Lecture 4

Question 1

What are steady state or equilibrium problems?

Answer 1

Situations where a quantum particles probability distribution does not vary with time.

Question 2

In regions of constant potential where E>V, what form does the solution have?

Answer 2

Propagating wave. To show this, rearrange the TISE to follow the format of a second order differential equation in terms of u(x). Use the substitution k^2 = 2m(E-V)/hbar^2 and then recognise that you have the equation for simple harmonic motion.

Question 3

In regions of constant potential where E=V, what form does the solution have?

Answer 3

No solution because you get the second dervative equal to 0

Question 4

In regions of constant potential where E<V, what form does the solution have?

Answer 4

Exponential growth/decay. Follow the same steps as for when V is greater but use an effective wavenumber q (E and V are reversed).

Question 5

What are the four boundary conditions for dealing with potential steps?

Answer 5

1. The wave function is continuous
2. Singularities in the wavefunction are forbidden
3. The first derivative of the wavefunction must be continuous
4. In a region in which the potential is infinite, the wavefunction is zero