Hermitian Matrices and Orbital Angular Momentum

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.
http://brane-space.blogspot.com/2010/07/space-quantization-further-simple-qm.html

and

http://brane-space.blogspot.com/2010/07/solution-of-quantum-mechanics-problems.html

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.

http://brane-space.blogspot.com/2012/02/uses-of-hermitian-matrices-in-physics.html

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

and

(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

(0...1)
(0....0)

and:

(L(x) - i L(y)) m.m' = ħ [(½ - m) (l½ + m + 1)]^½ dm',m+1

= ħ x

(0...0)
(1....0)

Problems:

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 =

(i……….i)
(i………-i)

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.