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
- (a) For 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 0xa/4.(b) Calculate this probability for n=1,2, and 3.arrow_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_forward(5) Demonstrate that the function p=-; d dx y = exp(timx) is simultaneously an eigenfunction of and p²arrow_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_forward5. 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_forward
- 2. 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_forwardFind 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_forward
- 3) 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_forward1s (1) 1s (2) (a (1) B (2) – a (2) B (1)) an eigenfunction of the For a two electron system, is b (1, 2): operator Sz = S1z + $22? If so, what is the eigenvalue? (s1z and s2z are the operators for the z-component of the electron spin for electrons 1 and 2, respectively.) O No, because different types of spin are mixed into a superposition state. O Yes, and the eigenvalue is Oh. O Yes, and the eigenvalue is 1h. O No, because the spin wavefunctions for the two electrons are scrambled.arrow_forwardImagine 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) = sinxarrow_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