QHO: Time dependant expectation value of the potential energy

In summary: where $m$ is the mass of the particle, $\omega$ is the speed of the particle, and $\hat x$ is the position vector of the particle.
  • #1
phb1762
6
0
Homework Statement
Calculating expectation values
Relevant Equations
$$<\lambda(t)> = \int_{-\infty}^{\infty}\Psi^*(x,t)\hat{O}\Psi(x,t)dx$$
Summary:: Linear Quantum harmonic oscillator and expectation value of the potential energy (time dependent)

Hello, I have attached a picture of the full question, but I am stuck on part b). I have found the expectation value of the <momentum> and the <total energy> However I am struggling with finding the expectation value of the potential energy. I’m not too sure what the operator would be.

My workings out to b) for the expectation of the total energy:

$$<E(t)> = (\frac{1}{\sqrt{2}})^2 * \frac{\hbar w}{2} + (\frac{1}{\sqrt{2}})^2 * \frac{3\hbar w}{2}$$
so:
$$<E(t)> = \frac{1}{2} * \frac{\hbar w}{2} + \frac{1}{2} * \frac{3\hbar w}{2}$$
so:
$$<E(t)> = \frac{1}{4} * \hbar w + \frac{3}{4} * \hbar w$$
so
$$<E(t)> = \hbar w$$
I know i have to use:

$$<\lambda(t)> = \int_{-\infty}^{\infty}\Psi^*(x,t)\hat{O}\Psi(x,t)dx$$,
for any observable $$\lambda$$, but what operator do I use for the potential energy?
Is there a shorter method?
I also know that $$<E(t)> = <T(t)> + <V(t)>$$,
where $$<T(t)>$$ is the kinetic energy.

I’m fairly new to quantum mechanics, as I only just started doing it, so any help would be greatly appreciated. I just want to learn the proper method of doing these types of questions as I expect similar ones will be on the summer exams! Thank you!
 

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  • #3
Do you know Virial Theorem in Classical Mechanics and Ehrenfest Theorem in Quantum Mechanics?
 
  • #4
phb1762 said:
Homework Statement:: Calculating expectation values
Relevant Equations:: $$<\lambda(t)> = \int_{-\infty}^{\infty}\Psi^*(x,t)\hat{O}\Psi(x,t)dx$$

Summary:: Linear Quantum harmonic oscillator and expectation value of the potential energy (time dependent)

Hello, I have attached a picture of the full question, but I am stuck on part b). I have found the expectation value of the <momentum> and the <total energy> However I am struggling with finding the expectation value of the potential energy. I’m not too sure what the operator would be.

My workings out to b) for the expectation of the total energy:

$$<E(t)> = (\frac{1}{\sqrt{2}})^2 * \frac{\hbar w}{2} + (\frac{1}{\sqrt{2}})^2 * \frac{3\hbar w}{2}$$
so:
$$<E(t)> = \frac{1}{2} * \frac{\hbar w}{2} + \frac{1}{2} * \frac{3\hbar w}{2}$$
so:
$$<E(t)> = \frac{1}{4} * \hbar w + \frac{3}{4} * \hbar w$$
so
$$<E(t)> = \hbar w$$
I know i have to use:

$$<\lambda(t)> = \int_{-\infty}^{\infty}\Psi^*(x,t)\hat{O}\Psi(x,t)dx$$,
for any observable $$\lambda$$, but what operator do I use for the potential energy?
Is there a shorter method?
I also know that $$<E(t)> = <T(t)> + <V(t)>$$,
where $$<T(t)>$$ is the kinetic energy.

I’m fairly new to quantum mechanics, as I only just started doing it, so any help would be greatly appreciated. I just want to learn the proper method of doing these types of questions as I expect similar ones will be on the summer exams! Thank you!

The expected value of potential energy is, by definition, the expected value of the potential operator. In this case the potential operator is given by:
$$\hat V = \frac 1 2 m \omega^2 \hat x^2$$
 

Related to QHO: Time dependant expectation value of the potential energy

1. What is QHO?

QHO stands for Quantum Harmonic Oscillator. It is a model used in quantum mechanics to describe the behavior of a particle in a harmonic potential.

2. What is the time dependant expectation value of the potential energy?

The time dependant expectation value of the potential energy is a measure of the average potential energy of a particle in the QHO model over a certain period of time. It takes into account the uncertainty in the particle's position and momentum.

3. How is the time dependant expectation value of the potential energy calculated?

It is calculated by taking the integral of the potential energy operator over the wave function of the particle. This integral is then multiplied by the complex conjugate of the wave function and normalized to account for the uncertainty in the particle's position and momentum.

4. What is the significance of the time dependant expectation value of the potential energy?

The time dependant expectation value of the potential energy is significant because it allows us to make predictions about the behavior of a particle in the QHO model over time. It gives us insight into the average potential energy of the particle and how it changes over time.

5. How does the time dependant expectation value of the potential energy relate to the uncertainty principle?

The time dependant expectation value of the potential energy is related to the uncertainty principle in that it takes into account the uncertainty in the particle's position and momentum. This means that the more accurately we know the potential energy of the particle, the less accurately we can know its position and momentum, and vice versa.

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