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Aleksey Kocherzhenko Lesson Plan Physical Chemistry I: Quantum Mechanics (this is a sophomore/junior-level course) Prerequisites: General Chemistry, Introductory Physics, Calculus, Differential Equations Week 1: The Concept of Orbitals • Student Learning Objectives: 1) Students will be aware of some key experimental findings and theoretical assumptions that stimulated the development of quantum mechanics: existence of electrons (J.J. Thomson’s cathode ray experiments), structure of the atom (Ernest Rutherford’s alpha-particle scattering experiments), oscillator quantization in the black body radiation problem (Max Planck), the discrete hydrogen atom spectrum (Johannes Rydberg), and field quantization in the photoelectric effect problem (Albert Einstein). 2) Students will be able to derive Bohr’s model of the hydrogen atom and describe its achievements and deficiencies. 3) Students will know the concept of particle-wave duality and the formula for de Broglie’s wavelength. 4) Students will be able to explain the concept of orbitals. • Lesson Outline ü Setting the context Chemistry students are sometimes skeptical about the need of learning quantum mechanics, since they might see it as a physics course that is not directly related to their prospective work. To counteract this, it is necessary to immediately address the relevance of quantum mechanics to synthetic chemistry. 1) Ask students if they are familiar with the ortho-meta-para rule (they should be: it is studied in freshman organic chemistry, almost universally a required course). Ask a volunteer to state that rule. 2) Say that the ortho-meta-para rule is usually presented in organic chemistry as an empirical finding, but physical chemistry can easily provide an explanation for it. After completing the course they will be able to theoretically show how this rule arises. By calculating the molecular orbitals, one easily finds that there is very little electronic density at the meta position, unlike the ortho and para positions. Ask what this means. (Since electrons are required for chemical bonding – students should know this from general chemistry! – it is very difficult to achieve meta-coordination.) 3) Conclusion: reactivity and functional properties are largely determined by orbital structure and symmetry, so learning to calculate molecular orbitals is worthwhile. At this point I should have the students’ interest and attention. I should also be able to gauge how well they remember the basics of general and organic chemistry. ü Key developments in early 20th century physics that led to the establishment of quantum mechanics 1) J.J. Thomson discovers the electron in experiments with cathode rays. He measures their deflection by electromagnetic (EM) fields and finds that the rays consist of particles that are roughly 1000 times lighter than the smallest of atoms, hydrogen. (Discuss: how does one get the mass of a particle from its deflection in the EM field?) Thomson proposes the “plum pudding” model of the atom. 2) Ernest Rutherford studies scattering of alpha particles on thin gold films. He finds that the scattering pattern is inconsistent with the plum pudding model. (Discuss: how did Rutherford conclude this? This is related to the discussion we just had about the deflection of particles in EM fields…) Rutherford proposes the “planetary” model of the atom. 3) Discuss: what is the problem with the planetary model? (It assumes that electrons are accelerated charged particles, since they are moving in a circular orbit around the nucleus – Newton’s First Law. Thus, electrons must radiate, lose energy and fall onto the nucleus, but that’s not what happens: atoms are stable particles. This is also inconsistent with Johannes Rydberg’s findings concerning the hydrogen spectrum, which is not continuous, as it would be if electrons were losing energy continuously). Show hydrogen spectrum and present Rydberg’s formula. 4) To explain the hydrogen spectrum, Niels Bohr makes the assumption that stable orbits in an atom are quantized: when electrons are in orbits of certain radii, they do not emit EM radiation. This assumption is unphysical, but it helps explain the hydrogen spectrum. Guide students through the derivation of the Bohr model. (What is the centripetal force for Bohr electrons? It is the Coulomb force. Does anyone know the formula? Students should, from introductory physics. Introduce the quantization condition. How do we derive the energy levels? Depending on the time, a volunteer could help me do it on the board: tests basic algebra knowledge.) Transitions between the derived energy levels correspond to transitions predicted by Rydberg’s formula. Homework: show this (very easy, shouldn’t take much more than 5 minutes). 5) We introduced the concept of quantization: how did Bohr think of it? He wasn’t the first one. In 1900, Max Planck quantized oscillators to describe black body radiation. (Do you remember Planck’s law of black body radiation from introductory physics? If not, show it and briefly explain how it can be derived; refer students to an excellent resource for reviewing physics topics, http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html, for more details.) If there is time, let students relax a little: tell them the story of how Planck was advised against studying physics. 6) Planck quantized oscillators, but Einstein was the first to quantize fields. Ask the students what they know about the photoelectric effect (hopefully, they know what it is and someone might mention that there is a threshold frequency for it – ask them to explain this, if not – explain myself). Present Einstein’s explanation: light consists of particles (photons or quanta) that carry a specific amount of energy. The photon energy depends on its frequency (Planck’s formula). If the frequency/energy transmitted to an electron by a photon isn’t high enough, the electron cannot leave its atom, and there is no photoelectric effect – this agrees with experiments. 7) But there is a problem. (Discuss: what is it? Light is known to be an EM wave. How is this known? Students should remember Young’s experiments and Fresnel’s theory from introductory physics.) So, sometimes light seems to behave as if consisting of particles, and sometimes – as a wave. How is this possible? There is no real classical analogy (except, maybe, the surfer and the wave). Homework: try and see how far that analogy can be carried! (This should take 10 – 15 minutes. It will give a good idea of whether the students remember Fresnel’s theory and of how imaginative they are.) Louis de Broglie postulated the principle of particle-wave duality. Introduce the de Broglie wavelength. Even if we don’t understand it, we should accept that it works very well to describe experiments. (Homework: do a back of the envelope calculation to estimate the de Broglie wavelength of an electron in the hydrogen atom and of a basketball. Briefly comment on the result: why don’t we see diffraction of basketballs? This shouldn’t take much more than 5 minutes.) 8) Now, remember: Bohr assumed that an electron was a particle moving in a centrosymmetric potential, and there was a problem with that. (What was it? We made an unphysical assumption that accelerating electrons do not emit EM radiation.) But what if we assume that the electron is a wave in a centrosymmetric potential? If the atom is in a steady state, the state of an electron isn’t changing. What is the type of a wave for which the locally stored energy doesn’t change with time? (It is a standing wave. Students should know this from introductory physics. Hopefully, someone describes what a standing wave is – maybe, draws it on the board, demonstrates it on a string, or even writes its equation. Discuss the properties of standing waves and see how well the assumption that an electron behaves like one describes its properties.) ü Summary and Outlook Epiphany: stable electronic states (orbitals) are nothing else but standing waves! But Christiaan Huygens mathematically developed wave theory already in the 17th century, so we know how to deal with waves. Now it’s just a matter of learning to apply wave theory to particles, like electrons and photons. This is what quantum mechanics (sometimes alternatively referred to as “wave mechanics”) does, and what we will be learning to do throughout the semester. • Assessment The student’s prior knowledge and understanding of the material will be assessed based on their participation in the discussion and on the homework assignments. Analysis In terms of its mathematical formalism, quantum mechanics may get relatively complex at times, but the principal challenge in teaching it is that it is a very counterintuitive theory. Students tend to have trouble accepting the concept of particlewave duality, the uncertainty principle, and other ideas that have no classical analogies. To make these concepts seem more logical, it is useful to follow the line of reasoning of the founders of quantum mechanics (who knew what the students already know at this point), to explain the challenges that they faced, and to show how quantum theory helped deal with physical phenomena that classical physics could not describe. After providing chemistry students with some motivation for learning quantum mechanics (which they don’t always have, since sometimes they cannot see how it relates to chemistry) and checking if they remember what they learned in freshman general and organic chemistry classes, I follow the developments in quantum physics more or less chronologically. We mostly focus on the problem of atomic structure, but make a couple detours to see how physicists working on this problem (particularly, Niels Bohr) borrowed concepts that arose at the time in other areas of physics (particularly, the concept of quantization introduced by Max Planck and applied to electromagnetic fields by Albert Einstein). Analyzing the atomic structure problem allows me to demonstrate the shortcomings of classical physics: if we are unwilling to go outside its boundaries, explaining atomic-level phenomena requires making unreasonable assumptions. So far, most of the physics we have used was fully classical. Apart from the orbital radius quantization assumption, Bohr’s atomic model does not employ anything that is unknown to students, so they should be able to derive it themselves, with a little help from me. This is why it is possible to use a mixed lecture-discussion format here. My approach is to present students with experimental findings and then to ask them to try and explain the experiments. Discovering the solution to the atomic structure problem on their own will help students see that this solution is, indeed, logical. Observing their thought process also allows me to assess how much students remember from the introductory physics course that is usually a prerequisite for physical chemistry and how well they think in general. The first homework assignment (showing that Bohr’s model explains Rydberg’s formula) serves a related purpose: to make sure that the students are comfortable with the basic math and physics that they will need throughout the course. Once our semiclassical solution hits a dead end, I introduce Louis de Broglie’s breakthrough idea of particle-wave duality. This is new information that students need to come to grips with. To let it sink in a little, I ask students to play around with it at home (second homework assignment: the electron and the wave-surfer complex). Students should immediately see that by making his assumption, de Broglie resolved many problems in early quantum theory: in particular – the problem that Bohr’s atomic model was facing. This should make accepting the wave-like nature of electrons and other particles easier. By discussing the behavior of waves in a centrosymmetric potential, we arrive at the understanding of the nature of atomic orbitals. At this point, we have covered all student learning objectives. To let students practice the de Broglie wavelength formula and to prepare them for Heisenberg’s uncertainty principle (to be discussed in the following lecture), I give a third homework assignment (electron vs. basketball).