Concept explainers
Normalize the following wavefunctions over the range indicated. You may have to use the integral table in Appendix 1.
(a)
(c)
(e)
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Chapter 10 Solutions
EBK PHYSICAL CHEMISTRY
- Bohr’s model can be used for hydrogen-like ions—ions thathave only one electron, such as He + and Li2+ . (a) Why isthe Bohr model applicable to He + ions but not to neutral Heatoms? (b) The ground-state energies of H, He + , and Li2 + aretabulated as follows: By examining these numbers, propose a relationship betweenthe ground-state energy of hydrogen-like systems and thenuclear charge, Z. (c) Use the relationship you derive in part(b) to predict the ground-state energy of the C5+ ion.arrow_forwardBohr’s model can be used for hydrogen-like ions—ions thathave only one electron, such as He + and Li2 + . (a) Why isthe Bohr model applicable to He + ions but not to neutral Heatoms? (b) The ground-state energies of H, He + , and Li2 + aretabulated as follows:By examining these numbers, propose a relationship betweenthe ground-state energy of hydrogen-like systems and thenuclear charge, Z. (c) Use the relationship you derive in part(b) to predict the ground-state energy of the C5 + ion.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
- Suppose 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_forwardImagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In eachcase, give your reasons for accepting or rejecting each function. (i) Ψ(x)=x2; (ii) Ψ(x)=1/x; (iii) Ψ(x)=e-x^2.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 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_forward(1) (2) (b) Look very carefully at the picture below. Give the relevant quantum numbers. Explain your answer. y-axis (c) (1) What is a wavefunction? (ii) What are the two parts of a wavefunction?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_forward
- 5) Determine the y*y for the following wavefunctions: a) y(x) = sin x + icos x b) y(x) = eilxarrow_forwardWrite the normalized form of the ground state wavefunction of the harmonic oscillator in terms of the variable y and the parameter α. (a) Write the integral you would need to evaluate to find the mean displacement <y>, and then use a symmetry argument to explain why this integral is equal to 0. (b) Calculate <y2> (the necessary integral will be found in the Resource section). (c) Repeat the process for the first excited state.arrow_forwardConsider 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
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