130_notes.dvi

19.4 Homework

1. The 1D model of a crystal puts the following constraint on the wave numberk.

cos(φ) = cos(ka) +

ma^2 V 0

̄h^2

sin(ka)

ka

Assume thatma

(^2) V 0
̄h^2 =
3 π
2 and plot the constraint as a function ofka. Plot the allowed energy
bands on an energy axis assumingV 0 = 2 eV and the spacing between atoms is 5 Angstroms.

2. In a 1D square well, there is always at least one bound state. Assume the width of the square
   well isa. By the uncertainty principle, the kinetic energy of an electron localized to that width
   is h ̄

2

2 ma^2. How can there be a bound state even for small values ofV^0?

3. Att= 0 a particle is in the one dimensional Harmonic Oscillator stateψ(t= 0) =√^12 (u 0 +u 1 ).
   Isψcorrectly normalized? Compute the expected value ofxas a function of time by doing
   the integrals in thexrepresentation.


4. Prove the Schwartz inequality|〈ψ|φ〉|^2 ≤ 〈ψ|ψ〉〈φ|φ〉. (Start from the fact that〈ψ+Cφ|ψ+
   Cφ〉≥0 for anyC.


5. The hyper-parity operatorH has the property thatH^4 ψ=ψ for any stateψ. Find the
   eigenvalues ofHfor the case that it is not Hermitian and the case that it is Hermitian.


6. Find the correctly normalized energy eigenfunctionu 5 (x) for the 1D harmonic oscillator.


7. A beam of particles of energyE >0 coming from−∞is incident upon a double delta function
   potential in one dimension. That isV(x) =λδ(x+a)−λδ(x−a).

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 and

the probability to be transmitted.