Please correct answer and don't use hand rating A and B are commuting operators. If your basis functions are eigenfunctions of A,explain how this condition will facilitate the calculation of the eigenvalues of B.Question 3[20 (4, 5, 11)](i)(ii)(iii)If we measure L₂ of a particle that has angular momentum quantum number 1-2, whatare the possible outcomes of the measurement?For a hydrogenlike atom in a stationary state with quantum numbers n, 1, and m, provethat =for³| Rnı |²dr.

(i) A and B are commuting operators. Two operators A and B share a common set of eigenfunctions when…

Answered: A and B are commuting operators. If…

Physical Chemistry

2nd Edition

ISBN:9781133958437

Author:Ball, David W. (david Warren), BAER, Tomas

Publisher:Ball, David W. (david Warren), BAER, Tomas

Chapter10: Introduction To Quantum Mechanics

Section: Chapter Questions

Problem 10.40E

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Similar questions

Indicate which of these expressions yield an eigenvalue equation, and if so indicate the eigenvalue. a ddxcos4xb d2dx2cos4x c px(sin2x3)d x(2asin2xa) e 3(4lnx2), where 3=3f ddsincos g d2d2sincosh ddtan
What is the physical explanation of the difference between a particle having the 3-D rotational wavefunction 3,2 and an identical particle having the wavefunction 3,2?
Indicate which of these expressions yield eigenvalue equations, and if so indicate the eigenvalue. a ddxsinx2b d2dx2sinx2 c iddxsinx2d iddxeimx, where m is a constant e ddx(ex)f (22md2dx2+0.5)sin2x3 g ddy(ey2)
Consider a one-dimensional particle-in-a-box and a three-dimensional particle-in-a-box that have the same dimensions. a What is the ratio of the energies of a particle having the lowest possible quantum numbers in both boxes? b Does this ratio stay the same if the quantum numbers are not the lowest possible values?
How is the Bohr theory of the hydrogen atom inconsistent with the uncertainty principle? In fact, it was this inconsistency, along with the theorys limited application to non-hydrogen-like systems, that limited Bohrs theory.
Based on the trend shown in Figure 11.5, draw the probability distribution of a harmonic oscillator wavefunction that has a very high value of n. Explain how this is consistent with the correspondence principle.
Show that the normalization constants for the general form of the wavefunction =sin(nx/a) are the same and do not depend on the quantum number n.
For a particle in a state having the wavefunction =2asinxa in the range x=0toa, what is the probability that the particle exists in the following intervals? a x=0to0.02ab x=0.24ato0.26a c x=0.49ato0.51ad x=0.74ato0.76a e x=0.98ato1.00a Plot the probabilities versus x. What does your plot illustrate about the probability?
Imagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In each case, give your reasons for accepting or rejecting each function. (1) Þ(x) = x²; (iv) y(x) = x 5. (ii) ¥(x) = ; (v) (x) = e-* ; (iii) µ(x) = e-x²; (vi) p(x) = sinx
(5) Demonstrate that the function p=-; d dx y = exp(timx) is simultaneously an eigenfunction of and p²
4) Is the function = B sin (bx) an eigenfunction of the operator d/dx = = d/dx if B and b are constants and real numbers? Show your work in detail.
Consider a particle of mass m confined to a one-dimensional box of length L and in a state with normalized wavefunction ψn. (a) Without evaluating any integrals, explain why ⟨x⟩ = L/2. (b) Without evaluating any integrals, explain why ⟨px⟩ = 0. (c) Derive an expression for ⟨x2⟩ (the necessary integrals will be found in the Resource section). (d) For a particle in a box the energy is given by En = n2h2/8mL2 and, because the potential energy is zero, all of this energy is kinetic. Use this observation and, without evaluating any integrals, explain why <p2x> = n2h2/4L2.

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