Chemistry 4 Problem Set 4 Spring 18 Solutions 1. V I II III a b c A one-dimensional particle of mass m is confined to move under the influence of the potential x a V V (x) = a < x b b x c elsewhere and pictured above. Give the Schrödinger equation for the particle in regions I, II and III, and impose the proper boundary conditions for the wavefunctions in each of the three regions. d ψ I m dx = Eψ I d ψ II + V m dx ψ II = Eψ II 1
d ψ III = Eψ m dx III ψ I () = ψ I (a) = ψ II (a) ψ II (b) = ψ III (b) ψ III (c) = ψ I(a) = ψ II(a) ψ II(b) = ψ III(b). V I II III a b A particle of mass m moves under the influence of the potential function pictured above and defined by x a V (x) = V a < x b x > b Give the Schrödinger equation for the particle in the three regions I ( x a), II (a < x b) and III(x > b) depicted in the figure, defining ψ I (x) to be the wavefunction for region I, ψ II (x) to be the wavefunction for region II and ψ III (x) to be the wavefunction for region III. Give the appropriate boundary conditions for ψ I (x), ψ II (x) and ψ III (x). d ψ I m dx = Eψ I d ψ II m dx d ψ III m dx + V ψ II = Eψ II = Eψ III
. Show that the wavefunction ψ I (a) = ψ II (a) ψ II (b) = ψ III (b) ψ I () = ψ(x) = e x / ψ I(a) = ψ II(a) ψ II(b) = ψ III(b) with = µω/ satisfies the one-dimensional Schrödinger equation for a harmonic oscillator of reduced mass µ and angular frequency ω. Be sure to identify the energy associated with ψ(x). Answer Ĥψ = d ψ µ dx + 1 µω x ψ = Eψ so that ψ (x) = xe x / 4. Show that the wavefunction ψ (x) = [ + x ]e x / Ĥψ = µ [ + x ]e x / + 1 µω x e x / [ µω = ( µ µ ω x µ + 1 )] µω e x / = ω e x / E = ω ψ(x) = x exp( mωx /) is an eigenfunction of the Hamiltonian operator for a harmonic oscillator of mass m and natural frequency ω. What energy is associated with this wavefunction? Ĥ = d m dx + 1 mω x Then d ψ dx = e mωx / ψ = xe mωx / [ dψ dx = / e mωx 1 mω ] x [ mω x mω x + ( mω ) x ] d ψ m dx + 1 [ mω x ψ = e mωx / ωx 1 mω x + 1 ] mω x = ωxe mωx /
= ωψ(x) Then E = ω 5. Consider the wavefunction given in problem 4. (a) Normalize the wavefunction. Answer N Then = N x e mωx / dx = 1 / mω [ ] mω / 1/ N = (b) Calculate the expectation value of the kinetic energy of the oscillator using this wavefunction. K = mω / xe mωx / ( d ) xe mωx / dx m dx Using the expression for the second derivative from problem 4 K = mω / [ xe mωx / mω m x mω mω x + x ] e mωx / dx = m = m mω / mω / mω e mωx / [ mω / mω = 4ω x mω x 4] dx mω ( 4 mω (c) As what point(s) in space is the oscillator most likely to be found? d dx x e mωx / [ = e mωx / x x mω x mω x = 4 ] = ) 5/
x = ± mω (d) What characteristics of the wavefunction of problem 4 indicate that the energy associated with the wavefunction is not the ground state energy? The wavefunction has a node, so that the wavefunction cannot represent the ground state. 6. Explicitly calculate V and K for the ground state of a harmonic oscillator of mass m and frequency ω, and show that K = V. = µω 1/4 ψ (x) = e x / Then 1/4 V = e x / 1 1/4 µω x e x / dx 1/ 1 = µω x e x dx 1/ 1 µω = µω = / 4 = µω 4µω = ω 4 K = 1/ e x / d / µ dx e x dx = 1/ e x / d / µ dx [ xe x ]dx = 1/ e x / [ + x ]e x / dx µ = 1/ [ ] e x dx + e x x dx µ = [ µ 4µ ] = 4µ K = V µω = ω 4 7. A quantum particle of mass m is subjected to the potential energy function { x < V (x) = 1 mω x x > where ω is a constant having units of frequency. Note that the domain of this system can be taken to be < x <. 5
(a) Give the boundary conditions that are appropriate for the wavefunctions which are solutions to the time-independent Schrödinger equation for the particle. lim x ψ(x) = lim ψ(x) = x (b) Show that the non-normalized wavefunction where ψ(x) = Nx exp( x / ) = 1/ mω satisfies the boundary conditions specified in part a. For ψ(x) = Nxe x / ψ() = and lim ψ(x) = x (c) Normalize the wavefunction given in part b. N x e x / dx = N 4 = 1 1/ 4 N = ψ(x) = 1/ 4 xe x / (d) Determine the value of x at which the particle in the quantum state defined in part b of this problem is most likely to be found. [ ] d dx x x / e = e x / x x = at x = (e) Calculate the expectation value of the momentum p for the particle in the quantum state defined in part b of this problem. p = 4 i 6 xe x / d dx / xe x dx
= 4 [ ( i xe x / e x / 1 x [ ] = 4 i 1 4 8. In problem 7 we have considered the half oscillator of mass m and natural frequency ω, and we verified that ψ 1 (x) = Nxe x / with = mω/ and N the normalization factor, is the ground-state eigenfunction of the Hamiltonian operator for x <. Determine an expression for the ratio x /x mp where x is the expectation value of x for the half oscillator in its ground state and x mp is the most likely location for the half oscillator in its ground state. Answer N x e x dx = N /4 = 1 N = 4/ 1/4 4 / x = 1/ x e x dx = 4/ 1 = 1/ = d dx x e x = (x x )e x = at x mp = 1 )] Then x x mp = 9. The normalized wavefunction for the first excited state of a harmonic oscillator of mass m and natural frequency ω having potential energy V (x) = 1/ mω x and total energy E 1 = ω/ is given by ψ 1 (x) = 4 1/4 xe x / where = mω/. Show that the probability that an oscillator in its first excited state is found in the tunneling region is given by P = 4 y e y dy ω = 1 mω x x = ω mω = mω = or x = ± 7
P = x = y ( 4 ) 1/ x e x x = y dx dx = dy P = ()() / 1 / y e y = 4 y e y 1. The normalized wavefunction and energy for a harmonic oscillator of mass m in its second excited quantum state are given by dy dy 1/4 ψ (x) = (x 1)e x / 4 E = 5 ω where = mω/, with ω being the natural angular frequency of the oscillator. Recalling that the potential energy function for a harmonic oscillator is given by V (x) = 1/ mω x, calculate an expression for the probability that an oscillator in its second excited state is found in the tunneling region. A numerical answer is not required, and you should leave your final expression in terms of exponentials and the complimentary error function [erfc(x)]. 5 ω = 1 ( 5 mω x x = ± mω ) 1/ 5 1/ = ± 1/ P = (x 1) e x 4 5/ y = x x = y dx = dy P = 1 5 (y 1) e y dy = 1 [e 5 (5 1/ + 5 / ) + erfc( 5)] 11. Consider a particle of mass m confined to move on the perimeter of a ring of radius R. Evaluate < L z > for the particle if the particle s wavefunction is given by dx (a) e iφ ; L z = 1 e iφ i φ e iφ dφ = dφ = 8
(b) e iφ ; (c) cos φ. L z = 1 e iφ i φ eiφ dφ = dφ = L z = N cos φ cos φdφ i φ = N i cos φ sin φdφ = 1. In spherical polar coordinates the operator corresponding to the square of the angular momentum ˆL is given by ˆL = ( 1 sin θ = ( cos θ sin θ θ sin θ θ + 1 sin θ θ + θ + 1 sin θ ) φ ) φ Show that the spherical harmonic Y1 (θ, φ) given in Eq. (7.61) on page 11 of the text is an eigenfunction of ˆL. Y1 (θ, φ) is independent of φ, and we need only consider the θ dependence of the angular momentum operator. We can also ignore all constants. Then cos θ sin θ θ + cos θ = cos θ cos θ = cos θ. θ 1. Consider a particle of mass m confined to move on the perimeter of a ring of radius R with associated Hamiltonian operator Ĥ = with the moment of inertia given I dφ by I = mr and the coordinate φ in the range φ <. Show that the wavefunction ψ(φ) = A sin φ satifies the time-independent Schrödinger equation for the system with A the normalization constant. Use the wavefunction to evaluate an expression for L z L z. The operator corresponding to the square of the z-component of the angular momentum of the system is just the square of ˆL z ; i.e. ˆL z = d dφ. ψ = A sin φ ψ = A cos φ ψ = A sin φ d ψ I dφ = I 9 d A sin φ = I ψ
[ φ A sin φ dφ = A 1 ] 4 sin φ = A = 1 A = 1 L z = 1 sin φ d i dφ sin φ dφ = 1 sin φ cos φ dφ = L z = 1 sin φ d sin φ dφ = sin φ dφ = dφ L z L z = 14. Consider a particle of mass m confined to move on the perimeter of a ring of radius R with associated Hamiltonian operator Ĥ = with the moment of inertia given I dφ by I = mr and the coordinate φ in the range φ <. Show that the wavefunction ψ(φ) = A(e iφ + e iφ ) satisfies the time-independent Schrödinger equation for the system with A the normalization constant. Normalize the wavefunction, and use the result to evaluate an expression for L z for the particle. d ψ (φ) = A(ie iφ ie iφ ) ψ (φ) = A( 9e iφ 9e iφ ) = 9ψ(φ) Ĥψ = I ψ (φ) = 9 I ψ(φ) A dφ (e iφ + e iφ )(e iφ + e iφ ) = A dφ (1 + 1 + e 6iφ + e 6iφ ) = 4A = 1 A = 1 L z = 1 dφ (e iφ + e iφ ) d 4 i dφ (eiφ + e iφ ) = 4 dφ (1 1 + e 6iφ e 6iφ ) = 15. Consider a particle of mass m confined to move on the perimeter of a ring of radius R with associated Hamiltonian operator Ĥ = I dφ = ˆL z with the moment of inertia I given by I = mr and the coordinate φ in the range φ <. Show that the wavefunction ψ(φ) = A cos 5φ satisfies the time-independent Schrödinger equation for the system with A the normalization constant. Normalize the wavefunction, and use the result to evaluate an expression for L z L z for the particle. ψ (φ) = 5A sin 5φ ψ (φ) = 5A cos 5φ = 5ψ 1 d
Normalization Ĥψ = 5 5ψ = Eψ or E = I I A cos 5φ dφ = 1 = A ψ(φ) = 1 cos 5φ A = 1/ L I z = I Ĥ = = 5 L z = i = 5 i cos 5φ d cos 5φ dφ I dφ cos 5φ dφ = 5 cos 5φ d cos 5φ dφ dφ cos 5φ sin 5φ dφ = L z L z = 5 11