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…
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.
= 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?
5.9. Show that if the operator Aop corresponding to the observable A is Hermitian then (4²) ≥ 0
6.1.5 Let X and Y be discrete random variables with joint PMF Px,y (x, y) = 0 0.01 x 1,2..., 10, = y= 1,2..., 10, otherwise.
2.2. (a) Verify explicitly the invariance of the volume element do of the phase space of a single particle under transformation from the Cartesian coordinates (x,y, z, px, Py, Pz) to the spherical polar coordinates (r,e,4,Pr.Pe,Po). (b) The foregoing result seems to contradict the intuitive notion of "equal weights for equal solid angles," because the factor sine is invisible in the expression for do. Show that if we average out any physical quantity, whose dependence on po and po comes only through the kinetic energy of the particle, then as a result of integration over these variables we do indeed recover the factor sin0 to appear with the subelement (de do).
4.7 Consider an infinite square-well potential of width a, but with the coordinate system shifted so that the infinite potential barriers lie at x = a/2 and x = a/2 (see diagram): V(x) x = =-= x = 12 X (a) Solve the Schrödinger equation for this case to calculate the normalized wave functions Un(x) and the corresponding energies En. (b) Explain why you get the same energies as for the square well between x = 0 and x = a, but a different set of wave functions.

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