#  Tensor ops.
# Tracelessness: Sum_m <m | T_k0 | m> = 0
# Wigner-Eckart theorem, selection rules.
# Analogy: {T_kq |jm>}  like {|kq> |jm>}
# Homework tips:
1) rotational transitions of homonuclear molecules: spin state of nuclei not affected,
hence parity of nuclear orbital angular momentum unchanged
(to preserve symmetry or antisymmetry of nuclear state).
2) Rotational energy = L^2/2I == L(L+1)hbar^2/2I.
# Zeeman effect in many electron atoms, Lande g-factor:

g = 3/2 + [S(S+1) - L(L+1)]/2J(J+1)

Note Baym (& Schwabl) say this is between 1 and 2 but that's not true!

#  E.g., when L = S + 1/2,  J=1/2:   g = 1 - 2S/3: it can be arbitrarily NEGATIVE.
When S = L + 1/2,  J=1/2:   g = 2 + 2L/3: it can be arbitrarily POSITIVE.
# Examples in the periodic table:

B :  2^P_1/2;   S = 1/2,  L = 1,  J = 1/2;   so g = 2/3 < 1

V:  4^F_3/2;    S = 3/2, L = 3, J = 3/2;  so g = 2/5 < 1

Nb:  6^D_1/2;  S = 5/2, L = 2,  J= 1/2;  so g = 10/3 > 2.

# hyperfine homework problem: write the hyperfine hamiltonian as V.I
where V is a vector operator wrt the degrees of freedom except nuclear spin.

# homework problem 5: note Y^5 has odd parity, so its expectation value vanishes
in any state of definite parity. This can be seen either from the integral itself, or abstractly:

Let P be the parity operator. Note PP=1, and PP*=1, since P is unitary.
Thus P*=P, i.e. P is also hermitian.

An operator odd under parity satisfies OP = - PO, or POP = - O.

Now let |v> be a state of definite parity, P|v> = p|v>, where p = +1 or -1. Then

<v |O|v> = - <v| POP|v> = - p^2 <v|O|v> = - <v|O|v>,   so  <v|O|v> = 0.

# Two electron atoms. Shift of ground state computed in perturbation theory,
and using variational method with effective nculear charge Z' as variational parameter.
Energy minimized when Z ' = Z - 5/16.