Chapter 3: Angular Momentum

Orbital Angular Momentum Operator

Laplacian in spherical polar coordinates

Orbital Angular Momentum Operator in spherical coordinates

Spherical polar coordinates

Spherical polar coordinates

Spherical polar coordinates

Spherical polar coordinates

Eigenvalues and eigenfunctions are given by

z-component is given by

Both L^2 and L^z assume

one set of discrete values

In terms of p^ and r^

In terms of p^ and r^

In terms of p^ and r^

L^2 commutes with

L^x, L^y, and L^z and L^+ and L^-

Raising Operator

Lowering Operator

Raising and lowering operators are not

Hermitian

Raising and lowering operators do not

represent physical quantities

Raising and lowering operators do

raise the eigenfunctions of L^z up or down the “ladder” of eigenvalues L^z

What is the effective current, i, for a particle with charge e-, mass me, moving in an orbit at a radius of a0 at an angular velocity ω?

Magnetic moment associated with effective current, i, for a particle with charge e-, mass me, moving in an orbit at a radius of a0 at an angular velocity ω

Orbital angular momentum, **L**

gyromagnetic ratio

Bohr magneton

What happens when you apply a **B** field?

The energy levels split by ΔE

ΔE

What did Stern Gerlach experiment find?

• An atom’s orbital angular momentum is quantised

• some particles possess an intrinsic angular momentum, called spin

What is spin, S?

Intrinsic angular momentum, possessed by some particles (e.g. electrons, protons, neutrons)

What are the spin quantum numbers for an electron?

s = 1/2

ms = +/- 1/2

Does orbital angular momentum require that there is an odd or even number of eigenstates of L^z?

Odd → 2l+1

**Orbital** angular momentum has an extra (?) condition that the wave function must be a well behaved function of position. What does this require?

n must be even and so l must have an integer value

Does an electron spin?

No but it has intrinsic angular momentum which is called spin