Problem 1:

Consider a neutron (mass m = 1.67*10^-27 kg) in the laboratory that is subjected to gravity (only the vertical motion in the z direction is of interest here) and a hard wall at z = 0.

(a)  What is the Hamiltonian that governs this system?
(b)  Make an estimate for the wave length of the quantum mechanical ground state of a gravitationally bound neutron.
(c)  Compute the action S(E) = ∮dz p(z) and quantize it to find the quantized energies.
(d)  How do energies scale with the principal quantum number?

Problem 2:

Submit this problem on Canvas as Assignment 5.  If you used an AI as a Socratic tutor, submit a copy of your session leading to your solution and reflect on your session.  If you did not need any help or worked with another student, explain your reasoning, do not just write down formulas. 

A particle is moving in a one dimensional potential,
U(x) = ∞  for x < 0 and x > π,
U(x) = 0  for 0 < x < π/2,
U(x) = U₀ for π/2 < x < π.
(Units:  Assume all quantities are expressed in terms of consistent small length and mass units.)

(a)  The variational method can be used to find an upper bound for the ground-state energy of the particle.
For the trial function Ψ = N(2/π)^½(sin(x) + a sin(2x)), find the optimal value of the variational parameter a.
(b)  For U₀ = 0.1*ħ²/(2m), estimate the ground state energy of the particle in terms of ħ²/(2m).
(c)  Find the wave function and energy of the ground state for the cases where U₀ --> 0 and U₀ --> ∞.

Problem 3:

A particle of mass m is bound in a 3D radially symmetric potential well which is weakly anharmonic, U(r) = ½mω²r² + εr⁴.
(a)  The ground state energy can be written as a power series E₀ = (3/2)ħω + O(ε) + O(ε²) + ... .   Use first order perturbation theory to determine the O(ε) term exactly.
(b)  The first order correction to the ground state wave function will mix the unperturbed Φ₀⁰(r) with a limited set of Φ₀ⁿ(r) of unperturbed basis states.  Which basis states are mixed at O(ε) in the ground state corrected to first order?  
Specify these using the Cartesian (n_x,n_y,n_z) labels for the harmonic oscillator.

[Note:  H₀ = H_0x + H_0y + H_0z.  H_0x = ħω (a^†a + ½),  a = αx + iβp,  a^† = αx - iβp, α = √(mω/(2ħ)),  β = 1/√(2mωħ).
For eigenstates |n> of H_0x we have a^†|n> = (n + 1)^½|n + 1>, a|n> = n^½|n - 1>.]