Copyright © 2012 jsd

Introduction to Quantum Mechanics
John Denker

1  Evidence of Quantum Behavior

There is a rather large amount of evidence that the world we live in is governed by the laws of quantum mechanics. Much of this evidence is understandable at the level of an undergraduate general-chemistry course or even a high-school chemistry course.

Some specific constructive suggestions:

  1. Put your palm against your forehead and push. The fact that atoms have any nonzero size and resist compression depends on the quantum nature of electrons. If they were classical particles, they would quickly spiral down into the nucleus, where their potential energy is lowest.

    The size of the hydrogen atom can be estimated quite easily, using little more than dimensional analysis, based on the value of Planck’s constant, Coulomb’s constant, the electron mass, and the electron charge (which are known from independent measurements).

  2. There is a widely-known high-school chemistry experiment that involves HCl diffusing in one direction and NH3 diffusing in the other direction in a 1 meter long 1 cm diameter glass tube. This (plus some algebra) provides rather decent quantitative information about the size of the molecules involved.

    Note that Avogadro died without ever knowing the value of Avogadro’s number, not within several orders of magnitude in either direction. It fell to Loschmidt to make the first serious measurement, based on transport data (speed of diffusion versus speed of sound). Again, there is no classical explanation for the nonzero size of atoms.

    A list of other lines of evidence for the size of atoms can be found in reference 1.

  3. Look at a sample of hydrogen and a sample of lithium. The fact that these are different depends on the quantum nature of electrons, including destructive interference of waves and identical-particle effects. More generally, the existence of the periodic table (and the periodicity thereof) cannot be explained in classical terms. This is fairly widely known at the high-school level, not to mention the college general-chemistry level.

    For more about interference of waves, see reference 2.

  4. Prepare a sample of liquid oxygen. This requires a little time, but not much effort, assuming you can lay hands on some liquid nitrogen. Observe the blue color of O2, quite unlike N2. Observe that O2 is paramagnetic, i.e. it sticks to a magnet, quite unlike N2. Either observation indicates the presence of unpaired electrons in O2. For that matter, the ferromagnetism of a bar magnet indicates the presence of lots of unpaired electrons.
    1. This proves there cannot possibly be filled Lewis octets in molecules. This is obvious for O2. It is less obvious but equally true for N2 and all other molecules.
    2. Obviously there is no classical explanation for paramagnetism or especially ferromagnetism. The correct explanation is beyond the scope of the introductory course, but simple experiments suffice to show that there are things going on that demand a non-classical explanation.
  5. Consider the sequence C=C, N≡N, O÷O, F−F, and (Ne Ne). Even the most qualitative consideration of bond order and bond strength as a function of the number of electrons can be taken as evidence for antibonding orbitals. Antibonding depends on destructive interference and on identical-particle effects. There is obviously no classical explanation. For the next level of detail on bonding and antibonding orbitals, see reference 3.
  6. Do some qualitative spectroscopy, using a neon lamp and/or sodium vapor lamp plus card-mounted diffraction gratings (less than 50¢ apiece). The spectral lines can be taken as evidence of transitions between energy eigenstates. The existence of the ground state and the existence of excited states depend on the quantum nature of the electron, including interference of waves as well as identical-particle effects.
  7. There are procedures for measuring the adiabatic exponent i.e. the “ratio of specific heats” for a gas, suitable for the undergraduate teaching lab. You can also use the readily-available published data. Observe that 2/(γ−1) i.e. the implied number of “degrees of freedom” is not an integer and is not even independent of temperature for some gases e.g. H2, Cl2, and CO2 at ordinary temperatures. There is obviously no classical explanation for this. It can be taken as evidence for quantization of the phase space for a rigid rotor. See reference 4.
  8. The Hall effect has been known since 1879. It can easily be measured using simple tabletop apparatus. The physics of p-type semiconductors is basically the same as the physics of antibonding orbitals. Obviously there is no classical explanation. See reference 3. Also, note that the technological and economic importance of the semiconductor industry can hardly be overstated.
  9. The resolving power of an electron microscope depends on the inverse wavelength (and hence on the energy) of the electrons.

    While we are on the subject: Grab a laser pointer or cat laser (available from the dollar store). Use it to exhibit speckle. Assert that the electrons in an electron microscope exhibit the same sort of speckle. As Feynman put it: “There is one simplification at least. Electrons behave in this respect exactly the same as photons; they are both screwy, but in exactly the same way.” That’s from reference 5 ... which I strongly recommend. Chapter 6 presents the fundamental ideas of quantum mechanics at a level that is accessible to bright middle-school students.

Other examples abound.

Bottom line: Evidence for the quantum nature of the world we live in is readily available.

2  Further Remarks

2.1  What is Quantized, Or Not

Sometimes things are quantized, and sometimes not. Non-experts tend to wildly overestimate how much quantization there is. In fact:

For more on this, see reference 6. The details are beyond the scope of the introductory course, but still – even at the introductory level – you don’t want to learn (or teach) stuff that is wrong and will have to be unlearned later.

For an introduction to what we mean by wavefunction, see reference 7.

2.2  Fields, Waves, and Particles

Everything in quantum mechanics can be described in terms of fields.

Some people like to argue about the distinction between waves and particles. Some people like to argue about the distinction between particles and fields. However, according to modern thinking, all such arguments are pointless. Let’s be clear:

You can create photons by stirring up excitations in the electromagnetic field.   You can create electrons (along with positrons) by stirring up excitations in the electron field.

These two processes are more similar than than they are different.

There is a definite pecking order: The concept of field comes first. Then comes waves. Particles come last, if at all.

More specifically:

For more on this, see reference 8.

2.3  Not Weird, Not Paradoxical

Keep in mind that 99% of what quantum mechanics predicts is not surprising. Mostly it explains stuff that you already knew, but didn’t have a good explanation for. This is what we would expect, in accordance with the correspondence principle. In the introductory course, it is important to emphasize the familiar aspects before delving into the less-familiar aspects.

Tangential remark: The same could be said about special relativity. Mostly it unifies and explains stuff you already knew. In the introductory course, it is important to emphasize the familiar non-weird aspects before delving into the less-familiar aspects. See reference 9.

It’s hard to decide to what extent quantum mechanics is “counterintuitive”. There’s a proverb that says that education is the process of cultivating your intuition. Gradually you become more familiar with what quantum mechanics says. With most things, the closer you look, the more imperfections you see. With quantum mechanics, it’s mostly just the opposite: the more closely you look, the more accurately the quantum mechanical predictions agree with experiments.

3  References

John Denker,
“Introduction to Atoms”

John Denker,
“Adding Waves and/or Vectors”

John Denker,
“How to Draw Molecules”

John Denker “Partition Function for Particle(s) in a Box”
(chapter 25 of Modern Thermodynamics)

Richard Feynman,
The Character of Physical Law

Note that the Messenger lectures (from which the book is derived) are available for free online. http://www.youtube.com/watch?v=Ja0HSFj8Imct=1m22s

John Denker,
“Coherent States”

John Denker,
“Models and Pictures of Atomic Wavefunctions”

Art Hobson,
“There are no particles, there are only fields”

John Denker,
“Welcome to Spacetime”
Copyright © 2012 jsd