PHYS212 : Lectures

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Subsections

# 7.3 Continuity and Probability Current

## 7.3.1 Defined

• Mass conservation is expressed with an equation of continuity identical to that describing conservation of charge,
 (7.34)

where is the probability density in units of particles per unit volume and is the probability current with units of particles per unit area per unit time.7.3

• Integrating Eq. 7.34 over a volume , and applying the divergence theorem, we have
 (7.35)

where represents the area bounding volume , and is the number of particles described by in .

• Eq. 7.35 translates nicely into words as
The rate of change of the number of particles in volume is due entirely to the particle current flowing through the area bounding .

• In one dimension, the equation of continuity takes the form
 (7.36)

where the units of are particles per unit length, and the units of are particles per time (a true probability current).

• The Schrödinger equation (Eq. 7.9) enables us to evaluate the first term in Eq. 7.36, yielding

which simplifies to
 (7.37)

• The form of the probability current can be deduced from Eq. 7.37 if one thinks about the product rule a bit. A less enjoyable approach is integration by parts. We find

## 7.3.2 Plane Waves

• A plane wave (Eq. 7.19) has probability current
 (7.38)

which can be written alternatively as

 (7.39)

where is the velocity of a classical particle with momentum .

• Eq. 7.39 suggests a way of setting the normalization constant for a continuous beam of particles. In order to give a sensible current density, must be the number of particles per unit length. Hence,
 (7.40)

where is the linear particle density.

## 7.3.3 Infinite Square Wells

• The position dependent part of the infinite square well wavefunctions (Eq. 7.32) are real functions.

• Consider the general case
 (7.41)

where is a real function.

• If we evaluate the probability current, we find that the time dependence disappears ( ), and
 (7.42)

the complex conjugation of has no effect, giving a cancellation of the two terms in .

• Hence, if a solution to the time independent Schrödinger equation (Eq. 7.16) is real, it has no probability current.

• In fact, identical reasoning leads to the conclusion that purely imaginary wavefunctions also have zero probability current.

 Copyright © 2003-2009, Lewis A. Riley Updated Wed Jan 18 09:51:28 2006