Separating Schrodinger's Equation

[MiCasilla]

I'm looking for the mathematical demonstration that the Time Dependent Schrodinger Equation can be separated into time-independent Schrodinger equation and time evolution equation.

Does anybody know a link which such demonstration?

 

[jcsd]

okay:


iħ ∂Ψ/∂t = - ħ²/2m ∂²Î¨/∂x² + VΨ

Okay now assuming energy is conserved (i.e. a closed sytsem)and thus V is time-independant:

Ψ(x,t) = u(x)T(t)

subsituting that into the TDSE:

iħ 1/T dT/dt = - 1/u ħ²/2m d²u/dx²

The left hand side is noe independent of x and the right is now independant of t, yet this equation must hold true for all values of x and t therfore:

iħ dT/dt = ET

Where E is a constant, thus:

- ħ/2m d²u/dx² + V(x)u = Eu

 

[MiCasilla]

Thank you, very very much!