Separating Schrodinger's Equation

  • Thread starter MiCasilla
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  • #1
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?
 
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  • #2
jcsd
Science Advisor
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okay:


iħ ∂Ψ/∂t = - ħ2/2m ∂2Ψ/∂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
 
  • #3
MiCasilla
Thank you, very very much!
 

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