L(aψ+bφ) =aLψ+bLφwhereaandbare arbitrary constants andψandφare arbitrary wave-functions. A multiplicative
constant is a simple linear operator. Differential operators clearly are linear also.
An example of a non-linear operator (which we will not use) isNwhich has the property
Nψ=ψ^2.7.5.2 Probability Conservation Equation*.
Start from the probability and differentiate with respect to time.
∂P(x,t)
∂t=
∂
∂t(ψ∗(x,t)ψ(x,t)) =[
∂ψ∗
∂tψ−ψ∗∂ψ
∂t]
Use the Schr ̈odinger Equation
− ̄h^2
2 m
∂^2 ψ
∂x^2+V(x)ψ=i ̄h∂ψ
∂t
and its complex conjugate
− ̄h^2
2 m
∂^2 ψ∗
∂x^2+V(x)ψ∗=−i ̄h∂ψ∗
∂t(We assumeV(x) is real. Imaginary potentials do cause probability not to be conserved.)
Now we need to plug those equations in.
∂P(x,t)
∂t=
1
i ̄h[
̄h^2
2 m∂^2 ψ∗
∂x^2
ψ−V(x)ψ∗ψ+− ̄h^2
2 m
ψ∗∂^2 ψ
∂x^2
+V(x)ψ∗ψ]
=
1
i ̄h̄h^2
2 m[
∂^2 ψ∗
∂x^2ψ−ψ∗
∂^2 ψ
∂x^2]
=
̄h
2 mi∂
∂x[
∂ψ∗
∂xψ−ψ∗
∂ψ
∂x]
This is the usual conservation equation ifj(x,t) is identified as the probability current.
∂P(x,t)
∂t+
∂j(x,t)
∂x= 0
j(x,t) =̄h
2 mi[
ψ∗∂ψ
∂x−
∂ψ∗
∂xψ]
7.6 Examples
7.6.1 Solution to the Schr ̈odinger Equation in a Constant Potential
Assume we want to solve the Schr ̈odinger Equation in a region in whichthe potential is constant
and equal toV 0. We will find two solutions for each energyE.