Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 247

Substituting (S.5.3.2) into (S.5.3.9) and using theresult of (S.5.3.7), we

obtain

Finally,using(S.5.3.5)yields

It is easy to show that asrequired by particle conservation. If

then sincethere is nochange, and the particle must stay

in the boundstate.

5.4 Attractive Delta Function Potential II (Stony

Brook)

a) In order to construct thewave function for the boundstate, we first

review its properties. Itmust vanish at the point At the point

it is continuous and its derivative obeys an equation similar to

(S.5.3.3):

Awayfrom thepoints it has an energy and

wavefunctionsthat arecombinations of and Theseconstraints

dictate thatthe eigenfunction has theform

At the point we match thetwo eigenfunctions and their derivatives,
using (S.5.4.1). This yields two equations, which are solved to find an
equation for