Ehrenfest's Theorem
Not what you're looking for?
Consider a 1-D free particle, describable as a wave packet at initial time t0.
a) Show, applying Ehrenfest's theorem, that <X> is a linear function of time and <P> is a constant.
b) Write the equations of motion for the mean values <X^2> and <XP + PX>. Integrate these equations.
c) Show that, with a suitable choice of the time origin, the rms deviation <(X-<X>)^2> is given by...
Please see attached for full question.
Purchase this Solution
Solution Summary
Expressions for d<x>/dt, d<x^2>/dt, d<x p+p x>/dt are derived using Ehrenfest's theorem.
Solution Preview
a) Ehrenfest's Theorem:
d<x>/dt = <p>/m (1)
d<p>/dt = -<grad V> (2)
d<p>/dt = 0, because grad V is zero for a free particle. Therefore p is constant. From (1) it then follows that <x> is a linear function of t.
b) For a general time independent operator A:
d<A>/dt = i/h-bar <[H,A]>
In our case H = p^2/2m:
d<A>/dt = i/(2 m h-bar) <[p^2,A]> (3)
By using that [x,p] = i h-bar, you find after some ...
Purchase this Solution
Free BrainMass Quizzes
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.
Intro to the Physics Waves
Some short-answer questions involving the basic vocabulary of string, sound, and water waves.
Basic Physics
This quiz will test your knowledge about basic Physics.
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.
The Moon
Test your knowledge of moon phases and movement.