University of California at San Diego – Department of Physics – TA: Shauna Kravec
Quantum Mechanics C (Physics 130C) Winter 2014
Worksheet 2
Please read and work on the following problems in groups of 3 to 4.
Solutions will be posted on the course webpage.
Announcements
• The 130C web site is:
http://physics.ucsd.edu/∼mcgreevy/w14/ .
Please check it regularly! It contains relevant course information!
• Hello everyone! This week we’re going to think about relating unitary to hermitian
operators. Remember my office hours are Wed. 1100-1200 in Mayer 4206.
Problems
1. A Useful Form
Let U = e−iH where H is some hermitian operator. Show that U is unitary. You may
make the assumption this series converges.
A useful identity is the Baker-Campell-Hausdorff formula:
et(A+B) = etA etB e
−t2
[A,B]
2
t3
e 6 (2[B,[A,B]]+[A,[A,B]]) ...
(1)
Which reduces to the usual form of multiplication for [A, B] = 0.
An important corollary is that any unitary operator has such a form as here!
2. Talking ’bout My Generation Let’s think about translations.
(a) Define the operator T (a) by the following:
T (a)|xi = |x + ai
Show that T (a) is unitary by considering the action of T † (a)
1
(2)
(b) Given the above we can express 2 by T (a) = e−iKa where K is hermitian.
Let |ki be an eigenbasis of K. What is the action of T (a)|ki?
Denote hx|ki = φk (x). Using the above equation the following expressions:
hx|T (a)|ki = hx|T † (−a)|ki
(3)
What does 3 imply about φk (y) where a = −y and x = 0?
(c) Recall that plane waves have momentum related to their de Broglie wavelength:
p = ~k
Rewrite T (a) with the implied expression for the operator P . This is why we say
momentum generates translations!
∂
T (a)|a=0 = − ~i P
We can use a clever trick to get P on its own: ∂a
What’s the expression for hx0 |P |xi? You will get a derivative of a delta function.
Recall that by some integration by parts:
Z
Z
0
dxδ (x − y)f (x) = 0 − dxδ(x − y)f 0 (x) = −f 0 (y)
(4)
Use 4 to derive the familiar expression for P by considering:
P ψ(x) = hx|P |ψi
2
(5)