Electron in Free Space

As a first example of applying the Schrodinger's wave equation, consider the motion of an electron in free space. If there is no force acting on the particle, then the potential function V(x) will be constant and we must have E > V(x). Assume, for simplicity, that the potential function V(x) = 0 for all x. Then, the time-independent wave equation can be written as

(1)

The solution to this differential equation can be written in the form

(2)

Recall that the time-dependent portion of the solution is

(3)

Then the total solution for the wave function is given by

(4)

This wave function solution is a traveling wave, which means that a particle moving in free space is represented by a traveling wave. The first term, with the coefficient A. is a wave traveling in the +x direction, while the second term, with the coefficient B. is a wave traveling in the -x direction. The value of these coefficients will be determined from boundary conditions. We will again see the traveling-wave solution for an electron in a crystal or semiconductor material.

Assume, for a moment, that we have a particle traveling in the +x direction. which will he described by the +x traveling wave. The coefficient B = 0. We can write the traveling-wave solution in the form

(5)

where k is a wave number and is

(6)

The parameter A is the wavelength and, comparing Equation (5) with Equation (4), the wavelength is given by

(7)

From de Broglie's wave-particle duality principle, the wavelength 1s also given by

(8)

A free particle with a well-defined energy will also have a well-defined wavelength and momentum.

The probability density function is Ψ(x, t)Ψ*(x, t) = AA*, which is a constant independent of position. A free particle with a well-defined momentum can be found anywhere with equal probability. This result is in agreement with the Heisenberg uncertainty principle in that a precise momentum implies an undefined position.

A localized free particle is defined by a wave packet, formed by a superposition of wave functions with different momentum or k values. We will not consider the wave packet here.

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Equipartition and the quantum oscillator