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Problem Set 8 - Quantum Theory I - Fall 2008 | PHYS 321, Assignments of Physics

Material Type: Assignment; Class: Quantum Theory I; Subject: Physics; University: Mesa State College; Term: Fall 2008;

Typology: Assignments

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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 min an infinite well of width Lthe energy eigenvalues are
En=π2~2n2
2mL2
where n= 1,2,3,.... Denote the associated state by |φni.Suppose that a particle is in the
following superposition of energy eigenstates at t= 0:
|Ψ(0)i=A
X
n=1
1
n2|φni
where Ais 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 tand 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 wil l be essential:
X
n=1
1
n2=π2
6
X
n=1
1
n4=π4
90 .
2 Dirac Delta Function
a) Apply the definition of the Dirac delta function to evaluate
Z
−∞
x2δ(xx0) dx
where x0is any real number.
1
pf2

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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) =

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.