Determine if a wavefunction is sharp or fuzzy in energy

[wood]

Homework Statement

For a particle of mass m in a one-dimensional infinite square well 0 < x < a, the normalised energy eigenfunctions ψ_n and eigenvalues E_n (integer n = 1, 2, 3, ...) are

$$ \psi_n(x) =\sqrt{\frac{2}{a}} sin \left( \frac{n \pi x}{a} \right) \;$$ inside the well otherwise Ψ_n(x)=0

Consider a state A with wave function ΨA(x,t) which at time t = 0 is given by
$$\Psi_A(x,0) = \frac {4}{\sqrt{5a}} sin^{3}\left( \frac{\pi x}{a} \right) for\; 0<x<a\; otherwise\; 0. $$

1. Is the state A sharp or fuzzy in energy?
2. Check that Ψ_A(x, 0) is normalised to one particle.
3. If you think A is sharp in energy, what energy does it have? Otherwise, what are the possible results of repeated energy measurements for the state A, and with what probability do they occur?
4. What is the formula for Ψ_A(x,t) for all times t?

Homework Equations

The Attempt at a Solution

The only part of this question I can confidently do is 2. I have checked this and it is normalised to one particle.

Going back to 1 I am pretty sure that the state A is fuzzy in energy. To check this I think I need Ψ_A(x, 0) to be an eigenfunction of the K energy operator. Without doing the second derivative I can be pretty sure that function is not coming back anytime soon and is therefore not an eignefunction...

If I am on the right path I think I then need to find <E>_A for part 3.

I would really appreciate it is anyone can help point me in the right direction on this.

Thanks

 

[Orodruin]

To check this I think I need ΨA(x, 0) to be an eigenfunction of the K energy operator.

Correct. If it has a definite energy, it will be an eigenstate of the Hamiltonian.

If I am on the right path I think I then need to find <E>A for part 3.

No, you do not need to compute the expected energy and in fact it will tell you very little about the mix of eigenvalues. Instead, you should be writing the wave function as a linear combination of different eigenfunctions.

 

[wood]

Thanks. I'm not sure how to write my wave function as a linear combination of eigenfunctions, not sure if we have seen that. Is there another way perhaps?

Ps nice quote in your sig.

 

[Orodruin]

Thanks. I'm not sure how to write my wave function as a linear combination of eigenfunctions, not sure if we have seen that. Is there another way perhaps?

You have a function ##\sin^3(\pi x/a)##. You need to write it as a linear combination of the functions ##\sin(\pi n x/a)##. How would you go about doing this?

Edit: Of course, you will have to care about the constants too, but let us start here.

 

[wood]

Are you saying I just separate the ^3 terms into three sin(nπx/a)

 

[Orodruin]

Are you saying I just separate the ^3 terms into three sin(nπx/a)

Yes ... or whatever number of sin(nπx/a) would be the correct one ...

 

[wood]

And that's where the constants come in?