Physics 324
Autumn 2021
Homework Assignment # 3
Due Tuesday October 26, in class by 10AM
or in instructor’s mailbox by 945AM, or in Canvas by 945AM.
Turn in all problems and clearly note all constants and assumptions you use.
1. Half Oscillator
The potential energy of a particle of mass m is given by the following expression (here, a is a
positive constant):
(
ax2
if x > 0
2 ,
V (x) =
∞, if x ≤ 0
(a) Without solving Schrodinger’s equation, determine the quantized energy values for this particle.
(b) Sketch the potential V (x) and the wave functions of the lowest 3 energy values.
2. Double δ−function Potential
The potential energy of a particle of mass m is given by
V (x) = V0 δ(x − x0 ) + V0 δ(x + x0 ),
where V0 < 0 and x0 > 0 are some constants, and δ() denotes Dirac delta function.
(a) How many bound states are there in this potential given that V0 = −h̄2 /(mx0 )? You do
not need to solve for the exact values of the bound state energies, only derive the necessary
equations to show how many bound states there are.
(b) Sketch the corresponding wave function(s), noting all the important details.
3. Released Particle
A free particle of mass m at time t = 0 has wave function
2
Ψ(x, 0) = Ae−αx ,
where A and α are positive, real constants.
(a) Find A in terms of α.
(b) Find φ(k) and comment on its properties.
(c) Determine Ψ(x, t) and |Ψ(x, t)|2 . You may leave your answers in integral form.
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