Thermodynamics, Statistical Physics, and Quantum Mechanics

For there are three possible eigenvalues of which
gives an energy of
For there is one eigenvalue of and this state has zero
energy.

5.26 Spin-Dependent Potential (MIT)

a) The spin operator is For spin 1/2 the expression

becomes, for Pauli matrices, where is the

unit matrix. The total spin operator for the two-particle system is

For the spin singlet state then while for the spin
triplet state then

b) The potential is repulsive for the triplet state, and there are no bound
states. There are bound states for the singlet state since the potential is at-
tractive. For the hydrogen atom the potential is and the eigenvalues
are

Our two-particle bound state has instead of and the reduced mass
instead of the mass so we have the eigenvalues

5.27 Three Spins (Stony Brook)

a) We use the notation that the state with threespins up is This
is the state with We operate on this with the lowering operator
which shows that the states with lower values of M are

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