1. Consider the following t = 0 wave function of a particle in an infinite square well (0 < x < L):
(a) Find a value for A such that the wave function is normalized.
(b) Plot representative snapshots (one for each time) of P(x, t) for this system. (For you own benefit, it will be useful to view animations of these dynamics.)
N.B. For convenience, take m = ~ = 1, and use y = x/L to make plots with respect to a dimensionless length. The time variable can be taken to be τ = t/L2, so your plots should be of P(y, τ ≥ 0), 0 < y < 1.
2. Consider a particle of mass m subject to the following potential function (taking V0 and L to be positive):
(a) Derive the transcendental equation for energy eigenstates having an energy E > 2V0. N.B. The wave function must be continuous for all x, and it must have a continuous first derivative unless one encounters an infinite discontinuity (such as the walls of the infinite square well potential). So, for this question the first derivative of ψ(x) must be continuous at x = L/2.
To simplify the mathematics and numerics, for the rest of this question take ~ = 1, L = 1, m = 1, and choose (in this system of units) Vo = 1.
(b) Find numerical values of the energies of the first two states for E > 2Vo (that is, for the pure states with the lowest two energies subject to the constraint that E > 2Vo). Plot the probability densities for these two states.
Q3. Consider a particle that can move in one dimension. To simplify the algebra, set m=1 and ~ = 1.
a) Consider the potential function
V1 (x) = 1/2x2
Determine α such that
ψ(x) = A x e-αx2
is an eigenstate of the Hamiltonian
Hˆ = -12∂2∂x2+ V1(x)
What is the energy of this energy eigenstate?
(b) Sketch the probability density associated with the third excited state of this potential.
(c) Now consider the potential function (which you would be wise to sketch)3
Determine the value of |Vo| such that the ground state wave function in the region |x| < L is given by
ψ(x) = B e-(x2/2)
What is the energy of this eigenstate? Hint: First, you must determine the form of the wave function in the |x| > L regions.