Thermal physics acts as the bridge between the macroscopic world and the microscopic world, connecting the concepts and ideas of quantum mechanics to the macroscopic principles of heat. It does this by introducing three new concepts: entropy, temperature and free energy
Before embarking on our study of thermal physics, we must first understand some basic concepts of quantum mechanics and probability theory. Let’s start by reviewing some of the underlying principles of quantum mechanics.
Quantum mechanics is the study of the fundamental constituents of the universe, and how they interact with each other and us. There are many philosophical debates over the “rightness” of the theory, and how it is interpreted. The most important thing to remember about it is that all of the experimental evidence to date supports the predictions made by quantum mechanics. In science, this fact is the base upon which everything is built, and everything else is “just commentary”. Regardless of your personal feelings, this inescapable fact forces acceptance of quantum mechanics and its predictions.
We’ll not go into the details of quantum mechanics, but rather just hit the highlights here. At the heart of the theory is the idea that every object can be described in terms of a mathematical function known as a wave function. All of the information about the object is contained in this function. How do we use it to make predictions? We need a way of representing physical quantities as mathematical constructs. In classical physics, we accomplished this by using scalars, vectors and tensors. In every case, these quantities are numbers, or groups of numbers. In transitioning from classical to quantum mechanics, we replace these numbers with operators, which act on the wave function to generate a new wave function.
One way of interpreting this approach is to think of the wave function as a vector in some abstract (potentially infinitely dimensional) vector space. The new wave function generated by the operators associated with physical quantities can then be thought of as creating a set of coordinate axes in this vector space, with each axis corresponding to a specific measurement outcome. By projecting the wave function of interest (usually the original wave function) onto these axes, the probability of measuring a specific value can be determined.
Assume that the wave function associated with an electron constrained to one dimension is given by
where a is a constant associated with the width of the wave function and k0 is related to the momentum of the electron. What is the probability of finding the electron at x = +2a?
The operator associated with a position measurement is given by
where d(x–x0) is known as the Dirac delta function. It has the property that
The probability of finding the electron at x = +2a is then given by
Every physically measurable quantity has a corresponding operator. This is not as complicated as it may seem, since most measurable quantities can be written as a function of a few basic quantities. For example, the operator for momentum (in one dimension) is given by
Using this, the operator for total energy in one dimension (assuming that the potential can be written as a function of position only) becomes
For every operator, there is a special set of wave functions. These functions are those which satisfy the relationship
in other words, the effect of the operator on the wave function is that it returns a multiple of the same wave function. These wave functions are called the eigenfunctions of the operator, and the multipliers are known as the eigenvalues. For the energy operator, (1.4) becomes
This is know as the Schrodinger equation.
What are the energy eigenfunctions and eigenvalues associated with free space (V = 0)?
The Schrodinger equation for free space is
Since E is a constant, the solutions can be seen to be
where C1 and C2 are constants determined by normalization, and E can take on any value.
What are the energy eigenfunctions and eigenvalues associates with a potential well is defined by
We can break the problem into two parts, depending on the value of V. For 0 < x < 2a, the potential is zero. Thus, the solutions are given by the eigenfunctions in the previous example. Since the potential is infinite everywhere else, the only non-infinite solution is a zero function. For completeness, we require the eigenfunctions to be continuous. Thus, we require that the interior eigenfunction go to zero at x = 0 and at x = 2a. This leads to a solution of the form
where n = 1, 2, 3, …. Substituting this back into the Schrodinger equation, we find that the allowed energy values are
In addition to classical quantities, quantum mechanics introduces some new quantities. One of the more important ones (from a thermal physics standpoint) is that of spin. Spin can be thought of as a type of angular momentum; one which is integral to the particle. All of the fundamental particles in the universe (electrons, protons, neutrons, etc.) have a spin equal to ½. Particles associated with the carriers of force (photons, gluons, etc.) have a spin equal to 1. Note that here we are referring to the magnitude of the spin. The “z” component can take on integral values whose magnitude can not exceed the total spin magnitude. Thus, a spin ½ object can have components of +1/2 and –1/2. Composite objects, such as nuclei, atoms and molecules, have a spin which has maximal values equal to the total number of particles divided by 2. The wave function associated with these objects is found by multiplying together the wave functions of the individual particles then summing together all of the possible permutations of these particles.
One of the more interesting discoveries of quantum physics was that particles with half integer spin obey slightly different laws of physics than those with full integer spin. This can be described by realizing that the symmetry or antisymmetry of the composite wave function under the interchange of two particles is a characteristic of the object. In particular, objects consisting of identical particles of half-odd-integral spin (i.e. spin 1/2, 3/2, …) are described by antisymmetric wave functions. Such objects are called fermions, and are said to obey Fermi-Dirac statistics. Objects consisting of identical particles of integral spin (spin 0, 1, 2, …) are described by symmetric wave functions. Such particles are called bosons, and are said to obey Bose-Einstein statistics.
The requirement that systems consisting of half-odd-integral spin have an antisymmetric wave function has a unique side effect. Consider two electrons which are in the same spin state and have the same energy. Then the wave function which represents the system must be written as
Notice that if the two electrons are put in the same location, so that they become indistinguishable, the wave function vanishes. Thus a state of given energy, angular moment, parity, and so on, can be occupied by two electrons only if they have opposite spin, and then only by two electrons. This is known as the Pauli exclusion principle. This difference between fermions and bosons will become important later during our studies of thermal physics.
We now turn to understanding probability theory. We need probability theory for two reasons. First, we have learned from quantum mechanics that we can not ever get exact physical information from a system. Instead, we can only obtain a probability that the system will be in a specific configuration when we test it. Secondly, probability theory already includes the tool we need the most: how to take a large quantity of (quasi‑) independent states and treat them as a single system so that we may make predictions about it.
To this end, assume that we have an experiment E, which produces outcomes Xi, i = 1, 2, 3, …, n. What is the probability of getting outcome Xi? For our purposes, we define the probability to be
where N is the total number of trials and ni is the number of times Xi occurs. How do we combine probabilities? The probability that a result of Xi or Xj occurs is
Similarly, the probability that two independent measurements, X and Y, would produce results Xi and Yj is
so we see that probabilities are additive, and if the measurements are independent, commutative.
What is the probability of throwing a 7 on 2 dice?
On a single die, the probability of throwing any specific point is 1/6. Thus, the probability of rolling any specific combination on two dice is (1/6)(1/6) = 1/36. How many different combinations add up to 7? Considering each die independently, we have
1+6, 2+5, 3+4, 4+3, 5+2, 6+1
so there are 6 possible combinations, each with probability 1/36. Thus the total probability is 6/36, or 1/6.
What is the probability of rolling a 3?
The possible combinations are just 1+2 and 2+1, so the probability is 2/36, or 1/18.
When drawing two cards, what is the probability of drawing a 3 of hearts and a 5 of diamonds?
The probability of drawing a 3 of hearts on the first draw is 1/52. But now there are only 51 cards left, so the probability of drawing a 5 of diamonds on the second draw is 1/51. Thus, since the two draws are independent of each other, the total probability is
(1/52)(1/51) = 1/2652
If we ask what the probability is regardless of order, we see that the probability is just doubled to 1/1326.
What is the probability of drawing three hearts in a row?
P(3 hearts) = (13/52)(12/51)(11/50) = 1716/132600
So far we have talked about discrete probabilities, in other words the result could take only a discrete number of answers. What if the result was continuous? To handle continuous probabilities, we define a probability density, p(x), such that
We can use the probability density to calculate the weighed average value, or expectation value, of x:
Similarly, for a function f(x),