Given a wave function at t=0, how do you find the wave function at time t?

In summary, to find \psi(r,t) for a spherically propogating shell containing N neutrons with initial state \psi(r,0), we can apply the time-independent Schrodinger equation and add a factor of e^{-i\omega*t} to account for time evolution. This will work if the state is an energy eigenstate, but if it is a linear combination of energy eigenstates, we will need to multiply each term by a phase factor and it may be difficult to obtain a closed formula.
  • #1
Demon117
165
1
I am given the following:

A spherically propogating shell contains N neutrons, which are all in the sate

[tex]\psi[/tex](r,0)=4[tex]\pi[/tex]i[tex]j_{1}[/tex](kr)(3/[tex]\sqrt{34}[/tex][tex]Y^{0}_{1}[/tex]+5/[tex]\sqrt{34}[/tex][tex]Y^{-1}_{1}[/tex])

at t = 0.

How do we find [tex]\psi[/tex](r,t)?

My attempt:

I have a few thoughts; could you apply the time-independent schrodinger equation to find the energy of the state? If that is the case then you would simply tack on the factor of [tex]e^{-i\omega*t}[/tex]. Then you would know that [tex]\hbar*\omega[/tex]=E. . . . right?
 
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  • #2
I think that should do it. With the TISE, and the TDSE factor, I think you can it.
 
  • #3
This will do if your state is energy eigenstate. If it is a linear combination of energy eigenstates, then you will have to multiply each term by the appropriate phase factor. In this case summation of the new series to get a closed formula may not be easy.
 

1. What is a wave function?

A wave function is a mathematical representation of a quantum system that describes the probability of finding a particle at a given location and time.

2. How is a wave function at t=0 different from a wave function at any other time?

The wave function at t=0 represents the initial state of a quantum system, while the wave function at any other time represents the state of the system at that specific time.

3. What information do you need to find the wave function at time t?

To find the wave function at time t, you need the initial wave function at t=0, as well as the time evolution operator, which describes how the wave function changes over time.

4. Can the wave function at any time t be predicted with 100% accuracy?

No, according to the principles of quantum mechanics, the wave function at any time t can only be predicted with a certain level of uncertainty. This is due to the probabilistic nature of quantum systems.

5. How is the wave function at time t used in practical applications?

The wave function at time t is used in many practical applications, such as predicting the behavior of atoms and molecules, designing quantum computing algorithms, and understanding the behavior of subatomic particles. It is also used in fields such as quantum mechanics, quantum chemistry, and materials science.

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