Chapter 4 全同粒子 | XiangSiqi’s Blog

5.1 Two-Particle System

For two-particle system, is function of spatial coordinates of two particles and time:

Its time evolution is determined by Schrodinger equation:

where is Hamiltonian of the system:

and the normalization is:

and for the time-independent potentials, which means , then we have time-independent Schrodinger equation:

where .

5.1.1 Bosons and Fermions

If two particle are not in entangled state, we can decouple the joint wave function into the wave function of two particles:

What is different from classical mechanics is that you can’t tell the two particle apart because we can’t observe them or stamp identification numbers on them. So we just know there is a particle in the state of and one is in the state of , but we can’t tell which is which, We call them indentical particles.

We have two way to construct a joint function that is noncommittal as to which particle is in which state:

we call Bosons for which we use plus sign and Fermions for which we use minus sign. According to relativistic quantum mechanics, all particles with integer spin is Bosons and all particles with half integer spin is Fermions.

For Fermions, the Pauli exclusion principle tells that two particles can’t have same state, which is because that if we take , we have

shows that two Fermions particles can’t share same wave function.

For entangled state, we use exchange operator to define Bosons and Fermions, the operator is defined as:

apply to the equation again:

which means that the eigenvalues of is 1 and eigenvalues of is :

For identical particles, the Hamiltonion of particles are same: , leading to:

shows and are compatible observables. Hence we can find a complete set of functions that are eigenstate of both satisfy the requirement of .

And the system which starts out in such state will remain in such state, with plus sign for Bosons and minus sign for Fermions.

5.1.2 Exchange Forces

For indistinguishable particles we have:

and for identical Bosons and Fermions:

Let’s concern the square of separation distance between two particles:

It turns out that for indistinguishable particles:

But for identical particles:

where

For symmetric configuration, the average distance is shorter than the indistinguishable result and the antisymmetric configuration is longer than the indistinguishable result.

The former shows attractive force while the latter shows repulsive force. We call it exchange force, although it is not real actually. The force comes from the geometric consequence of symmetrization requirement.

Electrons are fermions, which means that the exchange force is repulsive force, but how are covalent bound formed? In fact we ignore the effect of spin:

and is antisymmetric, leading is antisymmetric.

5.4 Quantum Statistic Mechanics

5.4.1 Entropy of Indistinguishable particle, Fermions and Bosons

According to Boltzmann statistic, for N-particles system, a particle was on a specific state of , which determines the energy level . We can divide the particles by energy level , particles is on state of . So we have the numbers of particles conservation and energy conservation:

Then we consider for a energy level , we have eigenstates, which means is -fold degenerate.

So to form distribution of , we first assume the particles are distinguishable, so the numbers of distribution is:

and for every energy state we have boxes to put particles in, so we have for every energy state:

And finally we get:

Now we consider the identical fermions, it is indistinguishable, so there is one choices to form , and the antisymmetrization require one particle in one state:

And for identical Bosons, the symmetrization require the one state can hold multiple particles, so consider the particles is divided by crosses(which means classes), and consider the particles and crosses are identical, we have