Folders Welcome.pdf

DiffEq\SeqnIn1d(3).docx

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

Potential  .docx  

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  

Flats .docx

 

 The differential equation can be solved numerically, especially when the starting point is a potential minimum.
DiffEq .docx