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Verify that the following wavefunctions are indeed eigenfunctions of the Schrödinger equation, and determine their energy eigenvalues.
(a)
(b)
(c)
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Chapter 10 Solutions
EBK PHYSICAL CHEMISTRY
- P7D.8* A particle is confined to move in a one-dimensional box of length L. If the particle is behaving classically, then it simply bounces back and forth in the box, moving with a constant speed. (a) Explain why the probability density, P(x), for the classical particle is 1/L. (Hint: What is the total probability of finding the particle in the box?) (b) Explain why the average value of x" is (x")= , P(x)x"dx . (c) By evaluating such an integral, find (x) and (x*). (d) For a quantum particle (x)=L/2 and (x*)=L (}-1/2n°n²). Compare these expressions with those you have obtained in (c), recalling that the correspondence principle states that, for very large values of the quantum numbers, the predictions of quantum mechanics approach those of classical mechanics.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_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
- What are the allowed total angular momentum quantum numbers of a composite system in which j1 = 5 and j2 = 3?arrow_forward8a. What is the expectation value of the linear momentum for the 1-D wavefunction: -ax 2 (x) = Ne where -0arrow_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_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_forwardConsider a 1D particle in a box confined between a = 0 and x = 3. The Hamiltonian for the particle inside the box is simply given by Ĥ . Consider the following normalized wavefunction 2m dz² ¥(2) = 35 (x³ – 9x). Find the expectation value for the energy of the particle inside the box. Give your 5832 final answer for the expectation value in units of (NOTE: h, not hbar!). In your work, compare the expectation value to the lowest energy state of the 1D particle in a box and comment on how the expectation value you calculated for the wavefunction ¥(x) is an example of the variational principle.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_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_forwardThe radial wave function of a quantum state of Hydrogen is given by R(r)= (1/[4(2π)^{1/2}])a^{-3/2}( 2 - r/a ) exp(-r/2a), where a is the Bohr radius. (a) Show analytically that this function has an extremum at r=4a. (b) Sketch the graph of R(r) x r. For a decent sketch of this graph, take into account some values of R(r) at certain points of interest, such as r=0, 2a, 4a, and so on. Also take into account the extremes of the function R(r) and their inflection points, as well as the limit r--> infinity. (c) Determine the radial probability density P(r) associated with the quantum state in question. (d) Show that the function P(r) you determined in part (c) is properly normalized.arrow_forwardConsider the wave function V222(x, ỹ, z) for a particle in a cubic box. Figure 4.45a shows a contour plot in a cut plane at ž = 0.75. (a) Convince yourself that the contour plot in a cut at ž = 0.25 would have the same pattern, but each positive peak would become negative, and vice versa. (b) Describe the shape of this wave function in a plane cut at ỹ = 0.5.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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