# Separate variables schrodinger equation

1. Jun 1, 2013

### Cogswell

1. The problem statement, all variables and given/known data
[2 Dimensional infinite square well]
Show that you can separate variables such that the solution to the time independent schrodinger equation, $\hat{H} \psi (x,y) = E \psi (x,y)$ can be written as a product state $\psi (x,y) = \phi (x) \chi (y)$ where $\phi (x)$ is a function of only the x coordinate and $\chi(y)$ is a function of only the y coordinate.

2. Relevant equations

$$\hat{H} = -\dfrac{\hbar^2}{2m} \left( \dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2} \right) + V(x,y)$$

3. The attempt at a solution

So... putting in $\psi (x,y) = \phi (x) \chi (y)$ I'll get:

$$\hat{H} \phi (x) \chi (y) = -\dfrac{\hbar^2}{2m} \left( \dfrac{\partial^2 \phi (x)}{\partial x^2} \chi (y) + \dfrac{\partial^2 \chi (y)}{\partial y^2} \phi (x) \right) + V(x,y) \phi (x) \chi (y)$$

And then dividing both sides by $\phi (x) \chi (y)$ I'll get:

$$\hat{H} = -\dfrac{\hbar^2}{2m} \left( \dfrac{1}{\phi (x)} \dfrac{\partial^2 \phi (x)}{\partial x^2} + \dfrac{1}{\chi (y)} \dfrac{\partial^2 \chi (y)}{\partial y^2} \right) + V(x,y)$$

And then... I don't really know where to go from here. I don't get what I'm supposed to do next.

2. Jun 1, 2013

### voko

What is the explicit form of V(x, y)? Can it be written as a sum X(x) + Y(y)?

3. Jun 1, 2013

### Cogswell

For a 2D potential in the infinite square well:

$$V(x,y) = \begin{cases} 0, & \text{if } 0 \le x \le a & \text{and} & 0 \le y \le b \\ \infty, & \text{otherwise} \end{cases}$$

Does that mean V(x,y) can be neglected?

I just don't get what it's asking me to do.

4. Jun 2, 2013

### ehild

V=0 in the potential well, if 0≤x≤a and 0≤y≤b, and infinite outside. Solve the equation inside the well with V=0, and outside where V is infinite. Is it possible that the wavefunction of a particle with finite energy differs from zero when the potential is infinite?

ehild

5. Jun 2, 2013

### voko

Of course not. Think about $$V_c(z) = \begin{cases} 0, & \text{if } 0 \le z \le c\\ \infty, & \text{otherwise} \end{cases}$$