Computing Expectation values as functions of time.

[Ed Quanta]

Homework Statement

6) A particle in the infinite square well has the initial wave function
Ψ(x,0)= Ax when 0<=x<=a/2
Ψ(x,0)= A(a-x) when a/2<=x<=a

a) Sketch Ψ(x,0), and determine the constant A.
b) Find Ψ(x,t)
c) Compute <x> and <p> as functions of time. Do they oscillate? With what frequency?

Homework Equations

Schrodinger Equation, Equations for expectation values

The Attempt at a Solution

Part (a) wasn't a problem. I solved for the normalization constant A by setting the integral of the norm of the wave function equal to zero.
A = 2 * sqrt(3/a^3)

Part (b) also wasn't a problem as Ψ(x,t)=Ψ(x,0)exp(-iEt/h)

Part (c) confuses me a little. I know how to calculate <x> and <p> using the equations <x> = ∫ψ*(x)xψ(x)dx and <p> = ∫ψ*(x)(h/i)d/dx(ψ(x))dx. But the as functions of time confuses me. Won't the exp(-iEt/h) cancel out when multiplied by its conjugate. If this is true, then how can <x> and <p> oscillate? How could they oscillate with any frequency? Any help would be appreciated.

 

[Dick]

psi(x,0) is not an energy eigenfunction. So there is no single value of E to use for it. You'll have to represent it as a sum of energy eigenfunctions (a Fourier series) to determine its time evolution.

 

[Ed Quanta]

Ok. I wasn't expecting that. I am not sure what you mean. Do you mean a series that looks exp(-iE1t/h) + exp(-iE2t/h) + exp (-iE3t/h) +...

 

[JeffKoch]

What are the eigenfunctions of the infinite square well, and how would you represent your initial function Ψ(x,0) in terms of them?

 

[Ed Quanta]

I am not following at all. I don't really know what you are getting at. The only energy eigenfunction I have seen for the infinite square well is E=n^2(pi^2)(h^2)/(2ma^2).

 

[nrqed]

I am not following at all. I don't really know what you are getting at. The only energy eigenfunction I have seen for the infinite square well is E=n^2(pi^2)(h^2)/(2ma^2).

Those are the energies which are eigenvalues (of the Hamiltonian). Corresponding to each of those eigenvalue is an eigenfunction. What are they?

 

[Ed Quanta]

exp(-iEnt/h)

 

[Dick]

That's only the time dependence. How about the space dependence?

 

[nrqed]

exp(-iEnt/h)

That is just the time dependence. What is the x dependence?

 

[Ed Quanta]

I think it would be Ψ(x)=Asin(n*pi*x/a)

 

[nrqed]

I think it would be Ψ(x)=Asin(n*pi*x/a)

Right. *These* are the energy eigenfunctions, so only these wavefunctions vary in time with an exponential $e^{-iE_n t/ \hbar}$. You can't stick a time exponential to any function of x!

So first you must expand your wavefunction over those eigenfunctions using orthonormality. If you haven't seen an example of that in class, it will be long to explain and I have to go teach right now. Maybe I can help more later.

 

[nrqed]

That's only the time dependence. How about the space dependence?

We wrote almost exactly the same sentence!

 

[Ed Quanta]

Right. *These* are the energy eigenfunctions, so only these wavefunctions vary in time with an exponential $e^{-iE_n t/ \hbar}$. You can't stick a time exponential to any function of x!

So first you must expand your wavefunction over those eigenfunctions using orthonormality. If you haven't seen an example of that in class, it will be long to explain and I have to go teach right now. Maybe I can help more later.

Ok, so what happens to my initial wave function, Ψ(x,0)?

 

[Dick]

We wrote almost exactly the same sentence!

For that reason, I feel confident you can continue here. I've got to get to work.

 

[nrqed]

Ok, so what happens to my initial wave function, Ψ(x,0)?

You have to write
$$\Psi(x,0) = \sum_n c_n {\sqrt{\frac{2}{a}} sin(\frac{n \pi x}{a})$$
and find the the value of the c_n using orthonormlaity of the sin wavefunctions. Once you have the c_n , plug them back into
$$\Psi(x,t) = \sum_n c_n {\sqrt{\frac{2}{a}} sin(\frac{n \pi x}{a}) e^{-iE_n t/\hbar}$$
Your answer will have to stay this way..as an infinite sum.

Patrick

 

[Ed Quanta]

Thanks my main man.

 

[nrqed]

Thanks my main man.

You're welcome. The only difficulty is in finding th ec_n which will require doing an integral of the sin wavefunction times your original wavefunction (leaving "n" an arbitrary parameter).

I haven't addressed your question c) though but I have to go.

Best luck

Patrick

 

[Ed Quanta]

Ok, I think I'm cool with (b). Help with part (c) anyone?

 

[dextercioby]

Ok, I think I'm cool with (b). Help with part (c) anyone?

I don't see the problem, now that you have the $\Psi (x,t)$.