Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 251

Using (S.5.4.1), we match the difference in the derivatives at with

the amplitude of the delta function potential. This leads to the eigenvalue

condition

The quantity on the left of (S.5.7.6) has a minimum value of 1, which it
obtains at This limit produces the eigenvalue So we must
have for the zero eigenvalue, which is the answer to part (b). The
above eigenfunction, for values of gives the bound state energy E < 0
when

5.8 Particle in Expanding Box (Michigan State, MIT,

Stony Brook)

a) For a particle confined to a box the ground state

and the first excited state are

After the sudden transition the final eigenfunctions are

b) In the sudden approximation let denote the probability that the
particle starts in the ground state 0 and ends in the final state