Time-independent Schrodinger equation in term of the TDSE

[rwooduk]

Homework Statement

Write down the general solution of the time-dependent Schrödinger equation in terms of the solutions of the time-dependent Schrödinger equation.

Homework Equations

TDSE
TISE

The Attempt at a Solution

I'm really not sure how to interpret this question, I could write the general solution to the time dependent Schrödinger equation:

\Psi (r,t) = \sum_{n} a_{n}(t) \Psi _{n} exp (-\frac{iE_{n}t}{\hbar})

insert this into the time dependent Schrödinger equation and show that it is satisfied, i.e. equal at both sides

OR

Does it want me to write ψ(r,t) = R(r)T(t) insert this into the time dependent Schrödinger equation and use separation of variables to show that there are two solutions, one with time and one with position dependance.

Thanks for any help interpreting what the question is asking me to do.

 

[BvU]

Your problem statement has independent twice.
I think what you did is fine if you reconsider the $a_n(t)$ :

you want to give expressions for them
if they are functions of t the whole idea of separation of variables is up the creek​

 

[rwooduk]

Your problem statement has independent twice.
I think what you did is fine if you reconsider the $a_n(t)$ :

you want to give expressions for them
if they are functions of t the whole idea of separation of variables is up the creek​

Yes apologies it should have read:

Write down the general solution of the time-dependent Schrödinger equation in terms of the solutions of the time-dependent Schrödinger equation.

So if i insert

\Psi (r,t) = \sum_{n} a_{n}(t) \Psi _{n} exp (-\frac{iE_{n}t}{\hbar})

into the time dependent Schrödinger equation and show that it is satisfied, do you think that's what the question is asking me to do?

also how would i expand a_n(t)?

many thanks for the reply

 

[BvU]

Still won't work because you want $i\hbar\;{\partial \Psi\over \partial t}=E\;\Psi$ and $da\over d t$ spoils that...

[edit] sorry, that's for the $\Psi_n$, not for $\Psi (r,t)$. Nevertheless, I think you don't want the $a_n$ time dependent. No time to dig up a reasonable argument now. Have to think that over.

 

[rwooduk]

Still won't work because you want $i\hbar\;{\partial \Psi\over \partial t}=E\;\Psi$ and $da\over d t$ spoils that...

[edit] sorry, that's for the $\Psi_n$, not for $\Psi (r,t)$. Nevertheless, I think you don't want the $a_n$ time dependent. No time to dig up a reasonable argument now. Have to think that over.

That's fine, I appreciate the help anyway. I will also think on it some more and post if I get anywhere. I look forward to any further ideas you or anyone else may have on this one.

 

[BvU]

The TISE comes about by separation of variables: split $\Psi ({\bf r},t) = f(t) \Psi ({\bf r})$. For f(t) the separated eqn becomes $\displaystyle {i\hbar\over f}\; {df\over dt} = E$.
Linearity of the SE allows coefficients $a_n$, but there is no room in f(t) for functions $a_n(t)$.

 

[rwooduk]

The TISE comes about by separation of variables: split $\Psi ({\bf r},t) = f(t) \Psi ({\bf r})$. For f(t) the separated eqn becomes $\displaystyle {i\hbar\over f}\; {df\over dt} = E$.
Linearity of the SE allows coefficients $a_n$, but there is no room in f(t) for functions $a_n(t)$.

ahh, ok so it's what I was thinking in the second part of the original post. That's great, thanks very much for the help!

 

[BvU]

Depending on where you are in the curriculum, there's more that can be loaded onto this exercise:
showing that any $\Psi({\bf r},0)$ can be written as such a summation (i.e. that the $\Psi_n$ form a basis),
the expressions for $a_n$, etc

 

[rwooduk]

Depending on where you are in the curriculum, there's more that can be loaded onto this exercise:
showing that any $\Psi({\bf r},0)$ can be written as such a summation (i.e. that the $\Psi_n$ form a basis),
the expressions for $a_n$, etc

We have done both, which is why I found the question so disambiguous, if both methods show the same thing I'll go with the separation of variables method.

thanks again