Concept explainers
State whether the following functions are acceptable wavefunctions over the range given. If they are not, explain why not.
(a)
(b)
(c)
(d)
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Chapter 10 Solutions
Physical Chemistry
- 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?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_forwardA normalized wavefunction for a particle confined between 0 and L in the x direction is ψ = (2/L)1/2 sin(πx/L). Suppose that L = 10.0 nm. Calculate the probability that the particle is (a) between x = 4.95 nm and 5.05 nm, (b) between x = 1.95 nm and 2.05 nm, (c) between x = 9.90 nm and 10.00 nm, (d) between x = 5.00 nm and 10.00 nm.arrow_forward
- Consider a fictitious one-dimensional system with one electron.The wave function for the electron, drawn below, isψ (x)= sin x from x = 0 to x = 2π. (a) Sketch the probabilitydensity, ψ2(x), from x = 0 to x = 2π. (b) At what value orvalues of x will there be the greatest probability of finding theelectron? (c) What is the probability that the electron willbe found at x = π? What is such a point in a wave functioncalled?arrow_forwardIf two wavefunctions, Wa and Wb, are orthonormal and degenerate, then what is true about the linear combinations 1 1 w. +v.) a a and (a) y+ and y- are orthonormal. (b) y+ and y- are no longer eigenfunctions of the Schrödinger equation. (c) V+ and y- have the same energy. (d) V+ and Y- have the same probability density distribution.arrow_forward[p* L₂ da 0 HO H + →arrow_forward
- Calculate the probability that a particle will be found between 0.49L and 0.51L in a box of length L for (i) ψ1, (ii) ψ2. You may assume that the wavefunction is constant in this range, so the probability is ψ2δx.arrow_forward106. Combining two real wave functions ₁ and 2, the following functions are constructed: A = ₁ + $₂₂ B = = ₁ +i0₂, C = ₁ −i0₂, D=i(0₁ +0₂). The correct statement will then be (a) A and B represent the same state (c) A and D represents the same state (b) A and C represent the same state. (d) B and D represent the same state.arrow_forwardSuppose that 1.0 mol of perfect gas molecules all occupy the lowest energy level of a cubic box. (a) How much work must be done to change the volume of the box by ΔV? (b) Would the work be different if the molecules all occupied a state n ≠ 1? (c) What is the relevance of this discussion to the expression for the expansion work discussed in Topic 2A? (d) Can you identify a distinction between adiabatic and isothermal expansion?arrow_forward
- Consider the three spherical harmonics (a) Y0,0, (b) Y2,–1, and (c) Y3,+3. (a) For each spherical harmonic, substitute the explicit form of the function taken from Table 7F.1 into the left-hand side of eqn 7F.8 (the Schrödinger equation for a particle on a sphere) and confirm that the function is a solution of the equation; give the corresponding eigenvalue (the energy) and show that it agrees with eqn 7F.10. (b) Likewise, show that each spherical harmonic is an eigenfunction of lˆz = (ℏ/i)(d/dϕ) and give the eigenvalue in each case.arrow_forward(a) If  = 3x? and B = , then show that  and ß donot commute with respect to the function f(x) = sin x. Show, if the wave function, w) = A cos(kx) + iA sin(kx) is an Eigen-function of the linear momentum operator, P and if so, what is the Eigen value. (Note: A and k are constants). (b)arrow_forwardwavefunction iš E7D.5(b) Calculate the probability that a particle will be found between 0.65L and 0.671 in a box of length L for the case where the wavefunction is (i) Y, (ii) y.. You may make the same approximation as in Exercise E7D.5(a).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