Particle in Expanding Box

A particle of mass m is contained in a one-dimensional impenetrable box extending from x = -L/2 to x = +L/2. The particle is in its ground state.

a) Find the eigenfunctions of the ground state and the first excited state.

b) The walls of the box are moved outward instantaneously to form a box extending from –L ≤ x ≤ L. Calculate the probability that the particle will stay in the ground state during this sudden expansion.

c) Calculate the probability that the particle jumps from the initial ground state to the first excited final state.

SOLUTION

a) For a particle confined to a box –L/2 ≤ x ≤ L/2, the ground state ѱ₀(x) and the first excited state ѱ₁(x) are

(1)

(2)

After the sudden transition L → 2L, the final eigenfunctions are

(3)

(4)

b) In the sudden approximation let P_0j denote the probability that the particle starts in the ground state 0 and ends in the final state j:

(5)

where the amplitude of the transition I_0j is given by

(6)

The amplitude for the particle to remain in its ground state is then

(7)

The probability P₀₀ is given by

(8)

The same calculation for the transition between the initial ground state and final excited state ѱ'₁ is as follows:

(9)

where

(10)

The integral is zero by parity, since the integrand is an odd function of x, so

(11)