How Do You Determine the Time Evolution of a Wave Function Given \(\psi(x,0)\)?

[atomicpedals]

I'm getting bogged down in what is probably a very basic subject and it's holding me back. I'm not really sure how to determine the wave function \psi(x,t) given a function \psi(x,t=0); and since this is pretty much the under-pinning of every homework problem I've seen so far it's a huge issue for me. Can anyone explain, at least generally, how I get from one to the other?

 

[vela]

The Schrodinger equation tells you how the state evolves over time. Your textbook should cover this.

 

[atomicpedals]

The book does, but I'm coming up short with applying it beyond the limited example in the text.

So I understand (at least intellectually) for a particle in an infinite square well (between say 0 and a) I find the normalization constant and then the nth coefficient via integration from 0 to a. How do I go from this simple situation to the more general free particle of mass m? Is it simply a matter of adjusting my limits of integration?

(I promise, I'm not trying to get something for nothing with this question)

 

[vela]

Sorry, it's not clear at all to me what your question is. Could you please elaborate?

 

[atomicpedals]

Sorry, my own lack of clarity isn't helping...

The general solution can be found as a superposition of eigenfunctions

\psi(x,t)=\sumc_n\psi_ne^-i/\hbarEt

how do I apply this to a given situation? If I start with \psi(x,0)=e^{i/h px} , I should arrive at \psi(x,t)=e^{i/h (px-p2/2m t)}

how did I get from point a to point b?

 

[vela]

The Hamiltonian for a free particle is \hat{H}=\hat{p}^2/2m. Apply it to \psi(x,0), which is an eigenstate of \hat{H}, to find E.

 

[Benway_USMC]

Hmm, I agree with your explanation of Time Evolution of a Wave Function. I feel that any other explanation would be Shallow and Pedantic. Are you referring to a particle in a square well?

 

[mathfeel]

Sorry, my own lack of clarity isn't helping...

The general solution can be found as a superposition of eigenfunctions

\psi(x,t)=\sumc_n\psi_ne^-i/\hbarEt

how do I apply this to a given situation? If I start with \psi(x,0)=e^{i/h px} , I should arrive at \psi(x,t)=e^{i/h (px-p2/2m t)}

how did I get from point a to point b?

You are already there. If the time-independent Hamiltonian is H(\hat{p},\hat{x}) and its eigenstates are \varphi_n with energy E_n. Then each will evolve in time by only acquiring a phase factor related to its energy \varphi_n \to \varphi_n e^{-it E_n}.

So suppose you state start out at t=0 as some linear superposition of \varphi_n: \psi (0) = \sum_{n} c_n(0) \varphi_n, at time t later. Each of the states evolves independently: \psi(t) = \sum_{n} c_n(0) e^{-i t E_{n}} \varphi_n. Or another way of stating this, the coefficients of linear superposition change like this c_n(0) \to c_n (0) e^{-i E_n t}

The job is therefore to expand any general initial state into linear combination of energy eigenstates of the Hamiltonian.