(adsbygoogle = window.adsbygoogle || []).push({}); splitting the time independent schroedinger equation

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

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 [itex] E = E_x + E_y + E_z [/itex] ?

2. Relevant equations

[tex] \frac{-\hbar^2}{2m} \frac{ \partial^2 \Psi(x) }{\partial x^2} + V(x) \Psi (x) = E \Psi(x) [/tex]

assuming:

[tex] \Psi (x) = R(x) S(y) J(z) [/tex]

3. The attempt at a solution

i want to assume that V(x) remains a function of x alone and only [itex] \Psi(x) [/itex] is a function of x,y,z , im 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 [itex] \Psi (r) [/itex]

from the defenition of [itex] \nabla^2 [/itex] 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,

[tex] \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] [/tex]

dividing through by R(x) S(y) J(z) and multiplying by [itex] \frac{-2m}{\hbar} [/itex]

[tex] \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} [/tex]

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Splitting the tim eindependent schroedinger equation

**Physics Forums | Science Articles, Homework Help, Discussion**