Download Probability Current and the continuity equation.

Atomic and Molecular Quantum Theory
Course Number: C561
E Probability Current
1. Consider the time-dependent Schrödinger Equation and its complex conjugate:
∂
h̄2 ∂ 2


 ψ(x, t)
ıh̄ ψ(x, t) = Hψ(x, t) = −
+
V
∂t
2m ∂x2


(E.16)
h̄2 ∂ 2
∂ ∗

∗
+ V  ψ ∗(x, t)
− ıh̄ ψ (x, t) = Hψ (x, t) = −
2
∂t
2m ∂x
(E.17)

Chemistry, Indiana University
E-18

c
2003,
Srinivasan S. Iyengar (instructor)
Atomic and Molecular Quantum Theory
Course Number: C561
2. Multiply Eq. (E.16) by ψ ∗ (x, t), and Eq. (E.17) by ψ(x, t):
ψ ∗ (x, t)ıh̄
∂
ψ(x, t) = ψ ∗(x, t)Hψ(x, t)
∂t


2
2
∂
h̄

 ψ(x,(E.18)
+
V
t)
= ψ ∗(x, t) −
2m ∂x2
− ψ(x, t)ıh̄
∂ ∗
ψ (x, t) = ψ(x, t)Hψ ∗(x, t)
∂t


2
2
∂
h̄
 ∗
 ψ (x,
+
V
(E.19)
t)
= ψ(x, t) −
2m ∂x2
3. Subtract the two equations to obtain (see that the terms involving
V cancels out)
h̄2  ∗
∂2
∂
∗
ψ (x, t)
ψ(x, t)
ıh̄ [ψ(x, t)ψ (x, t)] = −
∂t
2m
∂x2

∂2 ∗
− ψ(x, t) 2 ψ (x, t)
(E.20)
∂x

or


∂
h̄ ∂  ∗
∂ ∗
∂
ρ(x, t) = −
ψ (x, t) ψ(x, t) − ψ(x, t) ψ (x, t)
∂t
2mı ∂x
∂x
∂x
(E.21)
Chemistry, Indiana University
E-19
c
2003,
Srinivasan S. Iyengar (instructor)
Atomic and Molecular Quantum Theory
Course Number: C561
4. If we make the variable substitution:


h̄  ∗
∂ ∗
∂
J =
ψ (x, t) ψ(x, t) − ψ(x, t) ψ (x, t)
2mı 
∂x
∂x

h̄  ∗
∂
=
I ψ (x, t) ψ(x, t)
(E.22)
m
∂x
(where I [· · ·] represents the imaginary part of the quantity in
brackets) we obtain
∂
∂
ρ(x, t) = − J
∂t
∂x
or in three-dimensions
∂
ρ(x, t) + ∇ · J = 0
∂t
(E.23)
(E.24)
5. Equation (E.24) looks like the continuity equation of a classical
fluid. Hence J is a flux. In fact J is the flux associated with the
probability density and is hence called the “probability current”.
6. (Continuity equation of a classical fluid is basically the following: If you have a small volume element in a fluid. The change
in density in that volume element is given by the amount of fluid
coming into the volume element minus the amount going out.
That is, flux in minus flux out. The mathematical form of this is
Eq. (E.24). Hence the time-dependent Schrödinger Equation is
a continuity equation.)
Chemistry, Indiana University
E-20
c
2003,
Srinivasan S. Iyengar (instructor)
Atomic and Molecular Quantum Theory
Course Number: C561
7. Homework:
(a) Prove that {∇ · J = 0} for a stationary state.
(b) Calculate the probability current for:
ψ=
exp {ıkx}
x
(E.25)
(c) Find a relation between the probability current J and the
action function S that we introduced in the previous extra
credit homework. Simplify this expression using Eq. (F.1).
Explain your result.
(d) For the case of the 1D barrier that we studied earlier, using
the scattering perspective, calculate the probability current
for ψ1 for x < 0, and for x > 0.
Chemistry, Indiana University
E-21
c
2003,
Srinivasan S. Iyengar (instructor)