Chem 120a spring 2006 1 3 a if you were to use the

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Chem 120A, Spring 2006 1
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3. a) If you were to use the trial function of the form φ ( x ) = ( 1 + c α x 2 ) e - α x 2 / 2 , where α = ( k μ ) 1 / 2 / ¯ h and c is a variational parameter to calculate the ground-state energy of a harmonic oscillator, without doing any calculations, what value of c minimizes the energy? Why? b) Suppose we were to use a trial function of the form φ = c 1 e - α r + c 2 e - β r 2 to carry out a vari- ational calculation for the ground-state energy of the hydrogen atom. Can you guess without do- ing any calculations what c 1 , c 2 , α , and E min will be? What about a trial function of the form φ = 5 k = 1 c k e - α k r e - β k r 2 ? 4. Suppose that the Hamiltonian for a 1D system is given by, ˆ H = - ¯ h 2 2 m d 2 dx 2 + 1 2 m ω 2 x 2 + γ 6 x 3 , (4) where γ is a constant. a) We want to use perturbation theory to solve the Schr¨odinger equation for this Hamiltonian. What is the zero-order (unperturbed) Hamiltonian, ˆ H 0 ? To what system does the unperturbed Hamiltonian correspond? Which part should you treat as the perturbation? b) Given the exact solutions for the unperturbed Hamiltonian, calculate the first and second order cor- rections to the ground state energy due to the perturbation. For the second order correction use only the first 4 eigenfunctions of the unperturbed Hamiltonian. Discuss your answer. c) Use the first 4 eigenfunctions of the unperturbed Hamiltonian to calculate the first order correction to the ground state wavefunction. ( Hint: You can use parity to simplify your calculations. ) 5. The deuterium atom (D) consists of 1 electron, 1 proton, and 1 neutron. The spin of the nucleus (proton + neutron) is 1. In all that follows ¯ I 2 and ¯ I z
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