Introduction to Quantum Mechanics
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:
- 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).
- 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.
- 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.
- 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
- 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.
- 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.
- 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
- 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.
- 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.
- 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
- 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
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.
Time is not quantized.
- Space is not quantized.
- The energy of a free particle is not quantized.
- In a harmonic oscillator, the states of definite
energy are quantized. These states can be used as a
basis to describe all the states of the system.
However: The energy states are not the only states.
They are not even the only basis states.
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.
- Sometimes when the field is doing something interesting people
call it a particle.
- Sometimes when the field is doing something interesting people
call it a wave.
- In all cases, whether the field is doing something interesting
or not, it pays to think of it as a field.
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
You can create electrons (along with
positrons) by stirring up excitations in the electron field.
These two processes are more similar than than they
There is a definite pecking order: The concept of field comes
first. Then comes waves. Particles come last, if at all.
- (Field ⇒ Wave): Starting with the concept of field,
we can explain waves. These are not however equivalent, because
sometimes the field does something that doesn’t conform to the usual
notions of what a wave is.
- (Wave ⇒ Particle): We can use waves to explain
essentially everything we know about particles. Read about the ducks
in reference 2 to get an idea of how this works.
- The converse does not hold. Starting from ordinary notions of
how particles behave, there is no good way to explain wavelike
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
“Introduction to Atoms”
“Adding Waves and/or Vectors”
“How to Draw Molecules”
John Denker “Partition Function for Particle(s) in a Box”
(chapter 25 of Modern Thermodynamics)
The Character of Physical Law
Note that the Messenger lectures (from which the
book is derived) are available for free online.
“Models and Pictures of Atomic Wavefunctions”
“There are no particles, there are only fields”
“Welcome to Spacetime”