# 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?

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#### jcsd

Gold Member
okay:

iÄ§ âˆ‚Î¨/âˆ‚t = - Ä§2/2m âˆ‚2Î¨/âˆ‚x2 + VÎ¨

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

Î¨(x,t) = u(x)T(t)

subsituting that into the TDSE:

iÄ§ 1/T dT/dt = - 1/u Ä§2/2m 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:

iÄ§ dT/dt = ET

Where E is a constant, thus:

- Ä§/2m d2u/dx2 + V(x)u = Eu

#### MiCasilla

Thank you, very very much!

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