In the Classroom
edited by
applications & analogies
Ron DeLorenzo Middle Georgia College Cochran, GA 31014
Quantum Analogies on Campus Ngai Ling Ma* Department of Applied Biology and Chemical Technology, Hung Hom, Hong Kong Polytechnic University, Hong Kong
Analogies are useful in helping students grasp abstract ideas in quantum chemistry. For example, a stairway was used to illustrate the idea of energy quantization (1). Neto (2) used dice-throwing to illustrate a few important quantum concepts. In this paper, some simple analogies based on the daily experience of students are used to illustrate concepts such as quantum number, state, transition, operator, and wave–particle dualism. The analogies evolved from my experience teaching quantum chemistry as a part of our physical chemistry course. Using these analogies not only lightens up the atmosphere in my classroom, they have also appeared to be helpful to the students. Student, the Electron Let us assume that each student is an electron, and it is class time so that the students should be somewhere on campus (atom). Like the electron, each student is associated with his/her wave function. This wave function, which is a function of where the student is (position) and what hour it is (time) fully describes the student. After making these assumptions, we can use them to illustrate some important concepts in quantum chemistry. Quantum Numbers, States, and Degeneracy Most students at the college level accept that the state of an electron in an atom is specified by four quantum numbers. However, when a model system such as the particle-in-a-box model is introduced, most students are puzzled about these new quantum numbers. This suggests that students have associated quantum numbers with atoms and molecules, but fail to see that quantum numbers arise naturally from a quantized system. Using an analogy, quantum numbers can be introduced as a more general concept. When students settle down for their classes, their whereabouts may be specified by the name of the building and the room number (quantum numbers) that they are in. This analogy can be carried further to illustrate the concept of degeneracy. Two rooms adjacent to each other should be of the same height from the ground; hence any object inside the two rooms would have the same potential energy. This illustrates clearly the concept of degeneracy: while the rooms are different (specified by different room numbers), their potential energy is identical. This corresponds to the idea that orbitals like px, py, p z, with the same principal quantum number but different angular momentum quantum number, can have the same energy.
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Transitions and Selection Rules The concepts of transition between states and selection rules are important in spectroscopy. Assume that a student is currently in a particular classroom (state) for a certain class. Students are excited (!) when a class is over. When the student goes from one classroom (state) to another classroom (state) for another class, an allowed transition occurs. A forbidden transition occurs when a student skips class. The selection rules, which describe what transitions are allowed, are fully described by the schedule of a student. Probability and Probability Density Students often mix up probability and probability density. While the probability density is given by the square of the wave function, the likelihood of locating the particle within a certain volume depends on both the probability density and how large the volume is. One can illustrate the relation between probability and probability density using the analogy below. First of all, we consider the whole campus as one area (Fig. 1). Since the student has to be somewhere inside the campus, the probability of locating the student within this area is one (normalized). Then we divide the campus into two areas of unequal size: the area of the library and the area outside the library (Fig. 1). Even though a student may visit the library very often (high probability density), the probability of finding him somewhere outside the library may still be higher because the area outside the library is much larger than inside the library. This is in exact accordance with the fact that though the probability density of the 1s electron is highest at the nucleus, the most likely place to find the electron is at one Bohr radius from the nucleus. Operator One of the most fundamental concepts in quantum mechanics is the operator. Students at college level should be familiar with the Heisenberg uncertainty principle, which states that one cannot simultaneously measure the exact position and exact momentum of a particle. This arises because the position and momentum operators do not commute. Usually, simple mathematical operations such as addition and subtraction are given as examples of commutative operators, and matrix operations are used as examples of noncommutative operators. Since most students are more familiar with operations like addition than with matrix operations, it is common for students to get the impression that most quantum operators com-
Journal of Chemical Education • Vol. 73 No. 11 November 1996
In the Classroom
mute. It might be of interest to remind them that, in fact, most of the operations in real life do not commute. ˆ as “goWe can define two operators: operator 1, 1, ing to the chemistry laboratory”, and operator 2, 2ˆ , as “going to the library”. Where the student ends up will obviously depend on which operation he does first. A more interesting analogy is to let the feet of the student be the wave function, ψ, 1ˆ as “putting the socks on”, and 2ˆ as “putting the shoes on”. It is obvious that 1ˆ (2ˆ ψ) ≠ 2ˆ (1ˆ ψ). Wave–Particle Dualism Wave–particle dualism is probably one of the most difficult concepts in quantum chemistry. Students often ask, “How can an electron be a particle and a wave?” The answer to this question is, the nature of electron depends on how it is measured. A simple analogy to the wave–particle dualism would be: how a student is known depends on his relationship to the person who knows him. I usually pick a student in the class, and ask him what his parents call him and what his friends call him. In this analogy, the “name-calling” is regarded as a form of measurement. His parents might call him by his first name, but his friends might call him by a nickname. Obviously it does not matter what he is called, he is still the same person. However, what he “is” would depend on the nature of the observation. At this stage, I would remind the student that this is exactly like the nature of an electron: electrons are waves if you study their interference using a double slit; they are particles if you study them using a phototube. The nature of the electron depends entirely on how it is observed.
Figure 1. The campus map of the Hong Kong Polytechnic University. The library area has been highlighted with a circle.
Literature Cited 1. Malone, L. J. Basic Concepts of Chemistry; Wiley: New York, 1994; pp 135–136. 2. Neto, B. de B. J. Chem. Educ. 1984, 61, 1044–1045.
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