The expected values of position and momentum are given by
In general, any dynamic variable, , can be expressed in terms of position and momentum, i.e., .
Mathematically, A is a self-adjoint operator on a Hilbert space. In the most commonly used case in quantum mechanics, σ is a pure state, described by a normalized vector ψ in the Hilbert space. The expectation value of A in the state ψ is defined as
If dynamics is considered, either the vector ψ or the operator A is taken to be time-dependent, depending on whether the Schrödinger picture or Heisenberg picture is used. The time-dependence of the expectation value does not depend on this choice, however.
This expression is similar to the arithmetic mean, and illustrates the physical meaning of the mathematical formalism: The eigenvalues aj are the possible outcomes of the experiment, and their corresponding coefficient is the probability that this outcome will occur; it is often called the transition probability.
A particularly simple case arises when A is a projection, and thus has only the eigenvalues 0 and 1. This physically corresponds to a “yes-no” type of experiment. In this case, the expectation value is the probability that the experiment results in “1”, and it can be computed as
You need the wavefunction to find the expectation value. If psi is your wavefunction, then the expectation value:
<Lx> = <psi*| Lx | psi>
If you weren’t given a wavefunction, grab your Libboff or Griffiths and look it up. Particle in a box? In a well? Quantum dot? They must have given you a clue, because you can’t find the expectation value without a state to apply it to.
Once you have a state, use the raising and lowering version of Lx to do the computation.
For Lx | l m>? OK, that’s easier. Lx translates to L+ and L-, as you already have.
First, split it up.
< l m | (1/2)(L+ + L-) | l m >
= (1/2) ( < l m | L+ | l m > + < l m | L – | l m> )
Then apply to the right first.
< l m | L+ | l m > = < l m | sqrt[ l ( l+1) – m(m+1) ]*hbar | l m+1 >
= sqrt(that stuff)*hbar <l m | l m+1> =0
and so on. Check the signs inside the sqrt, I may have reversed one. But if you’re applying it between the same l and m states, you’ll get zero.