PHY 7669: Quantum Field Theory II
Spring Term 2013
Homework Set 3
Due Monday, March 11, 2005
Reading: Peskin&Schroeder.
Problem 1. Path integrals in quantum mechanics. In this problem we will derive carefully eq. (9.12). Consider a general quantum system with Ndof degrees of freedom, which is described by a set of generalized coordinates {q i } and conjugate momenta {pi }, i = 1, 2, 3, ..., Ndof .
Let the system be described by a Hamiltonian H(q, p), where q or p without a superscript stands
for the whole set {q i } or {pi }, correspondingly.
Consider the transition amplitude
U(qa , qb ; T ) = hqb |e−iH(q̂,p̂)T |qa i.
Notice that the Hamiltonian here is an operator, which itself depends on the operators q̂ i and
p̂i . Our goal will be to rewrite this as a path integral involving a scalar function H(q, p) (i.e.
where the q’s and p’s are numbers).
(a) First discretize the time interval into N short slices of duration ǫ ≡ T /N, and split the
exponent into N factors as
hqb |e−iĤT |qa i = hqb |e−iĤǫ e−iĤǫ ...e−iĤǫ |qa i .
Now insert a complete set of intermediate states

1=

Ndof
Y Z
i=1
dqki  |qk ihqk |
at positions k = 1, 2, 3, ..., N − 1:
hqb |
e−iĤǫ e−iĤǫ ...e−iĤǫ |qa i
≡ hqN | e−iĤǫ e−iĤǫ ...e−iĤǫ |q0 i
=

hqN | e−iĤǫ 

e−iĤǫ 
...

e−iĤǫ 
−iĤǫ
e

Ndof
Y Z
i=1
Ndof
Y Z
i=1
Ndof
Y Z
i=1
|q0 i .
i

dqN
−1 |qN −1 ihqN −1 |

i

dqN
−2 |qN −2 ihqN −2 |

dq1i  |q1 ihq1 |
(1)
1
Expand the exponents for small ǫ as e−iĤǫ = 1 − iĤǫ + ... and rewrite this as

Ndof

Ndof


Y Z
i=1
...

Y Z
i=1

i

dqN
−1 hqN |1 − iĤǫ|qN −1 i

i

dqN
−2 hqN −1 |1 − iĤǫ|qN −2 i

Ndof

Y Z
i=1
dq1i  hq2 |1 − iĤǫ|q1 ihq1 |1 − iĤǫ|q0 i .
(2)
Note that this is a product of N matrix elements, which includes only N − 1 intermediate
integrations over q’s.
(b) In order to eliminate the operator symbol, consider separately the cases of coordinate
and momentum-dependence. Let Ĥ = f (q̂). Use the eigenvalue equation q̂|qk i = qk |qk i to show
that




Ndof
NY
dof Z
i
X
qk+1 + qk 
dpk 
i
hqk+1 |f (q̂)|qk i = f
exp i
pik (qk+1
− qki )
2
2π
i=1
i=1
(c) Now consider the case of momentum dependence: Ĥ = f (p̂). Insert a complete set of
momentum eigenstates


Ndof
1=

Y Z
i=1
dpik  |pk ihpk |
and use the eigenvalue equation p̂|pk i = pk |pk i as well as
PNdof i i
1
hq|pi = √ ei i=1 q p
2π
to show that

Ndof
hqk+1 |f (p̂)|qk i = 
Y Z
i=1



Ndof
X
dpik 
i
f (pk ) exp i
pik (qk+1
− qki ) .
2π
i=1
(d) Combine the results from parts (a-c) and derive eqns. (9.11) and (9.12). Notice that
there is one momentum integration for each of the N matrix elements in (2).
2