exchange e¬ects play no role at all.

Proof We prove that

n 3N/2

nm 2πmkB T

dr n’1 exp ’ 2 (r k ’ r k’1 ) 2

C(n) dr 1 . . . = .

h2

2β

k=1

First make a coordinate transformation from r k to sk = r k ’ r 0 , k =

1, . . . , n ’ 1. Since the Jacobian of this transformation equals one, dr can

be replaced by ds. Inspection of the sum shows that the integral I now

becomes

ds1 · · · dsn’1 exp[’±{s2 +(s2 ’s1 )2 +· · ·+(sn’1 ’sn’2 )2 +s2 }],

I= 1 n’1

where

nm

±= .

2 2β

The expression between { } in s can be written in matrix notation as

sT An s,

3.3 Path integral quantum mechanics 55

with An a symmetric tridiagonal (n ’ 1) — (n ’ 1) matrix with 2 along the

diagonal, ’1 along both subdiagonals, and zero elsewhere. The integrand

becomes a product of independent Gaussians after an orthogonal transfor-

mation that diagonalizes An ; thus the integral depends only on the product

of eigenvalues, and evaluates to

π 3(n’1)N/2

(det An )’3N/2 .

I=

±

The determinant can be easily evaluated from its recurrence relation,

det An = 2 det An’1 ’ det An’2 ,

and turns out to be equal to n. Collecting all terms, and replacing C(n) by

(3.47), we ¬nd the required result. Note that the number of nodes n cancels:

the end result is valid for any n.

In special cases, notably a free particle and a particle in an isotropic har-

monic potential, analytical solutions to the partition function exist. When

the potential is not taken constant, but approximated by a Taylor expansion,

quantum corrections to classical simulations can be derived as perturbations

to a properly chosen analytical solution. These applications will be treated

in Section 3.5; here we shall derive the analytical solutions for the two special

cases mentioned above.

3.3.7 The free particle

The canonical partition function of a system of N free (non-interacting)

particles is a product of 3N independent terms and is given by

Q = lim Q(n) ,

n’∞

Q(n) = (q (n) 3N

) ,

n/2

2πmn

dx0 · · ·

(n)

q = dxn’1

h2 β

n

exp ’a (xk ’ xk’1 )2 , (3.50)

k=1

nm

a= ; xn = x0 . (3.51)

2 2β

The sum in the exponent can be written in matrix notation as

n

(xk ’ xk’1 )2 = xT Ax = yT Λy, (3.52)

k=1

56 From quantum to classical mechanics: when and how

where A is a symmetric cyclic tridiagonal matrix:

⎛ ⎞

2 ’1 0 ’1

⎜ ’1 2 ’1 0⎟

⎜ ⎟

A=⎜ ⎟ (3.53)

...

⎜ ⎟

⎝0 ’1 2 ’1 ⎠

’1 0 ’1 2

and y is a set of coordinates obtained by the orthogonal transformation

T of x that diagonalizes A to the diagonal matrix of eigenvalues Λ =

diag (»0 , . . . , »n’1 ):

y = Tx, T’1 = TT , xT Ax = yT TATT y = yT Λy. (3.54)

There is one zero eigenvalue, corresponding to an eigenvector proportional

to the sum of xk , to which the exponent is invariant. This eigenvector, which

we shall label “0,” must be separated. The eigenvector y0 is related to the

centroid or average coordinate xc :

n’1

1 def

xc = xk , (3.55)

n

k=0

√

1

y0 = √ (1, 1, . . . , 1)T = rc n. (3.56)

n

Since the transformation is orthogonal, its Jacobian equals 1 and integration

over dx can be replaced by integration over dy. Thus we obtain

n’1

n/2

2πmn

dy1 · · · dyn’1 exp ’a

(n) 1/2 2

q = n dxc » k yk .

h2 β

k=1

(3.57)

Thus the distribution of node coordinates (with respect to the centroid) is

a multivariate Gaussian distribution. Its integral equals

n’1

π (n’1)/2 ’1/2

dy1 · · · dyn’1 exp ’a Πn’1 »k

2

» k yk = . (3.58)

k=1

a

k=1

The product of non-zero eigenvalues of matrix A turns out to be equal to

n2 (valid for any n). Collecting terms we ¬nd that the partition function

equals the classical partition function for a 1D free particle for any n:

1/2

2πm

(n)

q = dxc , (3.59)

h2 β

as was already shown to be the case in (3.49).

3.3 Path integral quantum mechanics 57

Table 3.4 Intrinsic variance of a discrete imaginary-time path for a

one-dimensional free particle as a function of the number of nodes in the

path, in units of 2 β/m

σ2 σ2 σ2

n n n

2 0.062 500 8 0.082 031 50 0.083 300

3 0.074 074 9 0.082 305 60 0.083 310

4 0.078 125 10 0.082 500 70 0.083 316

5 0.080 000 20 0.083 125 80 0.083 320

6 0.081 019 30 0.083 241 90 0.083 323

∞

7 0.081 633 40 0.083 281 0.083 333