Physical chemistry: Energy operator and eigenfunction

[ReidMerrill]

Homework Statement

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

Homework Equations

ĤΨ_n(x) = E_nΨ_n

The Attempt at a Solution

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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

 

[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.

 

[blue_leaf77]

time-dependent system

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"?

 

[BvU]

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

 

[ReidMerrill]

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

That was taken directly from the assignment.

 

[BvU]

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

 

[ReidMerrill]

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

I found something in the textbook that might be relevant

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

 

[Borek]

Much easier to use LaTeX for math. Just saying.

https://www.physicsforums.com/help/latexhelp/

 

[BvU]

I found something in the textbook that might be relevant

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

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.

 

[ReidMerrill]

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.

Probability density= / ψ/²dx

 

[BvU]

Now fill in $\psi$ as given

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

 

[ReidMerrill]

Now fill in $\psi$ as given

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

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.