Download Quantum Theory 1 - Home Exercise 6

Quantum Theory 1 - Home Exercise 6
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1. We define the commutator of two operators  and B̂ by Â, B̂ = ÂB̂ − B̂ Â. Prove the
following identities:
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(a) Â, B̂ = − B̂, Â
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(b) Â, Â = 0
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(c) Â, B̂, Ĉ + B̂, Ĉ, Â + Ĉ, Â, B̂ = 0
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(d) Â, B̂ Ĉ = Â, B̂ Ĉ + B̂ Â, Ĉ
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(e) ÂB̂, Ĉ = Â B̂, Ĉ + Â, Ĉ B̂
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(f) If Â, Â, B̂ = B̂, Â, B̂ = 0
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• Â, B̂ n = nB̂ n−1 Â, B̂
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• Ân , B̂ = nÂn−1 Â, B̂
and therefore if F (B̂) is an analytical function of B̂ we get :
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• Â, F (B̂) = ∂F (B̂) Â, B̂
∂ B̂
2. Given an operator Â, we define it’s hermitian conjugate, † such that hΨ1 , ÂΨ2 i = h† Ψ1 , Ψ2 i.
(a) Show that if  = a(just multiplication by a scalar) then † = a∗ .
(b) Find the hermitian conjugate of the derivative operator D̂ =
d
dx
An Hermitian operator is an operator that satisfies  = † .
(c) Let  and B̂ be two Hermitian operators. We define the operator Ĉ =  + iB̂. Is this
operator Hermitian? If not, find its hermitian conjugate Ĉ † .
Prove the following statements regarding Hermitian operators:
(d) Two eigenstates of an Hermitian operator that have a different eigenvalue are orthogonal.
(e) Given two commuting operators  and B̂, and let ϕ be an eigenstate of B̂ with eigenvalue
b. Show that Âϕ is also an eigenstate of B̂, with the same eigenvalue b.
(f) From the previous article, deduce that if b is non-degenerate(there is only one eigenstate
associated with it), then ϕ is also an eigenstate of Â.
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3. Consider an infinite square well of width L, with a particle of mass m moving in it (− L2 <
x<
L
2 ).
At time t = 0, the state of the particle is described by the wavefunction
Ψ(x, 0) = A (3ϕ1 (x) + 4ϕ2 (x))
(a) Find |A|.
(b) Find Ψ(x, t)
(c) We measure the particle’s position at time t. What is the probability of finding the
particle at the right half of the well?
(d) Find hxi(t) and hpi(t) . Notice that while these are periodic, they are very different from
the classical results. Discuss the reasons for this difference.
4. Consider an infinite square well of width L, with a particle of mass m moving in it (− L2 <
x<
L
2 ).
The particle is in the lowest-energy state.
Assume now that at time t = 0 the walls of the well move instantaneously so that its width
doubles (−L < x < L). This change does not affect the state of the particle, which is the
same before and immediately after the change.
(a) Write down the wave function of the particle at times t > 0. Calculate the probability Pn
of finding the particle in an eigenstate n of the modified system. What is the probability
of finding the particle in an odd eigenstate?
(b) Calculate the expectation value of the energy at times t > 0.
You may want to use the series:
∞
X
l=0
π2
(2l + 1)2
=
2
[(2l + 1) − 4]
16
.
(c) If we assume instead that the walls move outward with a finite velocity u, our assumptions
should still hold provided that this velocity is much greater then the characteristic velocity
of the system, i.e. u v0 . What is the characteristic velocity v0 ?
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