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




Live dangerously: Read Chapter 12!

Spin

In General Physics II you 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: $$U_m=-\myv m \cdot \myv B$$

If the magnetic field is $\myv B \uv z$, we could write this as: $$U_m=-Bm_z.$$ where $m_z$ can take on any value between $$-|\myv m| \leq m_z \leq +|\myv m|.$$

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

Stern Gerlach apparatus

A uniform magnetic field $\myv B=B\uv z$ will cause a torque on a current loop but exerts no net force on the loop.

But in an inhomogeneous magnetic field, there is a net force on an electron. This is the idea behind the...

Stern-Gerlach experiment with a beam of silver atoms

Two discrete beams form, instead of a continuous spread of silver atoms. The current explanation:

  • Each Silver atom has one unpaired electron.
  • In a magnetic field only two states are ever measured: "spin up" and "spin down": that is,
  • The $z$-component of an electron's magnetic moment can take on one of only two possible values, and no value in between, $$m_z=\pm\frac12g\mu_B$$ where $\mu_B$ is a constant called the "Bohr magneton", and the $g$-factor, $g$, is a number which is ever so slightly greater than 2.

Total energy

If we have a system consisting of $N$ silver atoms, let's say that $N_\uparrow$ are in the spin-up state and $N_\downarrow$ are in the spin-down state, where $N_\uparrow+N_\downarrow=N$. Then the total energy of the system is $$U_B=(-N_\uparrow+N_\downarrow)\frac12g\mu_B B=-N_{ex}\frac12g\mu_B B,$$ where the "spin excess" is defined as $N_{ex}=N_\uparrow-N_\downarrow$.

Macrostates and microstates

The concepts of "microstates" and "macrostates" are as important to statistical mechanics as the concept of a "system" was in thermodynamics.

  • A system consists of a collection of $N$ atoms (or photons, or phonons, or electrons, or...).
  • A microstate is a full specification of which particles are in which states. E.g. *which* silver atoms have their unpaired electron in the $\uparrow$ state, and which in the $\downarrow$ state.
  • A macrostate is a specification of some macroscopically measurable characteristic of the whole system. E.g. "These 100 silver atoms have a total energy of $14g\mu_B B$". One could also specify a macrostate in terms of $P$, or $V$, or any other macroscopic thermodynamic parameter.

    In what follows, we'll be using the energy of a system of particles as the way to specify the macrostate.

  • There are many microstates that are compatible with a given macrostate.
  • An energy level is the group of all the states that have the same energy. In our spin system, there are just two energy levels.
  • One way of specifying 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.

    [Notice $n$ is not # of kilomoles here!]
  • 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. A thermodynamic state is characterized by things like $P$ or $V$ or $G$ or $U$ which represent some average of characteristics of the atoms, the ultimate constituents of the system.

The fundamental assumption of statistical thermodynamics is that all possible microstates of an isolated system which are compatible with the given macrostate are equally probable.

A preview of the statistical mechanics meaning of entropy 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