Assume that two particles are placed at random into one of two boxes. What are the probabilities of finding the following distributions if the particles are classical particles, identical bosons, or identical fermions?
Details of the calculation:
Classical particles are distinguishable. The state that has particle 1 in box 1 and particle 2 in box 2 differs from the state that has particle 2 in box 1 and particle 1 in box 2. Bosons and Fermions are indistinguishable. There is only one state with one of the indistinguishable particles in box 1 and the other in box 2. Fermions obey the Pauli exclusion principle. No two fermions can be in the same box.
Assume three particles (1, 2, 3) and three distinct one particle states (ψα(x),
Describe in detail the possible three particle states can that be constructed if:
(a) The three particles are distinguishable,
(b) The three particles are identical fermions, and
(c) The three particles are identical bosons.
(b) If the particles are identical fermions only one state can be
The wave function must be anti-symmetric under exchange.
Notation |a, b, c>: Let a denote the state of particle 1, b the state of particle 2, and c the state of particle 3.
|ψ> = (1/6)½[|α,β,γ> + |γ,α,β> + |β,γ,α> - |α,γ,β> - |β,α,γ> - |γ,β,α>]
is antisymmetric under exchange of any two particles.
(c) If the particles are identical bosons, 10 linearly independent states
can be constructed.
The wave function must be symmetric under exchange.
The following states satisfy this requirement.
(i) All three particles can be in the same state. (3 possibilities)
|ψ> = |α,α,α> or |ψ> = | β,β,β> or |ψ> = | γ,γ, γ >.
(ii) Each particle occupies a different state. (1 possibility)
|ψ> = (1/6)½[|α,β,γ> + |γ,α,β> + |β,γ,α> + |α,γ,β> + |β,α,γ> + |γ,β,α>]
is symmetric under exchange of any two particles.
(iii) Two particles occupy the same state and one particle occupies one of the remaining two states. (3 possibilities times two possibilities = 6 possibilities)
Example: |ψ> = (1/3)½[|α,α,γ> + |γ,α,α > + | α,γ,α>]
Protons and neutrons making up a light nucleus move in an average potential
that resembles that of a harmonic oscillator.
U(r) = -U0 + Mω2r2/2
Find numbers of protons (or neutrons) corresponding to the first three closed shells (magic numbers).
The energy levels are degenerate.
E0 = (3/2)ħω, two undistinguishable spin ½ particles can occupy this state (nx, ny, nz = 0).
E1 = (1 + 3/2)ħω, 6 undistinguishable spin ½ particles can occupy this state (nx or ny or nz = 1 with the other ni = 0).
E2 = (2 + 3/2)ħω, 12 undistinguishable spin ½ particles can occupy this state (nx or ny or nz = 2 with the other ni = 0, or two of the ni = 1 and one of the ni = 0).
Magic numbers: 2, 2 + 6 = 8, 8 + 12 = 20.
A nucleus can be considered as fermions moving in a three-dimensional
harmonic oscillator potential. Each nucleon has potential energy U = ½ m(ωx2x2
+ ωy2y2 + ωz2z2).
Assume that ωx = ωy = 2ωz.
(a) Find the first five magic numbers for protons and neutrons.
(b) Will the magnitude of the total angular momentum have good eigenvalues for this potential?
(c) Will Lx, Ly, or Lz have good eigenvalues?
(b) The potential is not spherically symmetric, [L2, U] ≠ 0, [L2,
H] ≠ 0, the stationary states are not eigenstates of L2. The
magnitude of the total angular momentum does not have good eigenvalues.
The potential is not invariant under rotations about the x-axis and y-axis, but is invariant under rotations about the z-axis.
[Lx, H] ≠ 0, [Ly, H] ≠ 0, [Lz, H] = 0.
Lx and Ly do not have good eigenvalues, Lz has good eigenvalues.