Homework Assignment 06: Simple Harmonic Oscillator
(Due: 03/10/2016)
Problem 1 (25%)
(a) [10%] Show that for a harmonic oscillator in the state n , the following uncertainty product
holds:
ΔxΔp = (n + 1 / 2) ! .
(b) [15%] Evaluate the following matrix elements for the simple Harmonic oscillator:
m X 2 n and m P 2 n ,
where X and P are position and momentum operators, respectively.
Problem 2 (25%)
(a) [15%] Show that
†
†
⎡a,F(a,a † )⎤ = ∂F(a,a ) , ⎡a †,F(a,a † )⎤ = − ∂F(a,a ) .
⎢⎣
⎥⎦
⎢⎣
⎥⎦
∂a
∂a †
†
†
Here a and a are the lowering and raising operators, and F(a,a ) is a function of a and a † .
(b) [10%] Using (a) evaluate
e λH ae −λH and e λH a †e −λH .
Here λ is a constant.
Problem 3 (50%)
A coherent state of a one-dimensional simple harmonic oscillator is defined by an eigenstate of
the lowering operator a :
a λ =λ λ ,
where λ is in general a complex number.
(a) [12%] Prove that
λ =e
2
−λ /2 λa †
e
0
is a normalized coherent state.
(b) [8%] Write λ as
∞
λ = ∑ f (n) n .
n=0
2
Show that the distribution of f (n) with respect to n is of the Poisson form.
Find the most probably value of n, and hence of E.
(c) [10%] Show that a coherent state can also be obtained by applying the translation operator
e −iPx/ to the ground state 0 . Here P is the momentum operator, while x is the
displacement (not an operator).
(d) [20%] The displacement operator is defined as
1
2
†
D(λ) = e −|λ| /2e λa e βa ,
where β is a complex number. Show that (1) D(λ) is a unitary operator when β = −λ* ; (2) the
coherent state
λ = D(λ) 0 ;
and (3) the displacement operator can be written as
†
*
D(λ) = e λa −λ a .
2