Calculating Expectation Value for z component of angular momentum

[TLeo198]

Homework Statement

Calculate the expectation value for the z component of angular momentum (operator is (h/i)(d/dx)) for the function sinx*e^(ix).

Homework Equations

I think the only one relevant is the expectation value:
<a> = integral[psi*(a)psi] / integral[psi*psi] where psi* is the complex conjugate and a is the operator (in this case, the operator of the z component of angular momentum).

The Attempt at a Solution

I don't really know how to begin this one, but I assume that you have to find the <a> equation where <a> is the expectation value. In that case, do you just take the integral of psi*(a)psi over the integral of psi*psi? In this case, psi = sinx*e^(ix)

 

[kreil]

Calculate the expectation value for the z component of angular momentum (operator is (h/i)(d/dx)) for the function sinx*e^(ix).

The operator you wrote down is for momentum:

$$\hat p = \frac{\hbar}{i} \frac{\partial}{\partial x}$$

However, the angular momentum operator is different:

$$\hat L_z = -i \hbar \left (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \right )$$

I don't really know how to begin this one, but I assume that you have to find the <a> equation where <a> is the expectation value. In that case, do you just take the integral of psi*(a)psi over the integral of psi*psi? In this case, psi = sinx*e^(ix)

Yes. To be clear, you want to calculate the following:

$$<L_z> = \int \psi^*(x) \hat L_z \psi(x) dx$$

In order to use this definition, you will first have to normalize the wavefunction (so that the denominator in your expression is equal to 1)