Schrodinger's Wave equation
The various experimental results involving electromagnetic waves and particles.
which could not be explained by classical laws of physics, showed that a revised formulation
of mechanics was required. Schrodinger, in 1926. provided a formulation
called wave mechanics, which incorporated the principles of quanta introduced by
Planck, and the wave-particle duality principle introduced by de Broglie. Based on the
wave-particle duality principle. we will describe the motion of electrons in a crystal
by wave theory. This wave theory is described by Schrodinger's wave equation.
The Wave Equation
The one-dimensional, nonrelativistic Schrodinger's wave equation is given by
where
the wave function, V(x) is the potential function assumed to he independent
of time, fn is the mass of the particle, and, is the imaginary constant iota
There are theoretical arguments that justify the form of Schrodinger's wave equation.
hut the equation is a basic postulate of quantum mechanics. The wave function
will be used to describe the behavior of the system and, mathematically,
can be a complex quantity.
We may determine the time-dependent portion of the wave function and the
position-dependent, or time-independent, portion of the wave function by using the
technique of separation of variables. Assume that the wave function can he written in
the form
Since the left side of Equation (2.9) is a function of position x only and the right side
of the equation is a function of time r only, each side of this equation must he equal
to a constant. We will denote this separation of variables constant by
The time-dependent portion of Equation (2.9) is then written as
where again m is the mass of the particle, V(x) is the potential experienced by the particle,
and E is the total energy of the particle. This time-independent Schrodinger's
wave equation can also be justitied on the basis of the classical wave equation as
shown in Appendix E. The pseudo-derivation in the appendix is a simple approach
but shows the plausibility of the time-independent Schrodinger's equation
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which could not be explained by classical laws of physics, showed that a revised formulation
of mechanics was required. Schrodinger, in 1926. provided a formulation
called wave mechanics, which incorporated the principles of quanta introduced by
Planck, and the wave-particle duality principle introduced by de Broglie. Based on the
wave-particle duality principle. we will describe the motion of electrons in a crystal
by wave theory. This wave theory is described by Schrodinger's wave equation.
The Wave Equation
The one-dimensional, nonrelativistic Schrodinger's wave equation is given by
where
the wave function, V(x) is the potential function assumed to he independentof time, fn is the mass of the particle, and, is the imaginary constant iota
There are theoretical arguments that justify the form of Schrodinger's wave equation.
hut the equation is a basic postulate of quantum mechanics. The wave function
will be used to describe the behavior of the system and, mathematically, can be a complex quantity.We may determine the time-dependent portion of the wave function and the
position-dependent, or time-independent, portion of the wave function by using the
technique of separation of variables. Assume that the wave function can he written in
the form
Since the left side of Equation (2.9) is a function of position x only and the right side
of the equation is a function of time r only, each side of this equation must he equal
to a constant. We will denote this separation of variables constant by
The time-dependent portion of Equation (2.9) is then written as
where again m is the mass of the particle, V(x) is the potential experienced by the particle,
and E is the total energy of the particle. This time-independent Schrodinger's
wave equation can also be justitied on the basis of the classical wave equation as
shown in Appendix E. The pseudo-derivation in the appendix is a simple approach
but shows the plausibility of the time-independent Schrodinger's equation
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