Download Homework Set 3

Physics 411
Homework Set No. 3
1.
Consider the matrix σ x defined by:
⎛0 1⎞
σx = ⎜
⎟ .
⎝1 0⎠
Prove the relation:
eiασ x = I cos α + iσ x sin α ,
where I is the 2x2 unit matrix.
Note: This problem was fully set-up in class.
2.
Prove the theorems for unitary operators stated in Le Bellac, page 52, section 2.3.4.
Note: the proofs of a) and b) are quite simple, and are very similar to the proofs given
in class for the case of Hermitian operators. Part c) is actually worked out in the text!
It is important to note the final result, namely, that a unitary operator Û can always
be written in the form
ˆ
Uˆ = e iα A ,
where  is a Hermitian operator.
(This result is quite useful. Indeed, we will encounter several important examples of
unitary operators which perform physical transformations on quantum mechanical
systems, such as rotations, time evolution, etc.; all those operators are given by
exponential operators —i.e., are of the form given above.)
3.
Le Bellac, page 58, problem 2.4.11.
Note: In setting up Eq. (2.54) you are allowed to just expand to a few orders in t; and
in giving the proof for the operator identity stated below it, you are allowed to guess
the form of the whole series (leading to sines and cosines); that is, you are not
required to do a rigorous demonstration by mathematical induction. As for part 2,
you should make explicit use of part 1! Note that the series terminates after the
second term!