Purchase Solution

Harmonic Oscillator and Wave Packets

Not what you're looking for?

Ask Custom Question

Consider the 1-D harmonic oscillator, with Hamiltonian H=p²/2m+½mw²x²=hw(a¬a+1/2), and with energy eigenstates H|n}=En|n}=hw(n+1/2)|n}. The eigenvectors of the annihilation operator a are known as coherent states: a|z}=z|z}, where z is in general a complex number (a is not Hermitian, so z is not necessarily real). Take |z} to be normalized: {z|z}=1.

a) Find an expression for |z} as a linear combination of the eneergy eigenstates |n}. Make sure it is normalized. Calculate {z1|z2}.
b) If the oscillator is in the state |z} at the instant that the oscillator energy is measured, what is the probability of obtaining the result En? What is the expectation value for the measured energy, {z|H|z}?
c) Show that for this state, change in x*change in p = h/2, so that the state is a minimum uncertainty wave packet.

Please see the attached for the proper format and notations.

Purchase this Solution

Solution Summary

The answer provides an expression for the complex number z, given the energy eigenstates within the harmonic oscillator. Using probabilities and the Hamiltonian equation, this expression is determined, alongside the expectation value for the measured energy.

Solution Preview

Please see the attached file for the adequate notation.

From the problem, we have a|z> = z|z>

Suppose, |z> = ?n fn (z) |n> and a|z> = ?n fn (z) ?n |n-1>

z|z> = ?n fn (z) ?n |n-1>
z ?n fn (z) |n> = ?n fn (z) ?n |n-1>

So, we have the following:
z?n fn (z) = fn+1 (z) ?(n+1)
fn+1 (z) = (z/?(n+1)) fn (z) ...

Purchase this Solution

Free BrainMass Quizzes
Intro to the Physics Waves

Some short-answer questions involving the basic vocabulary of string, sound, and water waves.

Variables in Science Experiments

How well do you understand variables? Test your knowledge of independent (manipulated), dependent (responding), and controlled variables with this 10 question quiz.

Introduction to Nanotechnology/Nanomaterials

This quiz is for any area of science. Test yourself to see what knowledge of nanotechnology you have. This content will also make you familiar with basic concepts of nanotechnology.

Basic Physics

This quiz will test your knowledge about basic Physics.

The Moon

Test your knowledge of moon phases and movement.