Midterm Examination 1

Size: px

Start display at page:

Download "Midterm Examination 1"

Lynette King
5 years ago
Views:

1 CHEM 332 Physical Chemistry Spring 2014 Name: Answer Key Midterm Examination 1 1. Match the individual with their accomplishment: A. Millikan G Proposed a probabilistic interpretation for *. B. Planck D Discovered the nuclear structure of the atom. C. Young H Discovered electron diffraction. D. Rutherford A Confirmed Einstein s photoelectric theory. E. Balmer B Quantized energy transfer in a blackbody cavity. F. de Broglie C Confirmed the wave nature of light. G. Born F Postulated a wave-particle duality for matter. H. Germer E Developed a formula to predict the wavelengths of Hydrogen s visible spectral lines. 2. Discuss how the Copenhagen interpretation of QM differs from that of Einstein et al. From Levine: "The probabilistic nature of quantum mechanics has disturbed many eminent physicists, including Einstein, de Broglie, and Schrodinger. These physicists and others have suggested that quantum mechanics may not furnish a complete description of physical reality. Rather, the probabilistic laws of quantum mechanics might be simply a reflection of deterministic laws that operate at a subquantum-mechanical level and that involve 'hidden-variables.' " "The orthodox interpretation (often called the Copenhagen interpretation) of quantum mechanics, which was developed by Heisenberg and Bohr, denies the existence of hidden variables and asserts that the laws of quantum mechanics provide a complete description of physical reality."

2 3. What is the wavelength of an electron (m e = 9.11 x kg) with a Kinetic Energy of 2.18 x J? T = ½ mv 2 or v = = 2.19 x10 6 m/sec = h/mv = (6.626 x J sec) / (9.11 x kg) (2.19 x 10 6 m/sec) = 3.32 x m 4. A Postulate of Bohr's model of the Hydrogen atom states: Certain orbits are stable and correspond to Stationary States. a) What physical problem was this Postulate intended to fix? An orbiting electron appears to be an oscillating charge which should act like a small transmitter radiating electromagnetic waves, thus constantly loosing energy. So, the electron should spiral into the nucleus. b) How was this problem more naturally fixed by de Broglie? The electron behaves as a wave which forms a standing wave pattern about the nucleus and appears stationary. 5. Balmer s original analysis of the visible spectrum of Hydrogen yielded the following formula for the wavelength s of the spectral lines: = [nm] Show that this formula is equivalent to the Rydberg formula for Hydrogen s visible lines. Explicitly determine R H using this formula. 1/ = R H = nm -1

3 6. Let and be Hermitian operators. Show that is Hermitian. Why do we care whether or not our operators are Hermitian? = because and are Hermitian Quod Erat Demonstrantum 7. Compute the following: where f(x) = 2x Do the (+ 3) and d/dx operators commute? The operators do not commute. If this had equaled zero, you could not conclude the operators commute. You would need to demonstrate that the commutator always equals zero.

4 8. a) What do we expect from "well-behaved" Wave Functions? Continuous Continuous First Derivative Single Valued Square Integrable b) What is wrong with the following Wave Function for a Particle in a One Dimensional Box with a Finite Wall? The Wave Function is not Square Integrable. (It should decay in the classically forbidden region.) c) Correct the above sketch. Note both the exponential decay in the classically forbidden region where the particle is tunneling, which corrects the square integrability, and the continuous first derivative at the boundary.

9. For a particle in a one-dimensional box: = n = 1, 2, 3, 4, a) Provide a sketch of 2 and 3. b) Set-up the integral that would need to be solved to show that 2 and 3 are orthogonal.

5 9. For a particle in a one-dimensional box: = n = 1, 2, 3, 4, a) Provide a sketch of 2 and 3. b) Set-up the integral that would need to be solved to show that 2 and 3 are orthogonal. c) Argue, based on your above sketches, that 2 and 3 are orthogonal. The symmetry of the functions, with half of 2 negative and the other half positive, ensures the product 2 3 must integrate to zero.

6 Fundamental Constants Avogadro s Number N A x entities/mole Boltzmann s Constant k x Joule/K Fundamental Charge e x C Mass of the Electron m e x kg Mass of the Proton m p x kg Mass of the Neutron m n x kg Planck s Constant h x Joule sec Rydberg s Constant R H cm -1 Speed of Light c x 10 8 m/sec

Chapter 1. From Classical to Quantum Mechanics

Chapter 1. From Classical to Quantum Mechanics Classical Mechanics (Newton): It describes the motion of a classical particle (discrete object). dp F ma, p = m = dt dx m dt F: force (N) a: acceleration