Our wave function will be a solution of the free particle Schr ̈odingerequation providedE 0 = p
(^20)
2 m.
This is exactly what we wanted. So we have constructed an equationthat has the expected wave-
functions as solutions. It is a wave equation based on the total energy.
Adding in potential energy, we havethe Schr ̈odinger Equation
− ̄h^2
2 m
∂^2 ψ(x,t)
∂x^2
+V(x)ψ(x,t) =i ̄h
∂ψ(x,t)
∂t
or
Hψ(x,t) =Eψ(x,t)
where
H=
p^2
2 m
+V(x)
is the Hamiltonian operator.
Inthree dimensions, this becomes.
Hψ(~x,t) =
− ̄h^2
2 m
∇^2 ψ(~x,t) +V(~x)ψ(~x,t) =i ̄h
∂ψ(~x,t)
∂t
We will use it to solve many problems in this course.
So the Schr ̈odinger Equation is, in some sense, simply the statement (in operators) that the kinetic
energy plus the potential energy equals the total energy.
7.2 The Flux of Probability*
In analogy to the Poynting vector for EM radiation, we may want to know theprobability current
in some physical situation. For example, in our free particle solution,the probability density is
uniform over all space, but there is a net flow along the direction of the momentum.
We can derive an equation showing conservation of probability (See section 7.5.2), by differentiating
P(x,t) =ψ∗ψand using the Schr ̈odinger Equation.
∂P(x,t)
∂t+
∂j(x,t)
∂x= 0
This is the usual conservation equation ifj(x,t) is identified as the probability current.
j(x,t) =̄h
2 mi[
ψ∗∂ψ
∂x−
∂ψ∗
∂x
ψ]
This current can be computed from the wave function.