Problem Set 7
PHY 465 - Spring 2015
Assigned: Monday, Mar. 30 Due: Friday, Apr. 6
Reading: Shankar Ch. 2.8, Ch. 8
Problems 1-2: Properties of the Classical Action
Shankar 2.8.6-7.
Problem 3-6: Path Integral Formulation of Quantum Theory
Shankar 8.6.1-4.
Problem 7: Convolution Trick for the Free Particle Propagator
The path integral formulation of quantum theory shows that the propagator for a free particle must take the form
U ( x
0
, t
0
; x, t ) = F ( t
0
− t ) exp i
S cl
( x
0
, t
0
; x, t ) where S cl
( x
0
, t
0
; x, t ) is the action of the classical trajectory connecting the points t and ending at the point x
0 at time t
0
.
x at time a) Why is the prefactor a function of t
0
− t rather than t and t
0 separately?
b) Use the convolution property
U ( x
0
, t
0
; x, t ) =
Z
+ ∞ dy U ( x
0
, t
0
; y, t
00
) U ( y, t
00
; x, t )
−∞ where t < t
00 the algebra.)
< t
0 to deduce the form of F ( t − t
0
). (Hint: setting x = x
0
= t
0
= 0 will simplify
Problem 8: Convolution Trick for the Simple Harmonic Oscillator
Repeat part b) of the previous problem for the simple harmonic oscillator.
Please write down how many hours you spent on this problem set.