5.6 (a) Let Q be an operator which is not a function of time, and letH be the Hamiltonian operator. Show thatმih(q) = ([Q, H])ƏtHere (q) is the expectation value of Q for an arbitrary time-dependent wave function V, which is not necessarily an eigenfunc-tion of H, and [Q, H]) is the expectation value of the commutatorof Q and H for the same wave function. This result is known asEhrenfest's theorem.(b) Use this result to show thatმ(p)=ItavმეWhat is the classical analog of this equation?

Answered: 5.6 (a) Let Q be an operator which is…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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6
A. at pattern. Let's ssuming that the e angular spatial poral frequencies w, correspond to e is then W, 1) ir) [7.33] avelength of the quals the group eing amplitude- e waves of fre- of modulating and sum over is called the wer sideband. 7.19 Given the dispersion relation w = ak', compute both the phase and group velocities. -7.20* Using the relation 1/v = dk/dv. prove that 1 Vg I V₂ 7.21* In the case of lightwaves, show that V = 1 Ve v dv v² dv V 7.22 The speed of propagation of a surface wave in a liquid of depth much greater than λ is given by v dn c dv 11 -+- C solve 7.25 only where g = acceleration of gravity, λ = wavelength, p = density. Y = surface tension. Compute the group velocity of a pulse in the long wavelength limit (these are called gravity waves). 7.23* Show that the group velocity can be written as dv dλ 7.24 Show that the group velocity can be written as gλ 2πΥ + 2πT pλ Vg = V-A ( n + w(dn/dw) 7.25 With the previous problem in mind prove that dn (v) dv n₂ = n(v) + v.…
5.1. (a) Prove that the parity operator is Hermitian. (b) Show that the eigenfunctions of the parity operator corresponding to different eigenvalues are orthogonal.
Legrende polynomials The amplitude of a stray wave is defined by: SO) =x (21+ 1) exp li8,] sen 8, P(cos 8). INO Here e is the scattering angle, / is the angular momentum and 6, is the phase shift produced by the central potential that performs the scattering. The total cross section is: Show that: 'É4+ 1)sen² 8, .
Problem 3.7 (a) Suppose that f(x) and g(x) are two eigenfunctions of an operator Q, with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of Q. with eigenvalue q. (b) Check that f(x) = exp(x) and g(x) = exp(-x) are eigenfunctions of the operator d?/dx², with the same eigenvalue. Construct two linear combina- tions of f and g that are orthogonal eigenfunctions on the interval (-1, 1).
A eat pattern. Let's ssuming that the e angular spatial Doral frequencies , correspond to e is then w, 1) [7.33] avelength of the quals the group 51) eing amplitude- e waves of fre- of modulating and sum over is called the wer sideband. 7.19 Given the dispersion relation w = and group velocities. 7.20* Using the relation 1/v = dk/dv. prove that 1 Ve 7.21* In the case of lightwaves, show that 1 V V= 1 11 C Vg v dv 2² = + ak², compute both the phase V = 7.22 The speed of propagation of a surface wave in a liquid of depth much greater than A is given by v dn c dv solve 7.25 only dv where g = acceleration of gravity, A = wavelength, p = density. Y = surface tension. Compute the group velocity of a pulse in the long wavelength limit (these are called gravity waves). 7.23* Show that the group velocity can be written as Vg = V-A gλ 2πΥ + 2πT pλ dv dλ 7.24 Show that the group velocity can be written as C n + w(dn/dw) 7.25 With the previous problem in mind prove that dn (v) dv n₂ = n(v) + v i…
Hamiltonian of an axially symmetric rotator acting on a state |lm) is given by: H = 21 212 where I, and Iz are the moments of inertia and Ly, Ly and L, are components of the angular momentum operator. a) Find expectation values of H. b) In the case of a rigid rotator (i.e., I, = 1½ = I ), find the energy expression and the corresponding degeneracy relations. c) Calculate the orbital quantum number / and the corresponding energy degeneracy of the rotator where the magnitude of the total angular momentum is v30h.
3.7 Suppose that a wave function (r,t) is normalized. Show that the wave function e¹0 W(r,t), where 0 is an arbitrary real number, is also normalized.
4.7 Let (x.t) be the wave function of a spinless particle corresponding to a plane wave in three dimensions. Show that (x.-) is the wave function for the plane wave with the momentum direction reversed. b. Let x(n) be the two-component eigenspinor of a-n with eigenvalue +1. Using the explicit form of x(n) (in terms of the polar and azimuthal angles $ and y that characterizen) verify that -io₂x() is the two-component eigenspinor with the
= 5.5 The one-dimensional parity operator II is defined by П(x) (-x). In other words, II changes x into -x everywhere in the function. (a) Is II a Hermitian operator? (b) For what potentials V(x) is it possible to find a set of wavefunc- tions which are eigenfunctions of the parity operator and solutions of the one-dimensional time-independent Schrödinger equation?
Vasntben
5.9. Show that if the operator Aop corresponding to the observable A is Hermitian then (4²) ≥ 0

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