Hw4
This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA
Overview
Download & View Hw4 as PDF for free.
More details
- Words: 327
- Pages: 2
QUANTUM THEORY OF MATTER Homework set #4: Time evolution of expectation values, Ehrenfest theorem
Problem # 4.1 : Show that d 2 1 hx i = (hxpx i + hpx xi) dt m for a three-dimensional wave-packet.
Problem # 4.2 : If hxi and hpi are the expectation values of x and p formed with the wave-function ψ(x) of a one-dimensional system, show that the expectation value of x and p formed with the wave-function φ(x) = exp[−(i/¯ h)hpix] ψ(x + hxi) vanishes, i.e. hxiφ = hpiφ = 0.
Problem # 4.3 : The Hamiltonian of a one-dimensional harmonic oscillator is given by p2 m H= + ω 2 x2 . 2m 2 (a) Find the time-dependence of the expectation value of x and p. What is the period of oscillation? (b) Show that the expectation values of x2 and p2 satisfy d 2 1 hx i = (hxpi + hpxi) dt m
and
d 2 hp i = −mω 2 (hxpi + hpxi) , dt
respectively. Compare with problem # 4.1. (c) Use these relations to show that the expectation value of the energy is conserved.
Hint: The equations of motion can be obtained either via partial integration (using Gauss’ divergence theorem) or using commutators with the Hamiltonian.
Problem # 4.4 : Virial theorem (a) Use the equation of motion for the expectation value of an operator O with the wavefunction ψ(t), i¯ h to show that
D∂ E d hOi = h[O, H]i + i¯ h O , dt ∂t D dV E d hxpi = 2hKEi − x , dt dx
where KE is the kinetic energy, V is the potential energy and H = KE + V . (b) Why is dhxpi/dt = 0 for a stationary state? In that case D dV E . 2hKEi = x dx This is called the virial theorem. (c) Use this result to prove that hKEi = hV i for stationary states of the harmonic oscillator.
Problem # 4.1 : Show that d 2 1 hx i = (hxpx i + hpx xi) dt m for a three-dimensional wave-packet.
Problem # 4.2 : If hxi and hpi are the expectation values of x and p formed with the wave-function ψ(x) of a one-dimensional system, show that the expectation value of x and p formed with the wave-function φ(x) = exp[−(i/¯ h)hpix] ψ(x + hxi) vanishes, i.e. hxiφ = hpiφ = 0.
Problem # 4.3 : The Hamiltonian of a one-dimensional harmonic oscillator is given by p2 m H= + ω 2 x2 . 2m 2 (a) Find the time-dependence of the expectation value of x and p. What is the period of oscillation? (b) Show that the expectation values of x2 and p2 satisfy d 2 1 hx i = (hxpi + hpxi) dt m
and
d 2 hp i = −mω 2 (hxpi + hpxi) , dt
respectively. Compare with problem # 4.1. (c) Use these relations to show that the expectation value of the energy is conserved.
Hint: The equations of motion can be obtained either via partial integration (using Gauss’ divergence theorem) or using commutators with the Hamiltonian.
Problem # 4.4 : Virial theorem (a) Use the equation of motion for the expectation value of an operator O with the wavefunction ψ(t), i¯ h to show that
D∂ E d hOi = h[O, H]i + i¯ h O , dt ∂t D dV E d hxpi = 2hKEi − x , dt dx
where KE is the kinetic energy, V is the potential energy and H = KE + V . (b) Why is dhxpi/dt = 0 for a stationary state? In that case D dV E . 2hKEi = x dx This is called the virial theorem. (c) Use this result to prove that hKEi = hV i for stationary states of the harmonic oscillator.
