# Calculating Expectation Value for z component of angular momentum

• TLeo198
In summary, to calculate the expectation value for the z component of angular momentum for the function sinx*e^(ix), you need to use the formula <L_z> = \int \psi^*(x) \hat L_z \psi(x) dx after normalizing the wavefunction. The operator for angular momentum is different from the one provided in the problem and is given by \hat L_z = -i \hbar \left (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \right ).
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)

TLeo198 said:
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)

## 1. What is the formula for calculating the expectation value for the z component of angular momentum?

The formula for calculating the expectation value for the z component of angular momentum is:

⟨Lz⟩ = ∫Ψ* Lz Ψ dτ

Where Ψ* is the complex conjugate of the wavefunction and dτ represents the volume element in three-dimensional space.

## 2. How do you interpret the expectation value for the z component of angular momentum?

The expectation value for the z component of angular momentum represents the average value of the observable quantity of the z component of angular momentum for a given quantum state. It is a measure of the most likely outcome if the measurement is repeated multiple times.

## 3. Can the expectation value for the z component of angular momentum be negative?

Yes, the expectation value for the z component of angular momentum can be negative. This means that the most probable outcome for the measurement of the z component of angular momentum will be a negative value. However, it is important to note that the square of the expectation value (|⟨Lz⟩|²) represents the average squared value, which is always positive.

## 4. How does the wavefunction affect the expectation value for the z component of angular momentum?

The wavefunction directly affects the expectation value for the z component of angular momentum. The wavefunction describes the probability amplitude for the quantum state, and the expectation value is calculated using the wavefunction. A different wavefunction will result in a different expectation value for the z component of angular momentum.

## 5. Can the expectation value for the z component of angular momentum change over time?

Yes, the expectation value for the z component of angular momentum can change over time. This is because the wavefunction, and therefore the expectation value, is dependent on time. As the quantum state evolves over time, the expectation value for the z component of angular momentum may also change.

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