Splitting the tim eindependent schroedinger equation

[Mechdude]

splitting the time independent schroedinger equation

Homework Statement

how would i go about splitting the time independent Schroedinger equation into 3 separate 1-D problems of the main 3-d problem whose total energy is $E = E_x + E_y + E_z$ ?

Homework Equations

$$\frac{-\hbar^2}{2m} \frac{ \partial^2 \Psi(x) }{\partial x^2} + V(x) \Psi (x) = E \Psi(x)$$
assuming:
$$\Psi (x) = R(x) S(y) J(z)$$

The Attempt at a Solution

i want to assume that V(x) remains a function of x alone and only $\Psi(x)$ is a function of x,y,z , I am i right? and further more what do i do to the expression to show that the variables are independent of each other? in solving the heat equation we would differentiate the stuff and find that the two sides were independent (one on time the other on position) is that what is done here? or is there some analogous way to do that?



EDIT:


sorry about the [itex] \Psi (x) [/tex] it should be $\Psi (r)$

from the defenition of $\nabla^2$ and separating the solution into a product of three functions R(x)S(y)J(z) :

this is what i get as the equation in cartesian coordinates,
$$\frac{-\hbar^2}{2m} \left[ S(y) J(z) \frac {\partial^2 R(x)}{\partial x^2} + R(x)J(z) \frac {\partial^2 S(y)}{\partial y^2} + R(x)S(y) \frac{\partial^2 J (z) }{\partial z^2} + V(r) R(x)S(y)J(z) = E R(x)S(y)J(z) \right]$$
dividing through by R(x) S(y) J(z) and multiplying by $\frac{-2m}{\hbar}$
$$\left[ \frac{ \partial^2 R(x}{R(x) \partial x^2} + \frac{ \partial^2 S(y)}{S(y) \partial y^2} + \frac { \partial^2 J(z) }{ J(z) \partial z^2} \right] -\frac{2m V(r)}{\hbar^2} = -\frac{E 2m}{\hbar^2}$$

im stuck from here on how to show that it can be separated into two separate independent variables and equation.

 

[vela]

You need the potential to be of the form V(x,y,z)=V_x(x)+V_y(y)+V_z(z), e.g., the potential for the three-dimensional harmonic oscillator is V(x,y,z)=(k/2)(x²+y²+z²). Then you can rewrite the equation into three pieces, each depending on only one variable.

 

[Mechdude]

thanks vela , now if i modify it , what actually inspires the spit into separate equations? is it because the x ,y and z are not dependent on each other?
$$\left[ \frac{ \partial^2 R(x}{R(x) \partial x^2} + \frac{ \partial^2 S(y)}{S(y) \partial y^2} + \frac { \partial^2 J(z) }{ J(z) \partial z^2} \right] -\frac{2m V_x (x) V_y (y) V_z (z)}{\hbar^2} = -\frac{E 2m}{\hbar^2}$$

 

[vela]

Reread what I wrote about the form of the potential more carefully.

 

[Mechdude]

let me give it one more try:
is this what you meant,
$$\left[ \frac{ \partial^2 R(x}{R(x) \partial x^2} + \frac{ \partial^2 S(y)}{S(y) \partial y^2} + \frac { \partial^2 J(z) }{ J(z) \partial z^2} \right] -\frac{2m (V_x (x) + V_y (y) +V_z (z))}{\hbar^2} = -\frac{E 2m}{\hbar^2}$$

if that's what you meant ,then its my bad, i did not read you post carefully.