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.
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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 infinitepotential 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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