Problem Set 7

Problem Set 7

Thursday, April 10, uploaded to Blackboard by 11:59pm

Problem 1 [10 points total]
Particles in a Box: In the lecture notes 3-3 we used periodic boundary conditions to define the single-particle energy eigenstates of a free quantum particle in a box. We could instead have used fixed boundary conditions where the wavefunctions must vanish at the walls, ψ(0,y,z)=ψ(L,y,z)=ψ(x,0,z)=ψ(x,L,z)=ψ(x,y,0)=ψ(x,y,L)=0.
Find the single-particle energy eigenstates of a particle in a box using the fixed boundary conditions. Compute G(ε), the number of single-particle energy eigenstates per unit volume that have ε_i ≤ ε, in the large L → ∞ limit, and show that it is the same regardless of whether we use fixed or periodic boundary conditions. The derivative, g(ε)=dG/dε is the density of states, and will be an important quantity to characterize such systems.
Problem 2 [10 points total]
Consider photons of a given energy ε = ℏω. As will be discussed in Notes 3-6, photons of frequency ω can be thought of as excitations of a quantized harmonic oscillator of frequency ω (the transverse EM wave at frequency ω), where excitation to level n of the oscillator corresponds to having n photons. Using the results at the end of Notes 3-1, show the following:
(a) Using Eq. (3.1.38) of Notes 3-1 for the probability P_n to be excited to level n of the harmonic oscillator (and so the probability to have n photons), show that the fluctuation in the number of photons is,
<n²> - <n>² = - (1/ε) (d<n>/dβ) where β = 1/k_BT

(b) Using the forumula for the equilibrium value of <n>, apply the above result to determine the relative fluctuation in the number of photons
[<n²> - <n>²]/<n>²
Is this large or small?
Problem 3 [10 points]
Consider a point particle of mass m attached to a harmonic spring with spring constant κ=mω₀². The particle moves in only one dimension given by the coordinate x. The rest position of the particle (when the spring in unstretched) is at x=0. The particle is in equilibrium at temparature T.
Assuming that the particle behaves quantum mechanically, compute the root-mean-square fluctuation in the position of the particle, Δx = (⟨x²⟩ - ⟨x⟩²)^1/2.
Hint: recall that for the quantum harmonic oscillator, the expected value of the kinetic energy equals the expected value of the potential energy (this is the quantum virial theorem).
Show that in the limit k_BT ≫ ℏω₀ you recover the answer for a classical particle.