# Orbital Angular Momentum Eigenfunctions

### From FSUPhysicsWiki

We will now find the orbital angular momentum eigenfunctions in terms of position. Recall from the previous section that

If we act on the left with a position eigenvector then this becomes

or, introducing

We may now separate out the dependence from the and dependences; i.e.,

Solving for the dependence, we obtain

We may now determine the dependence by using the fact that

In the position basis, the raising and lowering operators are given by

We thus obtain

Solving the above equation, we find that the full wave function is

where is an arbitrary function of Note that this function may (and, as we will see in the next chapter, does) depend on We may now find the wave functions by repeated application of It turns out to be

where

is an associated Legendre function. One may also write this as

where

are the spherical harmonics.

In the next chapter, we will be considering particles in central potentials, which are potentials that depend only on the distance of the moving particle from a fixed point, usually the coordinate origin. Since the resulting forces produce no torque, the orbital angular momentum is conserved. In quantum mechanical terms, this means that the angular momentum operator commutes with the Hamiltonian. Therefore, the results developed throughout this chapter will be very useful in discussing such potentials.

## Problems

**(1)** A system is initally in the state,

Find the possible results of a measurement of and the probabilities of finding each value.

**(2)** Classically, the Earth revolves around the sun counter-clockwise in the plane with the sun at the origin. Quantum mechanically, what is the minimum angle that the angular momentum vector of the earth can make with the axis? Ignore the intrinsic spin of the Earth. The orbital angular momentum of the Earth is Compare the minimum angle with that of a quantum particle with

**(3)** A plane rotator (i.e., a particle confined to move on a unit circle) is in a state with a wavefunction where is the azimuthal angle.

**(a)** Determine the normalization constant,

**(b)** Find the probability of measuring different values of the component of the angular momentum

**(c)** Find the expectation values of and