Quantum Theory 1 - Home Exercise 8

Quantum Theory 1 - Home Exercise 8
1. Consider some physical system that is represented by a Hilbert space with the set of normalized
energy eigenstates {|ni,
n = 0, 1, 2...} such that Ĥ|ni = En |ni.
We define the projector operator on the state |ni by P̂n = |nihn| :
(a) Is P̂n Hermitian?
(b) What is the representative matrix of P̂n in this basis?
(c) What are the eigenvalues of P̂n ?
(d) Prove P̂n P̂m = P̂n δnm .
(e) Is P̂n unitary?
(f) Prove that Ĥ =
P
En P̂n
n
2. Consider a harmonic oscillator of mass m and frequency ω.
(a) Calculate the matrix elements of the momentum operator using the eigenfunctions of the
hamiltonian pkn = hϕk , p̂ϕn i for k, n = {0, 1, 2, 3}.
(b) Calculate the matrix elements of the momentum operator pkn using ladder operators
(â, ↠).
(c) Calculate the matrix elements of the squared momentum operator p2kn using ladder operators (â, ↠).
(d) Calculate the matrix elements of the squared momentum operator p2kn using matrix
multiplication.
(e) Calculate ∆x̂, ∆p̂ for some eigenfunction of the hamiltonian and calculate ∆x̂∆p̂. What
is the minimal value of ∆x̂∆p̂ and what is the corresponding eigenfunction?
(f) Find the mean kinetic energy and potential energy of the state |ni,. Write the solution
in terms of the eigenenergy En . Explain this result.
3. Consider a harmonic oscillator of mass m and frequency ω. We define Coherent states |zi by:
|zi ≡ e−
|z|2
2
∞
X
zn
√ |ni
n!
n=0
in terms of a complex number z.
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(a) Show that these states are normalized. Prove that they are eigenstates of the annihilation
operator (â) with eigenvalue z.
(b) Calculate the expectation value of the number operator N = ↠â = hN i and the uncertainty ∆N in such a state. Show that in the limit of large occupation numbers N → ∞
the relative uncertainty ∆N /N tends to 0.
(c) Suppose that the oscillator is initially in such a state at t = 0. Calculate the probability
of finding the system in this state at t > 0. Prove that the evolved state is still an
eigenstate of the annihilation operator with a time-dependent eigenvalue. Calculate hN i
and hN 2 i in this state and prove that they are time-independent.
4. A polar representation of the creation and annihilation operators for a simple harmonic oscillator can be written as:
q
â ≡ N̂ + 1eiφ̂ ,
↠≡ e−iφ̂
q
N̂ + 1,
where the operators N̂ and φ̂ are assumed to be hermitian.
(a) Starting with the commutation relation [â, ↠] = 1, show that
[eiφ̂ , N̂ ] = eiφ̂ ,
[e−iφ̂ , N̂ ] = −e−iφ̂ .
Similarly, Show that
[cos φ̂, N̂ ] = i sin φ̂,
[sin φ̂, N̂ ] = −i cos φ̂.
(b) Calculate the matrix elements
hn|e±iφ̂ |ki,
hn| cos φ̂|ki,
hn| sin φ̂|ki
(c) The generalized Heisenberg relation(see proof at the bottom) states that for any two
operators  and B̂.
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(∆Â)2 (∆B̂)2 ≥ |h[Â, B̂]i|2
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(1)
Write down the Heisenberg uncertainty relation between the operators N̂ and cos φ̂.
Compute the quantities involved for the state
2 1/2
|ψi = (1 − |c| )
∞
X
cn |ni
n=0
where c is some complex parameter. Show that the resulting inequality is always true.
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(d) Consider a coherent state
|zi ≡ e−
|z|2
2
∞
X
zn
√ |ni
n!
n=0
and calculate the quantities (∆N̂ )2 , (∆ cos φ̂)2 and hsin φ̂i in this state. Show that the
Number-Phase uncertainty inequality reduces to an equality in the limit of very large
occupation numbers (z → ∞).
You may use the asymptotic formulas
2 ∞
X
|z|2n
1
e|z|
√
1−
≈
+ ...
|z|
8|z|2
n! n + 1
n=0
∞
X
2
e|z|
p
≈
|z|2
n=0 n! (n + 1)(n + 2)
|z|2n
1
+ ... .
1−
2|z|2
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Short proof of the generalized Heisenberg relation: Say we have two operators  and
B̂ with commutator [Â, B̂] = iĈ. We want to estimate the uncertainty (∆Â)2 (∆B̂)2 . By definition
we know (∆Â)2 = ( − hÂi)2 . We now apply Schwarz inequality:
(Â − hÂi)2 (B̂ − hB̂i)2 ≥ |(Â − hÂi)(B̂ − hB̂i)|2
(2)
Next, for any operator we know
F̂ =
F̂ + F̂ †
F̂ − F̂ †
+i
2
2i
(3)
therefore
(Â − hÂi)(B̂ − hB̂i) + (B̂ − hB̂i)(Â − hÂi)
2
(Â − hÂi)(B̂ − hB̂i) − (B̂ − hB̂i)(Â − hÂi)
+i
2i
i
= Ô + Ĉ
2
(∆Â)2 (∆B̂)2 = ( − hÂi)(B̂ − hB̂i) =
(4)
(5)
(6)
With Ô and Ĉ hermitian. Anyway, we know put this back in (2) and get
i 2 1
(∆Â) (∆B̂) ≥ Ô + Ĉ ≥ |h[Â, B̂]i|2 .
2
4
2
2
(7)
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