Section 6.4: Probability and Wave Functions

dx = | n =

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In general, a wave function, Ψ(x, t), is a solution to the Schrödinger equation. There are also a special sub-set of wave functions, ψ(x), that are called energy eigenfunctions as they are solutions to the time-independent Schrödinger equation4

        [−(ħ2/2m)(d2/dx2) + V(x)] ψ(x) = E ψ(x) , (6.7)

in one-dimensional position space. The time-independent Schrödinger equation relates the energy eigenfunction and the energy eigenvalue, E.  While we can choose several different variables in which to represent energy eigenfunctions, like position or momentum, we usually choose position.

In this book the symbol, ψ(x), refers to energy eigenfunctions which are solutions to Eq. (6.7) and describe eigenstates of energy. All time-dependent energy eigenfunctions, ψ(x,t), are also wave functions, but not all wave functions are energy eigenfunctions. In Section 7.7 we will consider time evolution of wave functions and energy eigenstates.

Born's interpretation of the solutions of Eq. (6.8) and Eq. (6.7) is that wave functions, Ψ(x, t), represent a probability amplitude at the point x and at a time, t, where the representations Ψ(x) and ψ(x) imply t = 0. The probability density at the point x is the absolute square of the wave function: ρ(x,t) = Ψ*(x,t) Ψ(x,t) = |Ψ(x,t)|2.  Usually we consider this function at time t = 0, where ρ(x) = Ψ*(x) Ψ(x) = |Ψ(x)|2. In one-dimension, the probability that a particle between x and x + dx (at time t = 0) is simply related to the probability density as

ρ(x) dx = Ψ*(x) Ψ(x) dx = |Ψ(x)|2 dx . (6.8)

When Ψ(x) represents a localized, bound-state solution of the Schrödinger equation (at t = 0), the integral over the probability density

∫ Ψ *(x)Ψ(x) dx = ∫ |Ψ(x)|2 dx = 1, [integral from −∞ to +∞] (6.9)

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 functions and energy eigenfunctions are normalized. Once normalized, the wave functions and energy eigenfunctions remain normalized for all later times.5

The animation shows the probability density for a particle in an energy eigenfunction of 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 a wave function (energy eigenfunction) is a solution to the Schrödinger equation (time-independent Schrödinger equation) and has a probabilistic interpretation:

  • The wave function (energy eigenfunction) must tend to a finite value (zero for bound-state solutions and a constant for scattering solutions) as x →  ± ∞. This is a statement of the normalizablility of the wave function (energy eigenfunction).  Mathematically, this means that for bound-state wave functions (energy eigenfunctions)

∫ Ψ*(x)Ψ(x) dx = ∫ |Ψ(x)|2 dx < ∞ [integral from ∞ to +∞]

∫ ψ*(x)ψ(x) dx = ∫ |ψ(x)|2 dx < ∞ [integral from ∞ to +∞]

in order to maintain Born's probabilistic interpretation of Ψ (ψ).

  • The wave function (energy eigenfunction) must be continuous and single valued. This means that a valid wave function (energy eigenfunction) 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 (energy eigenfunction).
  • The wave function (energy eigenfunction) must be twice differentiable. In other words, the wave function's (energy eigenfunction's) first derivative must be continuous, which means that the wave function (energy eigenfunction) 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 (energy eigenfunction) 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. (6.7) 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 7.7.

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