PHYS 424 Notes

H = - [hbar²/(2m₁)] Ñ₁² - [hbar²/(2m₂)] Ñ₂² + V(r₁ - r₂)
separates using r = r₁ - r₂ and R = (m₁r₁ + m₂r₂) / (m₁ + m₂) since

d/r1i = dR/dr1id/dR + dr/dr1id/dr         = [m1 / (m1 + m2 )] d/dRi + d/dri

d/r2i = dR/dr2id/dR + dr/dr2id/dr         = [m2 / (m1 + m2 )] d/dRi - d/dri

and the cross terms d²/drdR cancel, leaving

H = - {hbar² / [2 m₁m₂/(m₁+m₂)]} Ñ_r² - {hbar²/[2(m₁+m₂)]} Ñ_R² + V(r).

Setting y(R, r) = y_R(R) y_r(r), m = m₁m₂ / (m₁ + m₂) and M = m₁ + m₂ yields
- (1/y_r){hbar² / [2 m)]} Ñ_r² y_r + V(r) - (1/y_R){hbar²/[2M]} Ñ_R² y_R = E
which implies the separated set of equations

1. Identical Bosons, S=0
   1. Symmetrization in product states
      Two-particle y's are products, since the probability of two independent events is the product of the probabilities of either alone. The wave function for particle 1 in state a and particle 2 simultaneously in state b is
      y_a(r₁) y_b(r₂),    

      However, since the particles are identical, we cannot tell the difference between this situation and the one where particle 1 is in state b and particle 2 is in state a:
      y_b(r₁) y_a(r₂)      

      Nature reflects this ambiguity by always having the wave fuction be the symmetric combination of the two possibilities,
      A[y_a(r₁) y_b(r₂) + y_b(r₁) y_a(r₂)],
      where A is a normalization constant. In principle, there could be a minus sign between the two terms, but in practice only the plus sign occurs. The use of the pluys sign for bosons can be proved in relativistic field theory to be necessary in order for energies to be bounded below.

2. Identical Fermions, S=1/2
   1. Symmetrization in product states
      The wave function for particle 1 in space state a and spin state a' while particle 2 is in space state b and spin state b' is
      y_a(r₁) c_a'(s₁) y_b(r₂) c_b'(s₂),
      

      while if particle 2 is in space state a and spin state a' while particle 1 is in space state b and spin state b', the wave function is
      y_b(r₁) c_b'(s₁) y_a(r₂) c_a'(s₁).

      Again, we cannot tell which of these two possibilities we have, and the wave function that actually occurs is
      A[y_a(r₁) c_a(s₁) y_b(r₂) c_b(s₂) - y_b(r₁) c_b(s₁) y_a(r₂) c_a(s₁)]

   2. Fermi Exclusion Principle
      If a=b and a'=b', the wave function vanishes. So two fermions cannot be simultaneously in the same state. This is the Fermi Exclusion Principle, which among other things makes atoms-as-we-know-them possible.

   3. Symmetrization in total-spin states
      If the potential is independent of spin, then any space state can be associated with spin up or spin down. In this case, and some others, it is useful to use eigenstates of total spin. Using S=S₁+S₂, you can show that the total spin states are ["u" means spin up and "d" means spin down]
      

      S=0, M=0: c_u(1)c_d(2) -c_d(1)c_u(2)

      S=1, M=1: c_u(1)c_u(2)
      S=1, M=0: c_u(1)c_d(2) +c_d(1)c_u(2)
      S=1, M=-1: c_d(1)c_d(2)

      The S=0 state is antisymmetric under interchange of the two particles. If the overall state is also to be antisymmetric, the space state must be symmetric. Hence the proper wave function is

      A[y_a(r₁)y_b(r₂) +y_b(r₁)y_b(r_a)] [c_u(1)c_d(2) -c_d(1)c_u(2)]

      Similarly, the S=1 state is an antisymmetric space state times one of the spin states. In particular, for S=1, M=0 the state is

      A[y_a(r₁)y_b(r₂) -y_b(r₁)y_b(r_a)] [c_u(1)c_d(2) +c_d(1)c_u(2)].

      If a=b, the antisymmetric space state vanishes and no spin-1 state is permitted [Pauli principle again]. Clearly the antisymmetrization requirement reduces the number of states two electrons can occupy.

1. Single-particle states
   y_n = sqrt(2/a)sin(npx/a)
   E_n = hbar²(np/a)²/(2m)

2. Space state
   1. Total Spin 0
      y = [sqrt(2)/a] [sin(npx₁/a) sin(mpx₂/a) +sin(mpx₁/a) sin(npx₂/a)]

   2. Total Spin 1
      y = [sqrt(2)/a] [sin(npx₁/a) sin(mpx₂/a) -sin(mpx₁/a) sin(npx₂/a)]
3. Energies
   

   2mEa2/p2	n	m	Spin 0 state?	Spin 1 state?	Degeneracy
   2	1	1	yes	no	1
   5	2	1	yes	yes	4
   8	2	2	yes	no	1
   10	3	1	yes	yes	4
   13	3	2	yes	yes	4
   17	4	1	yes	yes	4
   18	3	3	yes	no	1