A solution to time dependent SE but not Time independent SE?

In summary, the conversation discusses the relationship between time dependent and time independent solutions to the Schrodinger equation. It is explained that a wave function can be a solution to the time dependent equation but not the time independent equation, even if the same potential is used. This is because the time dependence factor makes the function different. It is also noted that the time independent equation only applies when separation of variables can be used, and that a solution to the time independent equation can be expressed in the form of e^(-iEt)y(x). Finally, it is clarified that a linear combination of separable states can be a solution to the time dependent equation, but not to the time independent equation.
  • #1
faen
140
0
A solution to time dependent SE but not Time independent SE??

How is it possible that a wave function is a solution to the time dependent schroedinger equation, but not to the time independent schroedinger equation (without time factors tacked on) with the same potential? I had this case on my quantum physics exam. I wrote that the time dependent set consisted of a linear combination of wavefunctions which were solutions from another potential which spanned hilbert space. But it still sounds contradicting that this function didnt fit into the time independent schroedinger equation when its basis functions span the space too.
 
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  • #2
well... yeah. if it is time dependent then it is not time-independent... so all time dependent solutions are not time-independent solutions.
 
  • #3
yea but without the time dependence factor (t=0), wouldn't they be time independent too?
 
  • #4
faen said:
yea but without the time dependence factor (t=0), wouldn't they be time independent too?

yes, but without the time-dependent factor then it is not the same function. in general the time-dependence doesn't factor.
 
  • #5
the time independant schrodinger equation only applies when you can use separation of variables, while you could theoretically apply it at t=0 and get basis states for that particular instant, this would be of little use as the time evolution would be completely different.

so if you have a solution to the time independant equation this means the solution can be put in the form of e^(-iEt)y(x) whereas you can't do this for a solution to the time dependant solution
 
  • #6
Asked my teacher and found out that linear combination of separable states as solution to SE don't apply to the time independent SE but only to the time dependent SE. Not even at t=0.
 
Last edited:

1. What is the difference between time dependent and time independent Schrödinger equations?

The time dependent Schrödinger equation (TDSE) describes the evolution of a quantum system over time, while the time independent Schrödinger equation (TISE) describes stationary states of a quantum system in which the energy does not change over time.

2. Why is a solution to the TDSE but not the TISE important?

Many physical systems, such as atoms and molecules, are time-dependent and therefore require the use of the TDSE to accurately describe their behavior. Additionally, understanding the TDSE allows for predictions of future behavior and the ability to manipulate quantum systems for practical applications.

3. What are some examples of systems that require a solution to the TDSE?

Some examples include time-dependent potentials, time-dependent electromagnetic fields, and time-dependent interactions between particles.

4. How is the TDSE solved?

The TDSE is typically solved using numerical methods such as finite difference or finite element methods, as analytical solutions are only possible for a limited number of simple systems.

5. What are the key challenges in solving the TDSE?

The main challenges in solving the TDSE include the high computational cost of numerical methods, the need for accurate initial conditions and boundary conditions, and the complex nature of many quantum systems which may require approximations or simplifications.

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