QHO: Time dependant expectation value of the potential energy

[phb1762]

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!

 

[PeterDonis]

Moderator's note: Thread moved to advanced physics homework.

@phb1762 please show what attempts you have made at a solution, and not by posting an image. PF rules require you to post your work directly, using the PF LaTeX feature for equations:

https://www.physicsforums.com/help/latexhelp/

 

[Abhishek11235]

Do you know Virial Theorem in Classical Mechanics and Ehrenfest Theorem in Quantum Mechanics?

 

[PeroK]

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:
$$\hat V = \frac 1 2 m \omega^2 \hat x^2$$