Download Quantum Mechanics (this is a sophomore/junior

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).