- #1

TheSodesa

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## Homework Statement

Let us look at a 3-dimensional potential box. Show, that the wave function in this situation can be written as the product of 3 single-argument functions.

## Homework Equations

The 3D Schrödinger equation:

\begin{equation}

-\frac{\hbar^2}{2m} \left( \frac{\partial^2 \Psi}{\partial x^2} + \frac{\partial^2 \Psi}{\partial y^2} + \frac{\partial^2 \Psi}{\partial z^2} \right) + V\Psi = E\Psi,

\end{equation}

where ##\Psi = \Psi(x,y,z)##

## The Attempt at a Solution

I'm not at all sure how I'm supposed to show that this is true. I have not worked with partial differential equations at all, so I'm probably not in the right with what I tried to do:

I assumed, that ##\Psi(x,y,z) = \Psi_x(x) \Psi_y(y) \Psi_z(z)##, and tried plugging it into the Schrödinger equation as follows:

\begin{gather*}

-\frac{\hbar^2}{2m} &\left( \frac{\partial^2 \Psi(x,y,z)}{\partial x^2} + \frac{\partial^2 \Psi(x,y,z)}{\partial y^2} + \frac{\partial^2 \Psi(x,y,z)}{\partial z^2} \right) + V\Psi(x,y,z) = E\Psi(x,y,z)\\

\\

-\frac{\hbar^2}{2m} & \left( \frac{\partial^2 \Psi_x(x) \Psi_y(y) \Psi_z(z)}{\partial x^2} + \frac{\partial^2 \Psi_x(x) \Psi_y(y) \Psi_z(z)}{\partial y^2} + \frac{\partial^2 \Psi_x(x) \Psi_y(y) \Psi_z(z)}{\partial z^2} \right)

+ V\Psi_x(x) \Psi_y(y) \Psi_z(z) = E\Psi_x(x) \Psi_y(y) \Psi_z(z)\\

\\

-\frac{\hbar^2}{2m} & \left( \Psi_y(y) \Psi_z(z) \frac{\partial^2 \Psi_x(x)}{\partial x^2} + \Psi_x(x) \Psi_z(z) \frac{\partial^2 \Psi_y(y)}{\partial y^2} + \Psi_x(x) \Psi_y(y) \frac{\partial^2 \Psi_z(z)}{\partial z^2} \right)

+ V\Psi_x(x) \Psi_y(y) \Psi_z(z) = E\Psi_x(x) \Psi_y(y) \Psi_z(z)

\end{gather*}

Dividing by ## \Psi_x(x) \Psi_y(y) \Psi_z(z)##:

\begin{gather*}

-\frac{\hbar^2}{2m} & \left( \frac{1}{\Psi_x(x)} \frac{\partial^2 \Psi_x(x)}{\partial x^2} + \frac{1}{\Psi_y(y)} \frac{\partial^2 \Psi_y(y)}{\partial y^2} + \frac{1}{\Psi_z(z)} \frac{\partial^2 \Psi_z(z)}{\partial z^2} \right)

+ V = E

\end{gather*}

But... Now what? Is there a way to proceed from here, or did I not pursue the correct path?

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