**Quantum Mechanics and Momentum**

Part 10: Ladder Operators

Last week, we introduced the quantum angular momentum operator **L**, and found some properties of its magnitude and components. Most notably, that only one component may be fixed at a time, and that component, usually the *z* component, and the magnitude squared may be fixed simultaneously. So now, let us look at the common eigenfunctions of *L*^{2} and *L*_{z}. We want eigenfunction *Y* so that and .

Now, let us define two “ladder operators” *L*_{+} and *L*_{–} by

Let us examine the operation of *L*_{+} on our *L*_{z} eigenvalue equation:

Now, from the definition of the commutator,

,

and from our definition of *L*_{+},

.

So , and

,

which tells us that is also an eigenstate of *L*_{z}, with “raised” eigenvalue . Thus, we call *L*_{+} the “raising operator.” Repeated application of this shows that for *n* applications of the raising operator,

Similarly, we can show that gives

,

and so it is called the “lowering operator.”

Now, let us examine how *L*^{2} behaves on these raised an lowered eigenstates. The commutators of *L*^{2} with our raising and lowering operators are

,

and so

,

and the ladder of eigenvalues …b-2ℏ, b-ℏ, b, b+ℏ b+2ℏ,… generated by the raising and lowering operators are all eigenstates of *L*^{2} with eigenvalue *a*.

We have eigenfunctions , and , with eigenvalues .

Thus, .

Subtracting this from , and recalling that , we find:

.

Note that the middle term above corresponds to a non-negative physical quantity; thus we expect that cannot be negative: . This, in turn, means that our ladder must be bounded both above and below; there exists and such that and .

So then we have . But

.

Therefore:

Analogously, using , and , one finds

.

Taking the difference of these two equations, and thus cancelling *a*, one finds:

,

as .

Combine this with the knowledge that all eigenvalues on the ladder are separated by units of ℏ, we see , for some integer *n*, and so

, ,

giving us our eigenvalue ladder .

And as , we find

.

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Tags: Angular Momentum, Eigenfunction, Eigenvalue, Friday Physics, Ladder Operators, physics, Quantum Mechanics

This entry was posted on January 2, 2009 at 12:13 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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