Angular Momentum Operators

• In classical physics the angular momentum of a particle is defined by

• We will use this, together with the momentum operators, to construct the corresponding quantum mechanical operators for the components of L.

• Substituting for the momentum operators

one can show by direct calculation that

• As we have seen, the non-commutativity of these operators means that in general no two components of L can be known simultaneously with infinite precision. (The only exception is that they can all be zero simultaneously.)

• On the other hand, one shows that

where

• Thus each component commutes with L², so the square of the angular momentum and any component can be simultaneously known.

Angular Momentum in Spherical Coordinates

• We have written the operators for the components of L in terms of cartesian coordinates. But it is much more useful to have them expressed in spherical coordinates.

• To make the direct transformation from (x,y,z) to (r,q,f) is possible but algebraically tiresome. We will sketch the procedure for the simplest one, which turns out to be L_z.

• First we write the spherical  set as functions of the cartesian set:

• Then we use the chain rule. For example

• After working all this out, we look at

• After a bit of algebra we find a simple answer:

• The other components of L have mor complicated operators. For this reason, one usually chooses to treat L_z as the one component that is precsely known.

• For L² the answer is not so simple, but it is important to us:

Central Forces and Angular Momentum

• The above expression for L² appeared earlier in the Hamiltonian for central forces.

• This is easy to understand from classical physics. The momentum of the particle can be decomposed into components along r and perpendicular to r. We call the former the "radial momentum" and denote it by p_r; the latter (denoted by p_^) is related to the angular momentum by L = rp_^.

• The kinetic energy then becomes

• The total energy is thus

• Now the TISE for this kind of situation was found earlier to be

• We see that the left side of this is exactly the quantum mechanical equivalent of the classical energy above. The first term on the left involves the square of the operator for radial momentum, which is

• (This is interpreted as multiplication by r, followed by the other operations.) This operator has the same commutation relation with r as p_x has with x:

• Earlier we had for the angular part of the separated solutions of the TISE:

• Referring to the expression above for L² we see that this equation is the same as

Thus the angular part of the separated solution of the TISE is an eigenfunction of L². It is important that we know what those eigenfunctions and eigenvalues are.

Eigenfunctions and Eigenvalues

• Since L² and L_z commute, they can be known simultaneously. This means that the system can be in a state described by a function that is an eigenfunction of both operators. We will look for such functions.

• We start with requiring the function to be an eigenfunction of L_z, since that operator is very simple. We write

where a is a real constant. The solution of this is



where m is related to the eigenvalue a  by

• Now we bring in a physical requirement: The wavefunction must be single-valued.This means that if we add 2p to f (which brings us to the same point in space) the wavefunction must be the same. But

This is the same only if m is an integer.

• The eigenvalues of L_z are thus restricted to

• This is a remarkable new property of angular momentum in quantum mechanics: it is quantized in units of h/2p (as Bohr guessed). Actually, this is the property of a component of L, not of the magnitude.

• We thus have that the simultaneous eigenfunction of L² and L_z has the form

• We substitute this in the original eigenvalue equation for L² and obtain

• This must be solved for P(q). It is one form of Legendre's equation, known from the 18th century.

• There are two solutions for any values of K and m. But most of the solutions become infinite for q either 0 or p. These are therefore unacceptable as a part of a wavefunction.

• There are well-behaved solutions, however, for special values of K and m. They take the form of polynomials in sinq and cosq. The special conditions on the constants are these:

• Since the eigenvalue of L² is (h/2p)²K, we have found the following restrictions on the values of the angular momentum magnitude and one of its components:

• These are general properties of (orbital) angular momentum, and have nothing to do with what system is being considered.

• Note that the magnitude of L is

while the maximum value of L_z is

This means that the magnitude of L is greater than any component can be. The reason is this: If L_z were the same as the magnitude of L, then both L_x and L_y would have to be zero. But no two components can be simultaneously known, so they cannot both be zero. (The only exception is when all components are zero, i.e., L is zero.)

• The eigenfunction we are seeking thus depends on the two quantum numbers, so we denote it by

• The polynomials P(q) are called "Associated Legendre Polynomials" and are tabulated. Some are given in the text.

• This analysis give a complete account of (orbital) angular momentum. We can use it wherever it is appropriate, which is usually in cases where angular momentum is conserved. That is the case for all central forces.

• We will find later that there is another kind of angular momentum, called spin, that has nothing to do with the particle's location or momentum, and is entirely non-classical. The rules for the eigenvalues, however, are very similar to the ones here.