# 1d free particle. How do I find the solution to the DE?

1. Feb 20, 2013

### richyw

1. The problem statement, all variables and given/known data

Sorry if this should be in intro section. I'm not sure the math required. So I have
$$\frac{d^2\psi}{dx^2}=-\frac{2mE}{\hbar^2}\psi$$

I would like to see how I can get the solution that my book gives me which I know is
$$\psi(x)=A\sin kx+B\cos kx$$

where A and B are constant and k is 2mE/h^2
2. Relevant equations

shown above

3. The attempt at a solution

Is this easy? I am rusty on my differential equations and keep getting the wrong answer. Would it be better to split it into a system of differential equations?

2. Feb 20, 2013

### Mute

How are you trying to solve it? It would help if you show us some work so we can get an idea of where you're going wrong.

To give you some hints, you probably don't need to split it up into two first order differential equations. Does the term 'characteristic polynomial' ring any bells? That's the typical method used to solve an autonomous ODE like this. (You'll have to be familiar with complex numbers to use this method)

3. Feb 20, 2013

### richyw

I just searched my textbook from my intro ODE course and characteristic polynomial is mentioned once in the systems of ODE section. I remember what a characteristic equation is from linear algebra...

4. Feb 20, 2013

### richyw

ok I think I know where I went wrong. I accidentaly wrote down y''=-ky'. Ill try it again real quick.

5. Feb 20, 2013

### richyw

aha, that was my problem HAHA.

just out of curiosity in this case would you call the "characteristic polynomial" the part that you have to solve for zero? like in this case $(\lambda^2+k)$

6. Feb 20, 2013

### richyw

oh my god :O, If I write it as a system it IS the characteristic polynomial from linear algebra!

7. Feb 20, 2013

### richyw

thank you very much for your help!