## APPLICATIONS OF SCHRODINGER'S WAVE EQUATION

We will now apply Schrodinger's wave equation in several examples using various

potential functions. These examples will demonstrate the techniques used in the solution

of Schrodinger's differential equation and the results of these examples will

provide an indication of the electron behavior under these various potentials. We will

utilize the resulting concepts later in the discussion of semiconductor properties

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 he written from Equation (2.13) as

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

Recall that the time-dependent portion of the solution is

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 +I direction, while the second term, with the coefficient B.

is a wave traveling in the x direction. The value of these coefficients will he 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 be described by the +x traveling wave. The coefficient B = 0. We can

write the traveling-wave solution in the form

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

and momentum.

The probability density function is Y(x, t)Y*(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.

The Infinite Potential Well

The problem of a particle in the infinite potential well is a classic example of a bound

particle. The potential V(x) as a function of position for this problem is shown in

Figure 2.5. The particle is assumed to exist in region II so the particle is contained

within a finite region of space. The time-independent Schrodinger's wave equation is

again given by Equation (2.13) as

where E is the total energy of the particle. If E is finite, the wave function must be

zero, or = 0, in both regions I and III. A particle cannot penetrate these infinite

potential barriers, so the probability of finding the particle in regions I and

III is zero.

The time-independent Schrodinger's wave equation in region II, where V = 0.

becomes

Click on image to enlarge

The total energy can then be written as

where the constant K must have discrete values, implying that the total energy of the

particle can only have discrete values. This result means that the energy of the particle

is quantized. That is, the energy of the particle can only have particular discrete

values. The quantization of the particle energy is contrary to results from classical

physics, which would allow the particle to have continuous energy values. The discrete

energies lead to quantum states that will be considered in more detail in this

and later chapters. The quantization of the energy of a bound particle is an extremely

important result.

potential functions. These examples will demonstrate the techniques used in the solution

of Schrodinger's differential equation and the results of these examples will

provide an indication of the electron behavior under these various potentials. We will

utilize the resulting concepts later in the discussion of semiconductor properties

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 he written from Equation (2.13) as

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

Recall that the time-dependent portion of the solution is

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 +I direction, while the second term, with the coefficient B.

is a wave traveling in the x direction. The value of these coefficients will he 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 be described by the +x traveling wave. The coefficient B = 0. We can

write the traveling-wave solution in the form

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

and momentum.

The probability density function is Y(x, t)Y*(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.

The Infinite Potential Well

The problem of a particle in the infinite potential well is a classic example of a bound

particle. The potential V(x) as a function of position for this problem is shown in

Figure 2.5. The particle is assumed to exist in region II so the particle is contained

within a finite region of space. The time-independent Schrodinger's wave equation is

again given by Equation (2.13) as

where E is the total energy of the particle. If E is finite, the wave function must be

zero, or = 0, in both regions I and III. A particle cannot penetrate these infinite

potential barriers, so the probability of finding the particle in regions I and

III is zero.

The time-independent Schrodinger's wave equation in region II, where V = 0.

becomes

Click on image to enlarge

The total energy can then be written as

where the constant K must have discrete values, implying that the total energy of the

particle can only have discrete values. This result means that the energy of the particle

is quantized. That is, the energy of the particle can only have particular discrete

values. The quantization of the particle energy is contrary to results from classical

physics, which would allow the particle to have continuous energy values. The discrete

energies lead to quantum states that will be considered in more detail in this

and later chapters. The quantization of the energy of a bound particle is an extremely

important result.

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