Problem Set 8

Problem Set 8

DQ 8 -- Due Tuesday, April 6, by 5pm

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.

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Problems -- Due Thursday, April 8, by 5pm.

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Problem 1 [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, 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 2 [25 points total]

In lecture we computed the two particle density matrix in real space, ⟨r₁,r₂|ρ₂|r₁,r₂⟩, for spinless particles obeying either Bose-Einstein or Fermi-Dirac statistics. Here you will calculate it for spin 1/2 fermions, where each particle i has an intrinsic spin that can take on one of two possible values, s_i = ↑ or ↓.

For two free fermions in a box, we can write the spin part of the energy eigenstates as |s₁⟩|s₂⟩, which specify the spin of each particle. However we can also write the spin part in terms of eigenstates of the total angular momentum, |ℓ, m_z⟩. These are then:

spin singlet: |0, 0⟩ = ( |↑⟩|↓⟩ − |↓⟩|↑⟩ )/sqrt(2) spin triplet: |1, 1⟩ = |↑⟩|↑⟩
|1, -1⟩ = |↓⟩|↓⟩
|1, 0⟩ = ( |↑⟩|↓⟩ + |↓⟩|↑⟩ )/sqrt(2)

For this problem you want to compute the two particle density matrix using the above basis for the spin states,

⟨r₁, r₂, ℓ, m_z|ρ₂|r₁, r₂, ℓ, m_z⟩

To compute the above matrix element you will need the matrix elements,

⟨r₁, r₂, ℓ, m_z|k₁, k₂ , ℓ′, m′_z⟩

where k₁ and k₂ give the wavevectors of the real space part of the energy eigenstates.

a) Compute the above matrix elements ⟨r₁, r₂, ℓ, m_z|k₁, k₂ , ℓ′, m′_z⟩ for all four of the spin basis states |ℓ, m_z⟩. You will have to take into account that the total wavefunction must be antisymmetric under the exchange of the two particles.

b) Using your results from (a), compute the density matrix ⟨r₁, r₂, ℓ, m_z|ρ₂|r₁, r₂, ℓ, m_z⟩ for all four of the spin basis states |ℓ, m_z⟩.

You should find that for any of the spin triplet states, the density matrix has the same dependence on |r₁ − r₂| as we found previously for two spinless fermions, i.e. there is an effective repulsion at small separations.

However for the spin singlet state, the density matrix has the same dependence on |r₁ − r₂| as we found previously for two spinless bosons, i.e. there is an effective attraction at small separations. Give a simple physical argument as to why this is a reasonable result.

This is the origin of Hund's rule of atomic physics. As one fills up a partially full energy subshell of an atom, the electrons first go in with parallel spins since this gives an effective repulsion between them that lowers the electrostatic repulsive energy.

Bonus: (no points, just for fun) Consider the two particle density matrix where we use the spin states |s₁, s₂⟩ for the basis, instead of the total spin states |ℓ, m_z⟩. Compute ⟨r₁, r₂, ↑, ↑ |ρ₂|r₁, r₂, ↑, ↑⟩ and ⟨r₁, r₂, ↑, ↓ |ρ₂|r₁, r₂, ↑, ↓⟩. Can you explain your results?