Concept explainers
Indicate which of these expressions yield an eigenvalue equation, and if so indicate the eigenvalue.
(a)
(c)
(e)
(g)
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Chapter 10 Solutions
Physical Chemistry
- (5) Demonstrate that the function p=-; d dx y = exp(timx) is simultaneously an eigenfunction of and p²arrow_forwardHow many eigenstates of a 3D particle in a box have eigenvalue of E=38h2/(8ma2) if a=b=c? Would changing c change this number?arrow_forwardWhich of the following functions can be normalized (in all cases the range for x is from x = −∞ to ∞, and a is a positive constant): (i) e-ax^2; (ii) e–ax. Which of these functions are acceptable as wavefunctions?arrow_forward
- 2.) The function, f(x) = 3X² - 1, is an eigenfunction of the operator, A = - (1- x)(d²/ dx²) + 2x(d /dx). Find the eigenvalue corresponding to this eigenfunction.arrow_forwardConsider 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.arrow_forwardQ.4 (a) Write down the operator A^2 for (x. d/dx) (b) The operator (x+d/dx) has the eigen value α. Determine the corresponding wave functionarrow_forward
- 5. Consider a particle constrained to move in one dimension described by the wavefunction v (x) = Ne2** (a) Determine the normalization constant (b) Is the wavefunction an eigenfunction of d? +16x? dx? (c) Calculate the probability of finding the particle anywhere along the negative x-axisarrow_forward2. Which of the following wavefunctions are eigenfunctions d? of the operator dx- For those that are eigenfunctions, what is the eigenvalue (a) Y = ex (b) Y = x? (c) Y = sin x (d) Y = 3 cos x (e) Y = sin x + cos xarrow_forwardFor a particle in the stationary state n of a one dimensional box of length a, find the probability that the particle is in the region 0 xa/4. (b) Calculate this probability for n = 1, 2, and 3arrow_forward
- Find the eigenvalue of operating on the function f(x) = Asin(nx) + Bcos(mx) with the following operator: P = d²/dx2 What must be the value of the constants A, B, m and n be to make the function an eigenfunction of this operator? 1.arrow_forwardWhich of the following functions can be normalized (in all cases the range for x is from x = −∞ to ∞, and a is a positive constant): (i) sin(ax);(ii) cos(ax) e-x^2? Which of these functions are acceptable as wavefunctions?arrow_forward3) Consider an arbitrary operator A. A operates on any function to its right. Here are the results of this operator on three different functions: Af(x, y, z) = hf(x, y, z) AY(0) = 04(0) ÂY(x) = Y(x) Which of the functions are eigenfunctions of Â?arrow_forward
- Physical ChemistryChemistryISBN:9781133958437Author:Ball, David W. (david Warren), BAER, TomasPublisher:Wadsworth Cengage Learning,Introductory Chemistry: A FoundationChemistryISBN:9781337399425Author:Steven S. Zumdahl, Donald J. DeCostePublisher:Cengage Learning