Homework #2
Please hand in by next Thursday (October 30), either during the lecture
or email to <[email protected]>
1. Campbell-Baker-Hausdorff formula: Show that
1
eX eY = eX+Y e 2 [X,Y ] + (3rd order terms).
2. Consider the free particle path integral (with m = 1 for simplicity)
Z tf
Z
1 2
q˙ dt
hqf , tf |qi , ti i = Dq(t) exp i
ti 2
(1)
(2)
Write a general path q(t) as the sum of the classical path1 qc (t) plus a Fourier
series with coefficients an , n ≥ 1.
3. Show that the action for such a general path is
∞
1 (qf − qi )2 1 X 1 (nπ)2 2
S=
+
a
2 tf − ti
2 n=1 2 tf − ti n
(3)
4. Perform the integral
Z
dak eiS
(4)
over a single Fourier mode.
5. Write the entire path integral as a constant, depending only on tf − ti , times
the classical action,
Z Y
∞
i (qf − qi )2
iS
(5)
an e = c(tf − ti ) exp
2 tf − ti
n=1
Does the constant have a finite value?
6. The actual path integral measure contains a normalization constant γ,
Z
Z Y
iS
hqf , tf |qi , ti i = Dqe = γ
dan eiS ,
(6)
such that the combination γ · c(tf − ti ) is a finite number. The requirement that
Z
dq hqf , tf |q, tihq, t|qi , ti i = hqf , tf |qi , ti i
(7)
implies a relation between γc(tf − t), γc(t − ti ), and γc(tf − ti ). Find it and
solve it. Hint: γc(τ ) ∼ τ −1/2 .
1
That is, motion at constant velocity.
1