Write down the eigenvalue equation

[Exulus]

Hi guys, I've been given this question as part of my homework assessment however i don't even know what its asking me to solve :( I am sure you have to apply it to a certain equation but it doesn't say what! The question is:

"Write down the eigenvalue equation for the total energy operator:

$$E_{tot} = i\hbar\frac{d}{dt}$$

and solve this equation for both the eigenfunctions and the eigenvalues." (it should be a partial differential but i can't find the right symbol for it).

If someone could give me a hint as to what this is going on about it would be greatly appreciated! We've only really just touched on operators/eigenfunctions etc, I've answered other questions on the same paper about expectation values of momentum and position for a harmonic oscilator and to show its consistent with the Heisenberg uncertainty principle, which i have done. Thanks in advance :)

 

[Exulus]

Hmm...at a guess it means something like this:

$$i\hbar\frac{d\psi(x,t)}{dt} = E\psi(x,t)$$

But that seems too obvious? And how do i solve that without knowing what the wavefunction is?

 

[sporkstorms]

Hmm...at a guess it means something like this:

$$i\hbar\frac{d\psi(x,t)}{dt} = E\psi(x,t)$$

But that seems too obvious? And how do i solve that without knowing what the wavefunction is?

That is indeed an eigenvalue equation. However, you'd generally write it as just $$\tilde{E}_{tot} \psi(x,t) = E\psi(x,t)$$. You can then expand it further as you did (it's just when i think of an eigenvalue equation, i like to see the operator).

Just FYI, in most books and most online resources, you'll see the "total energy operator" called the Hamiltonian, and written as $$\tilde{H}$$

And to fully solve it, you're right - of course you'd need to know the wave equation. Otherwise the best you can do is write a general expression like you did.

 

[sporkstorms]

(it should be a partial differential but i can't find the right symbol for it).

\partial: $$E_{tot} = i\hbar\frac{\partial}{\partial t}$$

 

[Exulus]

Thanks sporkstorms :) In that case i think the question is worded very badly...as it doesn't mention what wavefunction to use!

The only wavefunction that is mentioned in the entire assessment is in question 1 which is for the harmonic oscillator:

$$u_{0}(x) = ({\frac{\mu\omega}{\pi\hbar}})^\frac{1/4}$$$$exp(\frac{-\mu\omega x^2}{2\hbar})$$

Which doesn't have any time dependence

 

[sporkstorms]

Which doesn't have any time dependence

Well, a harmonic oscillator is a harmonic oscillator because of a specific type of potential applied to it. Therefore, your Hamiltonian is not going to be the same as it is in the other problem.

Typically (or always?), for 1-d it's going to be:
$$\tilde{H}=\frac{\tilde{p}^2}{2m} + \frac{1}{2}m\omega^2\tilde{x}^2$$
Or using the raising and lowering operators:
$$\tilde{H}=\hbar\omega(\tilde{a}^{\dagger}\tilde{a} + \frac{1}{2})$$

If this is the first time you're seeing it, let me point out that solving the quantum harmonic oscillator is no small task.
I've gone through it twice (two methods - once with operators, once without), and still don't fully understand it.

 

[Exulus]

Yeah, we have done it, but a very long time ago and used lots of substitutions and stuff. What gets me is that surely can't be right, because the question is only worth 5 marks. Infact, i didnt even type out the full question..the rest of it is:

"Comment on how your solution relates to de Broglie waves for free particles, and how it is used for transforming the TDSE into the TISE form."

Well strange!

 

[sporkstorms]

And to fully solve it, you're right - of course you'd need to know the wave equation. Otherwise the best you can do is write a general expression like you did.

I figured i should further qualify this, in case it's not apparent. You can break that expression down further, and into finer parts as well.

Specifically, you can expand the wave equation in terms of the |n> basis. That is, you can write it as a sum,
$$|\psi, t\rangle = \sum_n c_n(t)|n\rangle$$
plug it into the S.E. to find the coefficients (should come out as exp's), etc. and eventually find an expression for E.
Any textbook should work through that.

Notice your Hamiltonian has no V(x) - that is, it's for a free particle.

Or if you want to work in position space, you can write out the full S.E., and "guess" (or if you're like me, take a textbook author's word for it, and just remember it) a solution of the form A*exp(b*x)*exp(c*t) and work from there.