130_notes.dvi

Assume that we canfactorizethe solution between time and space.

ψ(x,t) =u(x)T(t)

Plug this into the Schr ̈odinger Equation.
(
− ̄h^2
2 m

∂^2 u(x)

∂x^2

+V(x)u(x)

)

T(t) =i ̄hu(x)

∂T(t)

∂t

Put everything that depends onxon the left and everything that depends onton the right.
(
−h ̄^2
2 m

∂^2 u(x)

∂x^2 +V(x)u(x)

)

u(x)

=

i ̄h∂T∂t(t)

T(t)

=const.=E

Since we have a function of onlyxset equal to a function of onlyt, theyboth must equal a
constant. In the equation above, we call the constantE, (with some knowledge of the outcome).
We now have an equation intset equal to a constant

i ̄h

∂T(t)

∂t

=E T(t)

which has a simplegeneral solution,

T(t) =Ce−iEt/ ̄h

and an equation inxset equal to a constant

− ̄h^2

2 m

∂^2 u(x)

∂x^2

+V(x)u(x) =E u(x)

which depends on the problem to be solved (throughV(x)).

Thexequation is often called theTime Independent Schr ̈odinger Equation.

− ̄h^2

2 m

∂^2 u(x)

∂x^2

+V(x)u(x) =E u(x)

Here,Eis a constant. Thefull time dependent solutionis.

ψ(x,t) =u(x)e−iEt/ ̄h

• See Example 7.6.1:Solve the Schr ̈odinger equation for a constant potentialV 0 .*

7.5 Derivations and Computations

7.5.1 Linear Operators

Linear operatorsLsatisfy the equation