Thermodynamics, Statistical Physics, and Quantum Mechanics

59

where is theexacteigenstate of the harmonicoscillatorwitheigen-

value

a) Give

b) What is the parity of this state? Does it change with time?

c) What is the averagevalue of the energy for this state?Does it change

with time?

5.19 Time-Dependent Harmonic Oscillator II

(Michigan State)

Consider a simpleharmonic oscillator in one dimension. Introduce the

raising andlowering operators, and respectively. TheHamiltonianH

and wave function at are

where denotes theeigenfunction of energy

What is wavefunction atpositive times?

What is the expectationvalue for the energy?

The position can be represented inoperators by

where is a constant. Derive anexpression for the

expectation ofthe time-dependentposition

5.20 Switched-on Field (MIT)

Consider a simple harmonic oscillator in one dimension with the usual

Hamiltonian

a) The eigenfunction of the groundstate can bewritten as

Determine theconstantsNand

You may needoperatorexpressions such asa and

a)

b)

c)

QUANTUM MECHANICS