Thermodynamics, Statistical Physics, and Quantum Mechanics

58

and define the operators

a) Suppose wedefine a state to obey

Showthat thestates

are eigenstates of thenumber operator, with eigenvalue n:

Showthat is also an eigenstate of theHamiltonian and compute

its energy.

Hint:You may assume

Using the aboveoperators, evaluate theexpectationvalue

in terms of and

5.17 Coupled Oscillators (MIT)

Two identical harmonicoscillators in one dimensioneach have mass and
frequency Let the two oscillators be coupled by an interactionterm
whereCis a constant and and arethe coordinates of the two
oscillators. Find the exactspectrum of eigenvalues for this coupled system.

5.18 Time-Dependent Harmonic Oscillator I

(Wisconsin-Madison)

Consider a simple harmonicoscillator in one dimension:

At the wavefunction is

b)

c)

PROBLEMS