Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 245

c) For a positive energy state set where is the wave vector
outside thesquare well. Inside the square well we againdefine awave vector
according to

Again we have therequirement that the eigenfunctionvanish at For
we have an eigenfunction with two unknownparameters Band
Alternatively, we maywrite it as

in terms of two unknownsCandD. The two forms areequivalent since
We prefer to write it with the phaseshift
Again we match the two wave functions and theirderivatives at

Dividing(S.5.2.10) by (S.5.2.11), weobtain

Since is aknown function of theonlyunknown inthis equation is
which is determined bythisequation.

d) From(S.5.2.12) we derive anexpression for thephaseshift:

5.3 Attractive Delta Function Potential I (Stony

Brook)

a) The boundstate is stationary in time: itseigenvalue isE(E <0), and
thetime dependence of the wave function is
The equation for the bound state is