Time-independent Schrodinger equation in term of the TDSE

In summary, the question asks for the general solution of the time-dependant Schrodinger equation in terms of the solutions of the time-independant Schrodinger equation. If the solution to the time-dependant Schrodinger equation is inserted into the solution to the time-independant Schrodinger equation, then the question asks for the solution of the time-dependant Schrodinger equation. However, if the expressions for the ##a_n## time dependent terms are not allowed in the solution to the time-independant Schrodinger equation, then the question asks for the solution of the time-independent Schrodinger equation.
  • #1
rwooduk
762
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Homework Statement


Write down the general solution of the time-dependant schrodinger equation in terms of the solutions of the time-independant Schrodinger 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 dependant schrodinger equation:

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

insert this into the time dependant schrodinger 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 dependant schrodinger 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.
 
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  • #2
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​
 
  • #3
BvU said:
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-dependant schrodinger equation in terms of the solutions of the time-independant Schrodinger equation.

So if i insert

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

into the time dependant schrodinger 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 an(t)?

many thanks for the reply
 
  • #4
Still won't work because you want ##i\hbar\;{\partial \Psi\over \partial t}=E\;\Psi## and ##da\over d t## spoils that...

[edit] :oops: 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.
 
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  • #5
BvU said:
Still won't work because you want ##i\hbar\;{\partial \Psi\over \partial t}=E\;\Psi## and ##da\over d t## spoils that...

[edit] :oops: 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.
 
  • #6
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)##.
 
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  • #7
BvU said:
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!
 
  • #8
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
 
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  • #9
BvU said:
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
 

What is the Time-independent Schrodinger equation?

The Time-independent Schrodinger equation (TISE) is a fundamental equation in quantum mechanics that describes the behavior of a quantum system as a function of time. It is a differential equation that relates the energy of a system to its wave function.

What is the difference between the Time-independent Schrodinger equation and the Time-dependent Schrodinger equation?

The Time-independent Schrodinger equation is used to find the stationary states of a quantum system, while the Time-dependent Schrodinger equation is used to describe the time evolution of a system. TISE does not take into account the changing external factors, while TDSE does.

What is the significance of the Time-independent Schrodinger equation in quantum mechanics?

The Time-independent Schrodinger equation is a cornerstone of quantum mechanics and is used to predict the behavior of quantum systems. It allows us to calculate the energy levels and wave functions of a system, which are essential for understanding various physical phenomena.

How is the Time-independent Schrodinger equation solved?

The Time-independent Schrodinger equation is solved using various mathematical techniques, such as separation of variables, perturbation theory, and numerical methods. The solution depends on the potential energy function of the system and the boundary conditions.

What are the limitations of the Time-independent Schrodinger equation?

The Time-independent Schrodinger equation is limited to describing non-relativistic systems and cannot fully explain the behavior of particles at high energies. It also cannot account for interactions between particles, making it unsuitable for describing complex systems. However, it remains a valuable tool for understanding the behavior of quantum systems in many cases.

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