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Introduction to Quantum Mechanics Homework #2 (Due by April 8, 2015) 1. Consider a box (-L ≦ x ≦ L) -normalized free particle wave function Ψ= N [(e3ikx + 2e-2ikx)]. 1) Express the normalization constant N in terms of L. 2) If one measure px, what value will be resulted? Repeat the same problem for E. 3) Since it is a mixed state, it is neither an eigenfunction of the momentum operator or of the energy operator Ĥ. Calculate the expectation values of the momentum and energy, respectively. 2. Consider a 1-D particle (mass m) in a box (size L). 1) Calculate the probability P of finding the particle in the region 0 <x< L/4 for the ground state. 2) Repeat the problem to calculate Pn for a state with a translational quantum number n. 3) Show that the ‘correspondence principle’ holds true as n approaches ∞. 3. The average translational energy of N2 is <Etrans> = 3/2 kT. 1) Calculate the quantum number n = (nx2 + ny2+ nz2)1/2 corresponding to <Etrans> = 3/2 kT for N2 at 298 K. 2) What can you learn from the magnitude of n about the energy gap between two adjacent energy levels. 4. An electron is approaching a rectangular potential barrier from the left, where L = 4 Å and V0 = 15 eV. The De Broglie wavelength of the electron is 8 Å. 1) Calculate the energy of the electron in eV. 2) Calculate the tunneling probability through the barrier. 5. The valence electrons of a conjugated linear hydrocarbon molecule with alternating double and single c-c bonds (ex: CH2=CH-CH=CH-CH=CH2) can be considered a 1-D particle in a box because the electron is delocalized over the entire carbon chain. The bond length of c=c is 1.3 Ǻ and that of c-c is 1.54 Ǻ. Neglect the edge effect. Calculate the wavelength of the light in Ǻ emitted in the electronic transition n=2→ n=1 for the molecule CH2=CHCH2=CH-CH2=CH2. 6. 1) Solve the differential equation y’ = 2xy by an ordinary method. 2) Repeat the problem by using the power series method. 7. The ground state wavefunction of a 1D H.O. is given by Ψ(x) = N0 exp (- x2/2α2), where α = (h2/4π2mk)1/4 1) Determine the normalization constant N0 . 2) Show that Ψ(x) is an eigenfunction of the Hamiltonian operator with an eigenvalue of E0 = hν/2 3) Show that <T> = <V> = E /2 for the first excited state. 8. The vibrational motion of 1H19F can be approximated as a harmonic oscillator. The force constant k = 966 N/m. 1) Calculate the reduced mass of 1H19F. 2) Calculate ω 3) Calculate the wavenumber in cm-1 unit of the infrared light emitted by a vibrationally excited HF* molecule. The selection rule is that only Δv = -1 is allowed in emission. 9. Show that the operator Lz =( h/2πi) ∂/∂Φ 10. Suppose that Ĥ1 (x ) Φ1 (x) = E1 Φ1 (x) and Ĥ2 (y) Φ2 (y) = E2 Φ2 (y). Let Ĥ(x,y)= Ĥ1(x) +Ĥ2(y). Then, show that Ψ(x,y) = Φ1(x)Φ2(y) is an eigenfunction of Ĥ and the corresponding eigenvalue is E = E1 + E2. 11. The bond length of H2 is 0.74 Ǻ. Assuming H2 is a rigid rotor, 1) Calculate the moment of inertia of H2 molecule. 2) Which rotational quantum number l is the most probable for H2 in thermal equilibrium at 300 K? The 3D rotational wavefunction Y2,1 = (15/8π) cosθ sinθ exp (iΦ) 1) Show that Y2,1 is an eingenfunction of the operators L2 and Lz. 2) What are the eingenvalues of E, L2, and Lz , respectively. 3) What is the magnitude of the angular moment for this state? 4) Calculate the angle between L and the z-axis? 12. A rotational state of a 3D rigid rotor is represented by Ψ = N (Y3,0 + 3 Y2,-1 + 2 Y1,1 + Y0,0) 1) Calculate the normalization constant N. Hint: <Yl,ml│Yl’,ml’> = δl,l’ δml, ml’ (orthonormal) 2) Show the results (probability) of the E, L2, and Lz measurements by a Histograms, respectively. Assume that the measurements are made many times. 3) What are the expectation values of E, L2, and Lz, respectively.