Homework #5

Q1: Wavefunctions in different potential wells

Below are energy eigenfunctions of particles in two different potential wells. Consider the potentials U(x) drawn below the functions. For every energy eigenfunction / potential well combination, tell me why it could be, or certainly could not be, be an energy eigenfunction corresponding to that potential well.

Energy eigenfunctions:

Potential energy functions:

The x-axes of the potential functions do not correspond to those of the wave functions. The potentials correspond to, from left to right, the infinite square well (ISW), the finite square well (FSW), the quadratic or harmonic oscillator potential (HO), and a 1-sided linear potential well (1LW), and a 2-sided linear well (2LW).

Q2: Expectation values, uncertainties for infinite well

McIntyre 5.5:

This is a good example of where you can use Mathematica to do integrals. The point of the question is not to demonstrate prowess at integrals, but to use integrals as a tool explore some interesting physics. Note that if an integral yields zero, or a strikingly simple expression, seek out the symmetry or physical argument behind the result.

Extra practice: Having found Δx and Δp , also calculate the uncertainty product ΔxΔp for the first two states and demonstrate that ΔxΔp > ℏ/2.

Q3: Probability density

McIntyre 5.6

(only for the first 2 states)

Q4: Finite Square Well

McIntyre 5.18:

Hints: Use Mathematica to plot Eq. 5.88 as a function of z (or the alternative formulation of the same equations that we used in class – you’ll get the same result). Zoom in to find intersection points and hence find energies. Show your working.

Q5: Reflection/Transmission of quantum particle

The potential energy part of the Hamiltonian operator (a potential step of height V_0) for a system is depicted as a solid line in the energy/position diagram. This is an example of an unbounded system, so there is no condition on the energy eigenvalue. A particle of mass m and energy E is incident from the left. There are 2 cases: E > V_0, and E < V_0. (As you work through this problem, think carefully about the problem we discussed where a wave in a stretched string is incident on a string of different mass density. There are important similarities and equally important differences.)

( a ) Set up a wave incident from the left, a wave reflected to the left, and one transmitted to the right, so that the total wave function is:

Verify that these are indeed solutions to the energy eignevalue equation for this problem, provided that

( b ) What are the boundary conditions that establish the relationships among the coefficients?

( c ) The probability to observe the particle reflected is


Remember we measure probabilities and not amplitudes. Find r for both cases: E > V_0, and E < V_0. Also find the probability of transmission t = 1 - r. (See note below*)

( d ) Interpret your results, and also discuss the limiting cases V_0 = 0, and E » V_0.

*Note: Do not try to define the transmission coefficient as |C/A|². You’ll get the wrong answer. Ask me about this if you want to know the details.