What are steady state or equilibrium problems?
Situations where a quantum particles probability distribution does not vary with time.
In regions of constant potential where E>V, what form does the solution have?
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.
In regions of constant potential where E=V, what form does the solution have?
No solution because you get the second dervative equal to 0
In regions of constant potential where E<V, what form does the solution have?
Exponential growth/decay. Follow the same steps as for when V is greater but use an effective wavenumber q (E and V are reversed).
What are the four boundary conditions for dealing with potential steps?
- The wave function is continuous
- Singularities in the wavefunction are forbidden
- The first derivative of the wavefunction must be continuous
- In a region in which the potential is infinite, the wavefunction is zero
