Thermodynamics, Statistical Physics, and Quantum Mechanics

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5.11 Square Well (MIT)

A particle of mass isconfined to a space in one dimension by

infinitelyhigh walls at At the particle is initially in the left

half of the well with constant probability

a) Find the time-dependent wave function

b) What is the probabilitythat theparticle is inthe nth eigenstate?

c) Write anexpression for the average value of theparticleenergy.

5.12 Given the Eigenfunction (Boston, MIT)

A particle of mass moves in one dimension. It is remarked that the exact

eigenfunction for the ground state is

where is a constant and A is the normalization constant. Assuming that
the potential vanishes at infinity, derive the ground state eigenvalue
and

5.13 Combined Potential (Tennessee)

A particle of mass is confined to in one dimension by the potential

where and are constants. Assuming there is a bound state, derive the
exact ground stateenergy.

Harmonic Oscillator

5.14 Given a Gaussian (MIT)

A particle of mass is coupled to a simple harmonicoscillator in one di-
mension. The oscillator has frequency and distance constant

PROBLEMS