Thermodynamics, Statistical Physics, and Quantum Mechanics

244 SOLUTIONS

Dividingthese two equations produces theeigenvalueequation

The equation given by the rightmost equals sign is an equation for the
unknown Solving it gives the eigenvalueE.

5.2 Shallow Square Well II (Stony Brook)

a) For the bound state wecan write theeigenvalue as

where is thedecayconstant of theeigenfunctionoutside the squarewell

(seeProblem5.1).Inside the square well wedefine a wavevector by

The infinitepotential at theorigin requiresthat all eigenfunctions vanish

at So the lowesteigenfunction must have the form

At the point we match the eigenfunctions and theirderivatives:

We eliminatethe constantsAandBby dividing these two equations:

Earlier we established the relationship between and Sothe only un-

knownvariable is which isdetermined bythis equation.

b) To find theminimumbound state, we take the limit as in

the eigenvalue equation. From (S.5.2.1) we see that goes toanonzero

constant, and the eigenvalue equation only makes sense as if

tan whichhappens at Using(S.5.2.1)gives

Thus, wederive theminimumvalue of for abound

state: