5
Quantum
Mechanics
One-Dimensional Potentials
5.1 Shallow Square Well I (Columbia)
The groundstate energyEmust be lessthan zero andgreater than the
bottom of thewell, From the expression
one can deduce the form for the eigenfunction. Denote the ground state
energy where is to be determined. The eigenfunction
outside the well(V =0) has the form Inside the well,define
where One canshowthat ispositivesince
Inside the well, theeigenfunction has theform so
Matching and its derivative at gives two expressions:243