We continue now with further examples of how Hermitian matrices and operators play a role in advanced physics. Of some use for the interested reader will be two previous blogs in which the concept of orbital angular momentum at the quantum-atomic level was introduced, including examples of problems based on it. See e.g.
For example, we can look at the orbital angular momentum vectors in the x and y -directions, given by L(x) and L(y). Since these are Hermitian, i.e.
L(x) = h/2π [σ_x ] = ħ [σ_x ] , and L(y) = h/2π [σ_y ] = ħ [σ_y ]
where ħ = h/2π and σ_x, σ_y are the Pauli spin matrices as given in the earlier blog, viz.
and h is the Planck constant (h = 6.62 x 10^-34 J-sec), then it follows that the forms:
(L(x) + i L(y)) and (L(x) - i L(y)) are Hermitan conjugates.
These mathematical facts can be used to obtain expressions in terms of the quantum numbers l and m (see the first two blog links), where we solve for the quantity ‖C^m‖:
‖C^m‖ = ħ [(l - m) (l + m + 1)]^½
To determine the allowed values of l and m one always begins with the fact that if one is given a wave function, U (m,l) one can always generate a wave function U(m+1), l or U(m-1),l by operating respectively with (L(x) + i L(y)) or (L(x) - i L(y)). If m and m' then refer to spin quantum numbers in columns and row rspectively and d references diagonals, then:
(L(x) + i L(y)) m.m' = ħ [(l - m') (l + m' + 1)]^½ dm,m'+1
(L(x) - i L(y)) m.m' = ħ [(l - m) (l + m + 1)]^½ dm',m+1
The case of l = ½ is especially easy to treat and we get:
(L(x) + i L(y)) m.m' = ħ [(½ - m') (½ + m' + 1)]^½ dm,m'+1
= ħ x
(L(x) - i L(y)) m.m' = ħ [(½ - m) (l½ + m + 1)]^½ dm',m+1
= ħ x
1) Solve for σ_x, σ_y and σ_z in terms of the orbital angular momenta, L(x), L(y) and L(z)
2) Solve for the spin-momentum operators s(x), s(y) and s(z) in terms of σ_x, σ_y and σ_z. (State any assumptions made)
3) Show that the TOTAL angular momentum vector, J(z) =
(ħ/i (@/@φ )+ ħ/2 ..... 0)
(0 ....... ħ/i (@/@ φ ) - ħ/2)
displays a Hermitian matrix, where φ is a spherical angle and @/@ φ denotes a partial derivative with respect to it.
4) Show by direct computation that:
σ_x σ_y + σ_y σ_z =
5) Compute the matrices for (L(x) + i L(y)) m.m' and (L(x) - i L(y)) m.m' if the angular momentum quantum number l = -1.