The Schrodinger Wave Equation
There are several ways to develop the wave equation by applying quantum concepts to various classical equations of mechanics. One of the simplest approaches
is to consider a few basic postulates, develop the wave equation from them, and rely on the accuracy of the results to serve as a justification of the postulates. In
more advanced texts these assumptions are dealt with in more convincing detail.
- Each particle in a physical system is described by a wave function Ψ(x, y, z,t). This function and its space derivative (dΨ/dx dΨ/dy dΨ/dz) are continuous, finite, and single valued.
- In dealing with classical quantities such as energy E and momentum p, we must relate these quantities with abstract quantum mechanical operators defined in the following way:
and similarly for the other two directions.
3. The probability of finding a particle with wave function Ψ in the volume dx dy dz is Ψ*Ψ dx dy dz. The product Ψ*Ψ is normalized
and the average value (Q) of any variable Q is calculated from the wave function by using the operator form Qop defined in postulate 2:
Once we find the wave function Ψ for a particle, we can calculate its average position, energy, and momentum, within the limits of the uncertainty principle.
The classical equation for the energy of a particle can be written:
For a one-dimensional problem
which is the Schrodinger wave equation. In three dimensions the equation is
where Δ2Ψ is