Thermodynamics, Statistical Physics, and Quantum Mechanics

57

At time the particle’s wavefunction is given by

The constant is unrelated to any otherparameters. What is the proba-

bility that ameasurement of energy at finds the value of

5.15 Harmonic Oscillator ABCs (Stony Brook)

Consider theharmonicoscillatorgiven by

Define

Showthat

Showthat

Showthat

Showthat if is aneigenstate of with eigenvalue

then are alsoeigenstates of Nwitheigenvalues

and respectively.

Define suchthat What is the energyeigenvalue of

How can one construct othereigenstates ofHstarting from

What is the energyspectrum ofH? Are negativeeigenvaluespossible?

5.16 Number States (Stony Brook)

Consider thequantummechanicalHamiltonian for aharmonicoscillator
with frequency

a)

b)

c)

d)

e)

f)

g)

QUANTUM MECHANICS