Plot the Expectation Value of Spin - Intro to Quantum Mechanics Homework

[Graham87]

Homework Statement

Problem e). See picture

Relevant Equations

See picture

I have found an answer to all of them (a-e) but I don’t know how to plot the function.

Thanks!

 

[vela]

It looks like you forgot to take the complex conjugate when writing down the adjoint.

But in any case, are you really asking how to plot a constant as a function of time?

 

[Graham87]

It looks like you forgot to take the complex conjugate when writing down the adjoint.

But in any case, are you really asking how to plot a constant as a function of time?

Aha ! Like this?

 

[vela]

Yes, if your original answer is correct, but like I said, I think you didn't calculate the adjoint correctly.

Also, doesn't $2\hbar$ seem awfully big for a spin-1/2 particle?

 

[malawi_glenn]

I can second that this is wrong

you need to take the complex conjugate here
Remember that the $\dagger$ (hermitian conjugate) in linear algebra means transpose $T$ and complex conjugate $*$, i.e.
a matrix $A$ when you do the hermitian conjugate, you do this $A^\dagger =( A^*)^T$

 

[Graham87]

Yes, if your original answer is correct, but like I said, I think you didn't calculate the adjoint correctly.

Also, doesn't $2\hbar$ seem awfully big for a spin-1/2 particle?

Aha, thanks. I get after correction $(2/3)\hbar$.

 

[PeroK]

Aha, thanks. I get after correction $(2/3)\hbar$.

Aren't you suspicious that the expectation value of spin in the x-direction is constant in a z-direction magnetic field?

 

[hutchphd]

Do you know the motion of a classical "rotating" charged particle in this situation? The correct solution is (perhaps surprisingly) similar.

 

[Graham87]

Aren't you suspicious that the expectation value of spin in the x-direction is constant in a z-direction magnetic field?

Aha, true. I think I get it. Should it look like a trigonometric function?
I will try again later.

 

[Graham87]

Do you know the motion of a classical "rotating" charged particle in this situation? The correct solution is (perhaps surprisingly) similar.

Not really. But I’m guessing something like sin(x)cos(y)?
I will try again later.
Thanks!

 

[PeroK]

Aha, true. I think I get it. Should it look like a trigonometric function?
I will try again later.

Also,the Hamiltonian drives the time evolution of the system. You should expect something to change over time in this case.

 

[malawi_glenn]

Aha, thanks. I get after correction $(2/3)\hbar$.

Still not correct.
Could you show how you did that calculation so maybe we can spot another error?

 

[Graham87]

Still not correct.
Could you show how you did that calculation so maybe we can spot another error?

I used Euler formula to convert e, but I get something messy. Might my d-answer be wrong too (see pic at beginning) ?

 

[PeroK]

That looks right, although if you had used the exponential form you would see that is equal to:
$$\frac{2\hbar}{9}\big (\cos(2\theta) - 2\sin(2\theta)\big )$$And note that $\theta$ is a function of $t$ here.

 

[Graham87]

That looks right, although if you had used the exponential form you would see that is equal to:
$$\frac{2\hbar}{9}\big (\cos(2\theta) - 2\sin(2\theta)\big )$$And note that $\theta$ is a function of $t$ here.

Ah! I got the same now. Big thanks !