130_notes.dvi

b) Determine the coefficients needed to satisfy the boundary conditions.

c) Calculate the probability for a particle in the beam to be reflected by the potential and

the probability to be transmitted.

5.*The Schr ̈odinger equation for the one dimensional harmonic ocillator is reduced to the

following equation for the polynomialh(y):

d^2 h(y)

dy^2

− 2 y

dh(y)

dy

+ (

E

α

−1)h(y) = 0

a) Assumeh(y) =

∑∞

m=0

amymand find the recursion relation for the coefficientsam.

b) Use the requirement that this polynomial series must terminate to find the allowed ener-

gies in terms ofα.

c) Findh(y) for the ground state and second excited state.

6. A beam of particles of energyE >0 coming from−∞is incident upon a potential step in one
   dimension. That isV(x) = 0 forx <0 andV(x) =−V 0 forx >0 whereV 0 is a positive real
   number.
   a) Find the solution to the Schr ̈odinger equation for this problem.
   b) Determine the coefficients needed to satisfy the boundary conditions.
   c) Calculate the probability for a particle in the beam to be reflected by the potential step
   and the probability to be transmitted.

7.*A particle is in the ground state (ψ(x) = (mωπ ̄h)

(^14)
e
−mωx2 ̄h^2
.) of a harmonic oscillator potential.
Suddenly the potential is removed without affecting the particle’s state. Find the probability
distributionP(p) for the particle’s momentum after the potential has been removed.
8.A particle is in the third excited state (n=3) of the one dimensional harmonic oscillator
potential.
a) Calculate this energy eigenfunction, up to a normalization factor, from the recursion
relations given on the front of the exam.
b) Give, but do not evaluate, the expression for the normalization factor.
c) Att= 0 the potential is suddenly removed so that the particle is free. Assume that the
wave function of the particle is unchanged by removing the potential. Write an expression
for the probability that the particle has momentum in the range (p,p+dp) fort >0. You
need not evaluate the integral.
9.The Schr ̈odinger equation for the one dimensional harmonic oscillator is reduced to the
following equation for the polynomialh(y):
d^2 h(y)
dy^2
− 2 y
dh(y)
dy

+ (

E

α

−1)h(y) = 0

a) Assumeh(y) =

∑∞

m=0

amymand find the recursion relation for the coefficientsam.

b) Use the requirement that this polynomial series must terminate to find the allowed ener-

gies in terms ofα.

c) Findh(y) for the ground state and second excited state.

10.*Find the energy eigenstates (and energy eigenvalues) of a particleof massmbound in the
1D potentialV(x) =−λδ(x).