic oscillator, evaluated by numerical solution of

(3.67) for several values of n. The approximation fairly rapidly converges to

the exact limit, but for low temperatures a large number of nodes is needed,

while the limit for T = 0 (A = U = 0.5 ω) is never reached correctly.

The values of U , as plotted in Fig. 3.3, were obtained by numerical dif-

ferentiation as U = A ’ T dA/dT . One can also obtain U = ’‚ ln q/‚β by

di¬erentiating q of (3.67), yielding

n’1

1 1 1

(n) 22

U = +β ω2 . (3.70)

β n »k

k=1

60 From quantum to classical mechanics: when and how

The ¬rst term is the internal energy of the classical oscillator while the

second term is a correction caused by the distribution of

nodes relative to the centroid. Both terms consist of two equal halves

representing kinetic and potential energy, respectively.

The intrinsic variance of the distribution of node coordinates, i.e., relative

to the centroid, is “ as in (3.60) “ given by

n’1

2β

1 1

2

σintr = . (3.71)

m n2 »k

k=1

We immediately recognize the second term of (3.70). If we add the “classi-

cal” variance of the centroid itself

1

2

σcentroid = , (3.72)

βmω 2

we obtain the total variance, which can be related to the total energy:

U

σ 2 = σcentroid + σintr =

2 2

. (3.73)

mω 2

This is compatible with U being twice the potential energy Upot , which

equals 0.5mω 2 x2 . Its value as a function of temperature is proportional

to the curve for U (case n ’ ∞) in Fig. 3.3. As is to be expected, the total

variance for n ’ ∞ equals the variance of the wave function, averaged over

the occupation of quantum states v:

∞ ∞

U qu

P (v)Ev

2 2

x = P (v) x = = , (3.74)

v

mω 2 mω 2

v=0 v=0

where P (v) is the probability of occurrence of quantum state v, because the

variance of the wave function in state v is given by

(v + 1 ) Ev

Ψ— x2 Ψ dx

2 2

x = = = . (3.75)

v v

mω 2

mω

Note that this relation between variance and energy is not only valid for a

canonical distribution, but for any distribution of occupancies.

We may summarize the results for n ’ ∞ as follows: a particle can

be considered as a distribution of imaginary-time closed paths around the

centroid of the particle. The intrinsic (i.e., with respect to the centroid)

spatial distribution for a free particle is a multivariate Gaussian with a

variance of 2 β/(12m) in each dimension. The variance (in each dimension)

of the distribution for a particle in an isotropic harmonic well (with force

3.3 Path integral quantum mechanics 61

intrinsic variance σ2 (units: h/mω)

1

0.8

h2

free particle σ2 =

12 m kT

0.6

0.4

0.2

harmonic oscillator

0.2 0.4 0.6 0.8 1

temperature kT/hω

Figure 3.4 The intrinsic variance in one dimension of the quantum imaginary-time

path distribution for the free particle and for the harmonic oscillator.

constant mω 2 ) is given by

U qu ’ U cl 1 ξ1

coth ’

2

σintr = = , ξ = β ω. (3.76)

mω 2 mω 2 2ξ

For high temperatures (small ξ) this expression goes to the free-particle

value 2 β/12m; for lower temperatures the variance is reduced because of

the quadratic potential that restrains the spreading of the paths; the low-

temperature (ground state) limit is /(2mω). Figure 3.4 shows the intrinsic

variance as a function of temperature.

3.3.9 Path integral Monte Carlo and molecular dynamics

simulation

The possibility to use imaginary-time path integrals for equilibrium quantum

simulations was recognized as early as 1962 (Fosdick, 1962) and developed in

the early eighties (Chandler and Wolynes, 1981; Ceperley and Kalos, 1981;

and others). See also the review by Berne and Thirumalai (1986). Ap-

plications include liquid neon (Thirumalai et al., 1984), hydrogen di¬usion

in metals (Gillan, 1988), electrons in fused salts (Parrinello and Rahman,

1984, using a molecular dynamics variant), hydrogen atoms and muonium

in water (de Raedt et al., 1984), and liquid water (Kuharski and Rossky,

1985; Wallqvist and Berne, 1985).

62 From quantum to classical mechanics: when and how

If the potential is not approximated, but evaluated for every section of

the path, the expression for Q becomes

3N/2 3(n’1)N/2

2πmkB T 2πmkB T

n3nN/2

Q= dr lim

h2 h2

n’∞

n

nm V (r k )

— dr 1 · · · dr n’1 exp ’β (r k ’ r k’1 )2 + , (3.77)

2 2β2 n

k=1

with r = r n = r 0 . The constant after the limit symbol exactly equals the