In the folders under 1-d the potential, V(x), is independent of time so that
For
The Schrodinger equation, ..\SchrodingerEqn.docx (3) in Hartree units is
So that becomes
Cancelling the common exponential factors
Bob *** stop
For V and z independent of x, both lines in are solved by
For V>EH
The two
solutions
are
For EH>V
The two
Solutions
are
The -iEt in means that the positive exponents represent probability moving from left to right, and the negative exponents represent probability moving from right to left. The coefficients AC,BC,CC, and DC are arbitrary complex coefficients that can be used to make the wave function and its derivative continuous at the point where V1 becomes V2. The four potentials are given in
The fourth potential is a visual fit of a harmonic
oscillator potential to the barrier potential.
The potential near any minimum, dV(xmin)/dx =0 can be expanded as.
The Harmonic Oscillator states are derived from raising
operators in
HarmonicOscillator/Welcome.htm These are the same as the ones used in
../AngularMomentum.pdf The states in this potential are
discussed in
HarmonicOscillator .docx
Each of the four potentials is considered as a series of Vi values in
The differential
equation can be solved numerically, especially when the starting point is a
potential minimum.
DiffEq .docx