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 Schrödinger 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)=$\sum$c_n$\psi$_ne^-i/$\hbar$Et

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)=$\sum$c_n$\psi$_ne^-i/$\hbar$Et

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.