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Unformatted text preview: Chemistry 120A, Spring 2009 Midterm 2 Solutions March 20, 2009 Problems Problem 1: Multiple choice problems: 7 × 3 = 21 points 1. Linear combination of solutions to the time-dependent Schr¨ odinger equation are solutions of the time-dependent Schr¨ odinger equation: (a) always (b) sometimes (c) never A 2. Linear combination of solutions to the time-independent Schr¨ odinger equation are solutions of the time-independent Schr¨ odinger equation. (a) always (b) sometimes (c) never B 3. As the force constant k associated witha diatomic molecule is increased, the uncertainty in the momentum in the vibrational ground state will: (a) increase (b) decrease (c) stay the same A 4. How do you expect the wavelength of the n = 1 to n = 2 transition in the H atom to compare with the same transition in the He + atom? (a) longer in H (b) longer in He + (c) same A 5. Is it possible to simultaneously know the total value of angular momentum and its component along the x axis? (a) always (b) sometimes (c) never A 6. In the linear variational method, can an approximated ground state energy be below the true ground state energy? (a) always (b) sometimes (c) never C 7. What is the symmetry of this spin wavefunction with respect to particle interchange? Ψ(1 , 2) = α (1) β (2)- α (1) β (2) (a) symmetric (b) antisymmetric (c) neither B 1 Problem 2 (20 Points): Perturbartion theroy: You know the soltuions to the problem: ˆ H (0) ψ (0) m E = E (0) m ψ (0) m E and you are going to try to solve for the eigenvalues and eigenvectors of a related problem: ˆ H ψ (0) m E = E m ψ (0) m E where the difference between the two problems is called the perturbation: ˆ H = ˆ H (0) + λ ˆ V (1) and the strength of the perturbation is measured by some parameter λ . (a) (8 points) By writing power series expanstions in λ for the eigenvalues and eigenvectors, derive an expression for the leading correction to the eigenvalues of the solved problem due to the presence of the perturbation....
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This note was uploaded on 02/22/2010 for the course CHEM N/A taught by Professor Head-gordon during the Spring '09 term at Berkeley.
- Spring '09