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Phys 321 Fall 2008
Quantum Theory I: Homework 8
Due: 22 October 2008
1 Particle states and measurements
For a particle of mass m in an infinite well of width L the energy eigenvalues are
En =
π^2 ℏ^2 n^2
2 mL^2
where n = 1, 2 , 3 ,.... Denote the associated state by |φn〉. Suppose that a particle is in the following superposition of energy eigenstates at t = 0:
|Ψ(0)〉 = A
∑^ ∞
n=
n^2
|φn〉
where A is a real constant.
a) Apply the normalization condition to determine A.
b) Determine an expression for the expectation value of energy at time t = 0.
c) Determine an expression for the state at a later time t and use this to determine an
expression for the expectation value of energy at time t. Verify that this does not depend
on t.
Hint: the following will be essential:
∑^ ∞
n=
n^2
π^2
6
∑^ ∞
n=
n^4
π^4
90
2 Dirac Delta Function
a) Apply the definition of the Dirac delta function to evaluate
∫ (^) ∞
−∞
x^2 δ(x − x 0 ) dx
where x 0 is any real number.
b) An approximate representation of the Dirac delta function is
δǫ(x − x 0 ) :=
0 if x < x 0 − ǫ/ 2
1
ǫ
if x 0 − ǫ/ 2 < x < x 0 + ǫ/ 2
0 if x > x 0 + ǫ/ 2.
where the notion is that
lim ǫ→ 0
δǫ(x − x 0 ) = δ(x − x 0 ).
Integrate (^) ∫ ∞
−∞
x
2 δǫ(x − x 0 ) dx
and verify that as ǫ → 0 the result is the same as that of the previous part.
c) Evaluate (^) ∫ ∞
−∞
3 x^2 sin x δ(2x − π) dx.
Hint: Perform a “u substitution” so that the delta function appears as δ(u) in the inte-
grand and then use the definition of the delta function to evaluate the integral.
3 Inner product between states for particles in one dimension
Consider the following states and corresponding wavefunctions for particles in one dimension.
|Ψ 1 〉 ↔ Ψ 1 (x) =
aπ
e−x
(^2) / 2 a and
|Ψ 2 〉 ↔ Ψ 2 (x) =
a^3 π
x e−x
(^2) / 2 a
a) Show that 〈Ψi|Ψj 〉 = δij for all combinations of i and j.
b) Let
|Ψ 2 〉 and
4 i √ 5
Show that 〈Ψ|Ψ〉 = 1 and 〈Φ|Φ〉 = 1 and determine 〈Φ|Ψ〉. Hint: Try to do these without
computing any integrals.
