# Quantum mechanics

We have been doing classical calculations. It is time to make the transition to a statistical theory that is compatible with quantum mechanics.

"Ludwig Boltzmann [right], who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. "

- David L. Goodstein, States of Matter

### The hydrogen atom

You know that a single electron in a hydrogen atom can be in any one of a number of different states.

The different states have different spatial probabilities for the electron. The image shows the spatial probability (brighter) for an electron to have a particular location in the $x$-, $z$- plane for different states.

The states are labelled by principal quantum number $n=1,\ 2,\ 3...$ and angular momentum index $l=s,\ p,\ d...$.

Each state has a characteristic energy. It turns out that, in the hydrogen atom, the energy only depends on $n$ and not $l$. We say the $l$ states that all share a common $n$ are "degenerate".

We can make an energy level diagram. If we assume that the electron may only exist in one of the allowed states, then transitions between states only happen for certain energies. This explains the emission (and absorption) spectra of atomic systems.

For example, the transitions when an electron drops from a higher level down to $n=2$ are known as the Balmer series and have photon energies in the visible region of the spectrum.

An atom with many electrons might have more than one electron in any particular state, and even more in any one "energy level".

### Spin

You have all calculated the torque on a current loop. The resulting forces twist the current loop so as to align $\myv m$ with $\myv B$. The magnitude of the magnetic moment is given by $$|\myv m |=I A.$$

There is an energy, $U_m$, associated with a magnetic moment in a magnetic field: $$U_m=-\myv m \cdot \myv B$$

An electron has a magnetic moment, as if it were a spinning charge and also experiences a torque in a magnetic field.

A torque on a current loop will cause no net force on a current loop. But in an inhomogeneous magnetic field, there is a net force on an electron. This is the idea behind the...

### Stern Gerlach apparatus

Stern-Gerlach experiment with a beam of silver atoms

Modern explanation:

• Silver has one unpaired electron.
• An electron has total spin 1/2 \hbar.
• In a magnetic field only two states are allowed: spin up and spin down.

### The features we need from QM

• A system consists of a collection of $N$ atoms (or photons, or phonons, or electrons, or...) with many different states.
• A microstate is a full enumeration of which particles are in which states. [E.g. in the hydrogen atom with one electron this would involve specifying the principal quantum number $n$ (1,2,3...), the angular momentum state $l$ (s, p, d..), and $m$ (spin up or spin down)]
• Each state has a characteristic energy.
• An energy level is the group of all the states that have the same energy. [E.g. all the states with the same principal quantum number]
• Caution: we're about to change the meaning of $n$...
• A macrostate (or configuration) consists of listing how many particles $N_j$ there are in each energy level $j$, e.g.
$$\{ N_1, N_2, N_3...N_n \}$$
for $n$ different energy levels.
• There is a constraint on the $N_j$ numbers: they must sum to the total number of particles
$$N=\sum_{j=1}^n N_j.$$

A macrostate is analagous to what we've been talking about as a thermodynamic state.

The fundamental assumption of statistical thermodynamics is that all possible microstates of an isolated system are equally probable.

E.g. An isolated system consisting of 3 independent harmonic oscillators, which have equally spaced energy levels...

A word from Richard Feynmann:

So we now have to talk about what we mean by disorder and what we mean by order. …

Suppose we divide the space into little volume elements. If we have black and white molecules, how many ways could we distribute them among the volume elements so that white is on one side and black is on the other? On the other hand, how many ways could we distribute them with no restriction on which goes where? Clearly, there are many more ways to arrange them in the latter case.

We measure “disorder” by the number of ways that the insides can be arranged, so that from the outside it looks the same. The logarithm of that number of ways is the entropy. The number of ways in the separated case is less, so the entropy is less, or the “disorder” is less.

#### Image credits

Michael Diderich, Charles Cowley, Theresa Knott