Midterm 1 Solutions - Chemistry 120A, Spring 2009 Midterm 1...

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Chemistry 120A, Spring 2009 Midterm 1 Solutions February 20, 2009 Problems Question 1: Multiple choice problems: 7 × 3 = 21 points 1. At a finite potential step, the 2nd derivative of the wavefunction Ψ ∂x , exhibits: (a) continuity (b) a finite step (c) an infinite step B 2. All else constant, a smaller force constant for the harmonic oscillator will cause the wavelength corresponding to the energy difference between levels to: (a) increase (b) decrease (c) stay the same A 3. Can a particle in quantum mechanics pass through a potential barrier that is smaller than its kinetic energy? (a) always (b) sometimes (c) never B 4. Which of the following is the quantity | φ ih ψ | ? (a) a number (b) a ket (c) an operator C 5. Which of the following is the quantity h φ | ψ i ? (a) a number (b) a ket (c) an operator A 6. All else being equal, if one compares two wavefunctions, the one with the smaller curvature ± 2 Ψ ∂x 2 ² will have an energy which is: (a) higher (b) lower (c) either is possible B 7. Which of the following has the longest de Broglie wavelength? (a) a 1 eV He atom (b) a 1 eV photon (c) a 1 eV electron B 1
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Question 2: (15 points) Consider applying the Heisenberg uncertainty principle, Δ p Δ x ~ 2 , to the problem of a particle in a one-dimensional box, with box length a . (a) (5 points) Estimate the uncertainties that are expected in the position (from box size) and momentum (from Heisenberg), Δ x and Δ p (1) Because the particle is in the box, we know that Δ x a (Any answer in the range 0 . 1 a Δ x a is acceptable) (2) Then, from the Heisenberg uncertainty principle, Δ p Δ x ~ 2 Δ p ~ x Δ p ~ 2 a (b) (5 points) From your estimate of Δ p estimate the lowest possible energy of the system. The lowest allowed energy level for the particle in a box has a non-zero kinetic energy. This
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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 University of California, Berkeley.

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Midterm 1 Solutions - Chemistry 120A, Spring 2009 Midterm 1...

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