# Eigenvalue Quantization

### From FSUPhysicsWiki

The motivation for exploring eigenvalue quantization comes form wanting to solve the energy eigenvalue problem for a particle in a central potential. It is not possible, in general, to specify and measure more than one component of the orbital angular momentum. It is, however, possible to specify simulataneously with any one component of since commutes with all of its Cartesian components, as we saw earlier. We typically choose A central potential yields a Hamiltonian that commutes with and thus the energy eigenstates of the system may be chosen to be eigenvectors of and one component of usually

The quantization of angular momentum follows simply from the commutation relations derived earlier. Recall that is given by

Let us now define the operators, Note that and are Hermitian conjugates of each other.

We choose to work with these operators because, as we will see shortly, the operators function as raising and lowering operators for eigenvectors of

We may use the commutation relations derived earlier to show that

and

Therefore,

We may also show that

We may also easily see that

Let be a normalized eigenstate of with eigenvalue and of with eigenvalue Let us first determine what the effects of the operators, are. By definition,

If we now act on this expression from the left with and use the above commutation relations, we find that

We therefore see that is also an eigenvector of but with an eigenvalue of In other words,

where is a normalization constant. We see now that, as asserted earlier, the operators are raising and lowering operators for eigenstates of

To find the normalization constant, we will evaluate the norms of these states. The norm for is, using the expressions derived above,

The right-hand side of this expression is equal to If we now take to be real, then

Similarly, the norm for is

and thus, again taking to be real,

In summary,

We may now restrict the possible values of as follows. Obviously, it is impossible for to be larger than This means that

This implies that there must be both a lower bound and an upper bound on the allowed values of for a given value of Let be the upper bound on Then

This means that

or

The value of that we obtained is therefore a suitable value, since it is larger than the assumed upper bound on If we now consider the lowering operator, then, using this value of

We see by direct substitution that, if there is a lower bound, then it must be equal to In order for us to be able to lower a state to by repeated application of the lowering operator, then and must differ by an integer:

This means that the possible values of are quantized in half-integer steps; i.e.,

If we chose any other values of then it would be possible to construct states with arbitrarily low values of which is impossible. The allowed values of are all integers such that for an integer value of or all half-integers satisfying the same constraint if is a half-integer.

From this point forward, we will use the notation, for an eigenstate of with eigenvalue and of with eigenvalue The quantum number is often called the orbital quantum number and the magnetic quantum number, the latter being so named because it characterizes the energy shift of, say, an atom in the presence of a magnetic field.

## Problem

Evaluate the following expressions:

**(a)**

**(b)**