Assessment of Quantum Mechanical Concepts

Detroit Mercy offers a comprehensive engineering program with degrees in mechanical, civil, robotics and mechatronic systems, electrical, computer, environmental, and architectural engineering. The College of Engineering & Science has a well-established co-operative education program with a long history of placing graduates into the workforce upon graduation. Located in the city of Detroit the college has close ties to the automobile industry, its numerous suppliers and local defense contractors. Detroit Mercy engineering students take a comprehensive physics sequence during the winter semester of their freshman year and fall semester of their sophomore year. The college offers PHY 3690 Modern Physics with Device Applications as a junior level physics course. The course is required of electrical engineers and offered as an elective to other engineering students. The class covers introductory topics in quantum mechanics leading to a basic understanding of the behavior of charge carriers in solids. A description of the course and the students will be presented later in the paper. Students are introduced to entanglement and quantum computation with computer simulations of quantum measurements. We believe that a brief introduction to these topics helps students understand the relationship between operators and the results of a measurement of the wavefunction. Over the past several years we have assessed students in the course with the Quantum Mechanics Conceptual Survey (QMCS).[1] This instrument was designed to be used as a general survey of students’ conceptual understanding. One of the interesting aspects of this instrument is that engineering students in modern physics courses were considered during its development and validation. In this paper we will analyze our students’ conceptual understanding of quantum mechanical concepts and compare them with those students that participated in the development of the survey. Responses to sample questions will be examined and student difficulties will be identified. We believe readers will be surprised as to how persistent certain student misconceptions appear to be. Course Description and Content Modern Physics with Device Applications PHY 3690 is a junior level course offered by the physics department. The class is required for electrical engineers and is a technical elective for other engineering or science majors—registration of non-electrical engineers is unusual. The class is offered in the winter term and for the past three years, the period over which the QMCS instrument was administered, the enrollment averaged 8 students per term; typically, one of those students was female. The prerequisite for the course is successful completion of one year of calculus-based general physics with the associated laboratories. The typical student has completed a course in differential equations with linear algebra. Engineering students are introduced to MATLAB [2] during their freshman year. We leverage this knowledge of the MATLAB environment along with their experience with linear algebra to manipulate vectors and matrices—the original language of quantum mechanics. The specific learning outcomes from the most recent syllabus are: Students will use distribution functions to describe physical systems and apply the concepts to blackbody radiation. They will analyze electromagnetic radiation in terms of the wave and particle models, and solve problems dealing with spontaneous and stimulated emission of radiation. Students will solve nonlinear equations using numerical techniques. They will apply the Bohr model to analyze electron energy levels in atoms and relate those levels to observed line spectra. Students will apply the de Broglie and Heisenberg hypotheses; analyze wave packets and recognize the probabilistic interpretation of the wave function. Students will use Dirac notation to represent quantum states and unitary matrices to represent operators. They will simulate quantum computation experiments utilizing MATLAB. Students will solve the Schrödinger equation in one dimension for various potentials. Students will identify cubic crystal lattices and use standard notation to identify planes and directions. They will identify dopant and impurity types; draw energy band diagrams and relate the structure of the bands to physical properties; develop the concepts of electrons and holes in materials and study the effects of their concentrations on space-charge and diffusion. They will analyze the statistics of electron occupation using FermiDirac statistics; identify and analyze current flow mechanisms in pn junction diodes, solar cells, and transistors. Students will analyze nanoscopic materials such as graphene and other interesting 2-dimensional materials. The course topics include: 1) Properties of Light a) Spectral Irradiance and Blackbody Radiation b) Photoelectric Effect and the Photon Concept 2) Nuclear Atom a) Atomic Spectra and the RutherfordBohr Model of Atomic Structure b) Spontaneous and Stimulated Emission of Radiation 3) Wave Properties of Matter a) The de Broglie Hypothesis and the Heisenberg Uncertainty Principle b) Wave Packets 4) Quantum Computation and Simulation a) Dirac Notation b) Matrices and Operators c) Mermin’s Device and Entanglement 5) The Schrödinger Equation a) One Dimensional Examples b) Expectation Values and Operators c) Quantum States and Superposition 6) Crystal Properties a) Hard Sphere Model and Density b) Crystal Lattices and Miller Notation 7) Quantum Theory of the Solid State a) Energy-Band Theory b) Quantum Statistical Mechanics 8) Charge carriers a) Donors and Acceptors b) Chemical Potential and Fermi Energy c) Drift and Diffusion Currents 9) Semiconductor Junctions a) Equilibrium Conditions b) Current-Voltage Characteristics c) Metal-Insulator-Semiconductor structures 10) Solar Cells and Lasers a) Optical Absorption and Gain b) Current-Voltage Characteristics 11) Nanoscopic Materials a) Graphene b) 2–dimensional electronic systems The Dirac notation [3], [4] and curriculum associated with simulated quantum computation [5] are treated throughout the course. Mermin’s Device [6] is discussed in the third week of class to introduce entanglement. After the publication of Mermin’s original paper in 1981 he developed other variants of his device [7], [8] that are not discussed in the class. The other thought experiments that Mermin subsequently developed are more appropriate for an advanced audience; the devices he describes do not require perfectly correlated particles. The concept of entanglement is fundamental to quantum mechanics and was first introduced by Schrödinger in 1935. However, as Schroeder [9] points out, the word has been virtually absent from publication until the 1980’s. Various aspects of quantum computation are revisited throughout the course as MATLAB projects. These projects escalate in complexity and are used to reinforce the value of the quantum simulations. The quantum computational simulations are based on the published work of Candela.[10] Assessment Instrument The QMCS 2.0 is a research-based instrument developed to survey students’ conceptual understanding of quantum mechanics. It is a 12-question multiple choice survey of student understanding of various topics in introductory quantum mechanics or modern physics courses. As discussed by the authors, it is written using everyday language, it is conceptual in nature with no need to memorize formulas, the distractors are believed to be effective at discriminating students’ preconceived notions, and most faculty believe that it is too easy. It is administered during the last week of the course and does not count against a student’s grade. Part of the validation of the survey involved interviewing faculty that have recently taught a modern physics or quantum mechanics course. Faculty buy-in is believed to be an important factor that can affect teaching practice. Faculty have absolutely no consensus about which topics are important in a quantum mechanics course. Some believe that concepts should be taught while others are of the mindset of “shut up and calculate.” The concepts that had the most overlap among faculty, listed from highest to lowest, were: i. wave function and probability, ii. wave-particle duality, iii. Schrödinger equation, iv. quantization of states, v. uncertainty principle, vi. superposition, vii. operators and observables, viii. tunneling, ix. measurement. Reviews of textbooks and syllabi showed a great deal of overlap in the topics covered and a surprising lack of discussion of measurement, wave function collapse etc. Our intent is to compare student responses from the published QMCS data to that of the students taking PHY 3690. The authors of the QMCS used input from faculty teaching modern physics for engineers in the design of the instrument so utilizing the instrument for our engineering students seems appropriate. To protect the fidelity of the QMCS we will not reproduce the test here. We do discuss some of the questions that were presented in the original manuscript. The QMCS authors recommend using the instrument as a formative assessment of student understanding of quantum mechanical concepts. The authors encourage faculty to administer the test in modern physics courses to inform their teaching and to publish results for the benefit of the broader community. Sample Questions from Quantum Mechanics Conceptual Survey Question 1 from the survey is shown below along with the percentage of responses from the QMCS group and from the modern physics courses at Detroit Mercy in Figure 1. The correct answer is given as selection D. Consider the distractors used for the problem. Clearly the problem is soliciting whether a student can recognize that the larger the energy difference between the electronic energy levels the larger the energy of the emitted photon and the greater the frequency (the shorter the wavelength) of the light.