# Physical chemistry: Energy operator and eigenfunction

1. Jan 20, 2017

### ReidMerrill

1. The problem statement, all variables and given/known data
The energy operator for a time-dependent system is iħ d/dt. A possible eigenfunction for the system is
Ψ(x,y,z,t)=ψ(x,y,z)e-2πiEt/h
Show that the probability density is independent of time

2. Relevant equations
ĤΨn(x) = EnΨn

3. The attempt at a solution

I understand the concept of eigenfuntions but I don't really know which side of the equation I would apply the operator to or how it would prove anything about probability density

2. Jan 20, 2017

### BvU

Hi,

Your relevant equation isn't . At least not here.

You need an expression for the probability density in terms of the probability amplitude $\psi$ and when you work that out you'll see the time dependence cancels; that's all.

3. Jan 20, 2017

### blue_leaf77

I don't think a time-dependent system (e.g. time dependent potential) always has such a separable eigenfunction, let alone the existence of the eigenfunctions. Are you sure it was not "time-independent"?

4. Jan 20, 2017

### BvU

Agree with BL. Could you render the problem statement exactly as is ?

5. Jan 20, 2017

### ReidMerrill

That was taken directly from the assignment.

6. Jan 21, 2017

### BvU

Well, then I would claim that the given function is a solution and write down the probability density.

7. Jan 21, 2017

### ReidMerrill

I found something in the textbook that might be relevant

Â(x) Ψn(x,t) = an Ψn(x,t)
Â(x) Ψn(x) e-it(E/ħ)= an Ψn(x) e-it(E/ħ)
Â(x) Ψn(x) = an Ψn(x)
The time cancels out

8. Jan 21, 2017

### Staff: Mentor

9. Jan 21, 2017

### BvU

Yes, fine. First thing you need is an expression for the probability density in terms of the probability amplitude $\psi$. This is the third time I ask. Please answer by posting that expression.

10. Jan 21, 2017

### ReidMerrill

Probability density= / ψ/2dx

11. Jan 21, 2017

### BvU

Now fill in $\psi$ as given

PS wasn't the probability density just $|\psi^2|$ ?

12. Jan 21, 2017

### ReidMerrill

That's from my notes but you're right. The dx makes no sense. It's not written in a comprehensible way in my textbook.