Constructing Models in Quantum Mechanics: Potential Energy Diagrams

1. SAMPLE METAL Tip V I I Constructing Models in Quantum Mechanics: Potential Energy Diagrams Sam McKagan JILA, University of Colorado at Boulder Claim: The potential energy diagrams most commonly used in quantum mechanics, such as infinite square wells and square barriers, are abstract models based on extremely sophisticated reasoning and approximations. Standard instruction teaches students to manipulate these models, but does not address how to build the models or what they represent. Thus, many of our students completely fail to make any connection between potential energy diagrams and the potential energy involved in real physical situations. Since these potential energy diagrams are a basic component of nearly all of quantum mechanics, these students essentially have no idea what they’re doing in quantum mechanics class. Claim: They’re not getting it: The tools we give them: Unspoken assumptions: Representations of Potential Energy in Introductory Physics: • What do experts know that we never talk about? • In QM, we use potential energy instead of forces to describe interactions between objects. • “The potential” in the Schrodinger equation refers to the potential energy of a particle as a function of position. • This potential describes the interactions of the particle with its environment. • We use simplified potentials because real systems are usually too hard to model. • These simplified potentials can sometimes be good approximations of real systems. • Determining an approximate potential for a real system requires knowing what you can ignore. Traditional Modern Physics Course1: Students fail to make sense of PE diagrams: Very familiar: Less familiar: Student Interviews: electron going through square barrier: Interviewer: Ifthis curve that you drewis the potential energy, then what isthis square thingthat’s drawn here? Student 1: I don’t know, that’s just the bump that it goes through. I don’t know what it means. I just see that and I know that it’s some kind of obstacle that it goes through. Interviewer: What does the potential energy looks like for this case? Student 2: For the electron? I guess it would be a straight line here, and then… well, it would have a certain potential energy, wouldn’t it? Going up to the gap? I’m not exactly sure. I don’t know what it would… I don’t know what the potential energy for the electron would look like. Interviewer: So this thing that’s being plotted here, U(x), what is that? Student 2: Potential energy. I guess it’s the potential energy of the, I’m not exactly sure. I know that the barrier, within the barrier, the potential energy increases. So I guess it would be a measure of the potential energy of the medium that it’s in, of some sort, I’m not exactly sure. Interviewer: But it’s not the potential energy of the electron? Student 2: Um, I don’t, not, that doesn’t ring a bell to me, why it would be. That doesn’t come to my mind. I don’t know, I guess it could be, but… All correspond to concrete physical systems! They come from somewhere! How do you determine the potential energy function for a given physical system? Representations of Potential Energy in Quantum Mechanics3: Reformed Modern Physics Course2: (Designed to address known student difficulties & make models explicit) Students still struggle to make sense of PE diagrams: • Example: Scanning Tunneling Microscope: • Potential is uniform inside a conductor, so V(x) is flat in tip and sample (only works if sample is conductor). • Complete circuit in steady state, so electron flow doesn’t change potential. • Because an electron is bound to a metal, it has a different potential energy in the metal than in the surrounding air. The difference between these two potential energies is given by the work function of the metal. • To analyze this system, we need to look at the potential energy of any one electron due to its interactions with all the other atoms and electrons in the metal of both the tip and the sample, and with the electric field of the applied voltage. • If there is a voltage across a region of space, the potential energy of an electron in that region is a linear function of position. • The potential difference between the tip and the sample tells you the potential difference between two points just outside the metals, not inside. • You can ignore collisions of the electron with other electrons and atoms. “the potential” V(x) Student Questions: • “I have trouble understanding what the potential is when we are looking at models of an electron in a wire, free space, finite square well, infinite square well. I am sort of getting this idea of it being similar to a work function in that once the potential (V) is less than the potential energy, the electron is out of the wire. I can usually follow the math/calc that follows the examples okay, but the overall concept of this potential (V) still confuses me, and so I still don't have a firm grasp of [what] the square well models mean/represent/whatever.” • “I cant find a general description of an infinite well, i understand what it does but not what it is or where its used.“ • “Voltage is used when we talk about electromagnetic forces, like the coulomb force. What I'm confused about is that we used a voltage well to show the strong force in effect. Is it accurate to show the strong force as a very deep voltage well?” Infinite Square Well Finite Square Well Sample Tip Harmonic Oscillator Hydrogen Atom Clicker Question: Student Responses: “free particle” V=0 Exam Question: An electron is tunneling from a scanning tunneling microscope (STM) tip to sample’s surface. The tip’s work function is 4 eV and the sample’s work function is 5 eV . a. Draw potential energy curve if no voltage between tip and surface. b. Hook up a 5 V battery. Draw a new curve. “step potential” “tunneling” All abstract mathematical constructs! No relation to real physical systems. 40% draw correct curve: 40% draw correct curve: In Intro, rarely draw diagrams of PE functions. In QM, rarely talk about physical systems.  No connection between the two courses. Example: modeling electron in wire as finite square well End Notes: Implications for Teaching: How to help students build models of PE diagrams: . References: S. B. McKagan and C. E. Wieman, 2005 PERC Proceedings (2006). S. B. McKagan, K. K. Perkins, and C. E. Wieman, 2006 PERC Proceedings (2007). Pictures of potential energy representations are taken from the Physics Education Technology Project simulations Quantum Bound States and Quantum Tunneling, available at: http://phet.colorado.edu. Acknowledgements: Thanks to the NSF for providing the support for this project, and to all the members of the PhET Team and the Physics Education Research at Colorado group (PER@C). • Minimum: • We must be more explicit about the models we are using in QM so students have some idea what they’re doing. • Ideal: • Teaching QM is a great opportunity to teach students how to build models! • Practice determining potentials for real physical situations and vice versa. • Analyze underlying assumptions • …