quiz8soln - 1 / 2 Z ∞-∞ x 2 e-γx 2 dx = ¯ h 2 γ 4 μ...

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1 Problem 1 In previous discussion, we have considered approximate methods for obtain- ing energies, wavefunctions, or both for quantum systems more complicated than the Hydrogen atom. In this exercise, consider the quantum harmonic oscillator and use the variational principle to evalute the ground state wave- function and the ground state energy. Use the following normalized trial wavefunction: χ = ± γ π ² 1 4 e - γx 2 / 2 γ = variational parameter The Hamiltonian for the one-dimensional quantum harmonic oscillator is: ˆ H = - ¯ h 2 2 μ d dx 2 + kx 2 2 You are free to use your handbook of equations for evaluating necessary integrals. 1
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1.1 Solution The one-dimensional harmonic oscillator: ˆ H = - ¯ h 2 2 μ d 2 dx 2 + kx 2 2 Let’s consider a trial wavefunction: ˆ H = - ¯ h 2 2 μ d 2 dx 2 + kx 2 2 Trial wavefunction: χ = ± γ π ² 1 / 4 e - γx 2 / 2 ; γ variable ( variational parameter ) E var = Z χ * ˆ Hχd r = ± γ π ² 1 / 2 Z e - γx 2 / 2 ³ - ¯ h 2 2 μ d 2 dx 2 + kx 2 2 ´ e - γx 2 / 2 dx = - ¯ h 2 2 μ ± γ π ² 1 / 2 µZ -∞ γe - γx 2 dx + Z -∞ ( γx ) 2 e - γx 2 dx + k 2 ± γ π ²
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Unformatted text preview: 1 / 2 Z ∞-∞ x 2 e-γx 2 dx = ¯ h 2 γ 4 μ + k 4 γ ≥ E o Now, allow γ to vary and optimize E var : dE var dγ = ¯ h 2 4 μ-k 4 γ 2 = 0 γ = ( kμ ) 1 / 2 ¯ h ≡ α This leads, with no surprise, to the exact solutions to the 1-D H.O. χ = ψ o ( x ) = ± α π ² 1 / 4 e-αx 2 / 2 → E var = ¯ hω 2 = E o 2 2 Problem 2 What is the general expression for the Frst order perturbation correction to the ground state energy of a system with perturbation Hamiltonian ˆ H 1 ? 2.1 Solution E 1 o = Z all space ψ * ˆ H 1 ψ d 3 r 3 Problem 3 What are the e±ective nuclear charges ”seen” by electrons in the 1 s , 2 s , and 2 p atomic orbitals for the carbon atom? What is the true nuclear charge for this system? 3.1 Solution 1s = 5.67 2s = 3.22 2p = 3.14 3...
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This note was uploaded on 02/02/2012 for the course CHEM 444 taught by Professor Dybowski,c during the Fall '08 term at University of Delaware.

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quiz8soln - 1 / 2 Z ∞-∞ x 2 e-γx 2 dx = ¯ h 2 γ 4 μ...

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