130_notes.dvi

7 The Schr ̈odinger Equation

Schr ̈odinger developed adifferential equation for the time development of a wave function.
Since the Energy operator has a time derivative, the kinetic energyoperator has space derivatives,
and we expect the solutions to be traveling waves, it is natural to try an energy equation. The
Schr ̈odinger equation is theoperator statement that the kinetic energy plus the potential
energy is equal to the total energy.

7.1 Deriving the Equation from Operators

For afree particle, we have
p^2
2 m

=E

−

̄h^2

2 m

∂^2

∂x^2

ψ=i ̄h

∂

∂t

ψ

Letstry this equation on our states of definite momentum.

−

̄h^2

2 m

∂^2

∂x^2

1

√

2 π ̄h

ei(p^0 x−E^0 t)/ ̄h=i ̄h

∂

∂t

1

√

2 π ̄h

ei(p^0 x−E^0 t)/ ̄h

The constant in front of the wave function can be removed from both sides. Its there for normaliza-
tion, not part of the solution. We will go ahead and do the differentiation.

−

̄h^2

2 m

−p^20

̄h^2

ei(p^0 x−E^0 t)/ ̄h=i ̄h

−iE 0

̄h

ei(p^0 x−E^0 t)/ ̄h

p^20

2 m

ei(p^0 x−E^0 t)/h ̄=E 0 ei(p^0 x−E^0 t)/ ̄h