|
Please wait for the animation to completely load.
The wave function, ψ(x), is a solution to the time-independent Schrödinger equation4
[−(ħ2/2m)(d2/dx2) + V(x)]
ψ(x) = E ψ(x)
,
(6.9)
in one-dimensional position space (at t = 0). The time-independent Schrödinger equation relates the wave function and the energy eigenvalue,
E. While we can choose several different variables in which to represent the wave function, like position
or momentum, we usually choose position. If we just say, the wave function, we almost certainly
mean the wave function in position space.
Born's interpretation of the solutions of Eq. (6.9) is
that the wave function, ψ(x), represents a probability
amplitude at the point x and at a time, t (here t = 0). The
probability density at the point x is the absolute square of the
wave function: ρ(x) = ψ*(x) ψ(x) = |ψ(x)|2. In one-dimension, the probability that a particle between
x and x + dx is simply related to the probability density as
ρ(x) dx = ψ*(x) ψ(x) dx = |ψ(x)|2
dx .
(6.10)
When ψ(x) represents a localized, bound-state solution of the time-independent Schrödinger
equation, the integral over the probability density
∫ ψ*(x)ψ(x) dx = ∫ |ψ(x)|2 dx = 1, [integral from
−∞ to +∞]
(6.11)
since the probability of finding a particle somewhere must be one.
In order to ensure
that the total probability is one, we must often check that the bound-state wave function we are
using is normalized. Once normalized, the wave function remains
normalized for all later times.5
The animation shows the probability density for a particle in
an infinite square well. The particle is confined between x = 0 and
x = 1. You can change the state, n, and the interval, dx, in
which the probability is calculated. Note that there are regions
of space in which you would not expect to find the particle. Set dx to 1 and see what happens.
In order to guarantee that the wave function is a
solution to the time-independent Schrödinger equation and has
a probabilistic interpretation:
∫ ψ*(x)ψ(x) dx = ∫ |ψ(x)|2
dx = 1
[integral from −∞ to +∞]
(6.12)
in order to maintain Born's probabilistic interpretation of ψ.
-
The wave function must be continuous and single valued. This means that a valid wave function
should not have any jumps in it (continuous) and at every point in space
have only one value associated with it (single valued). If a wave function was not continuous or not single valued,
it would have multiple values for the same position, thereby ruining the
probabilistic interpretation of the wave function.
-
The wave function must be twice differentiable. In other words, the wave function's
first derivative must be continuous, which means that the wave function itself must have no
kinks. This is true as long as the
potential energy function is itself not severely discontinuous. When the potential energy function has a severe discontinuity, the
wave function may have a kink. Examples of such severe
discontinuities in the potential energy include the infinite square well (Chapter 10) and the
attractive Dirac delta function well (Chapter 11).
___________________
4This terminology parallels Styer's [2] terminology that emphasizes
[ −(ħ2/2m)(∂2/∂x2) + V(x) ] ψ(x,t)
= iħ ∂ψ(x,t)/∂t
(6.8)
as the
Schrödinger equation and Eq. (6.9) as an energy-eigenvalue equation, which is a special time-independent
case of the Schrödinger equation. In general, the Schrödinger equation is three dimensional and has a time
dependence. In this chapter we only consider the time-independent case, leaving the time-dependent case to Chapter
7.
5This is a property of the
Schrödinger equation and the time evolution of states. Quantum-mechanical time development
is discussed in Section 7.6.
© 2006 by Prentice-Hall, Inc. A Pearson Company
|