San José State University |
---|
applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
---|
The Source of the Difference in Particle Statistics for Bosons and Fermions |
Bosons are particles for which there is no limit on the number which may occupy a particular quantum state. But at most only one fermion may occupy a particle quantum state. There is a cute image that helps one remember which behavior prevails for each. The image is of a tavern in which there is a communal table in the middle and booths around the walls. The communal table is occupied by bosons (allusion to gregarious bozo carousers) to any number. The booths have limited occupation and are occupied by fermions (allusion to firm uptight, upright customers). Given that fermions have two spin states the image of two to a booth fits.
This material is to establish the quantization of a field and the particles it contains. Let ψ(r, t) be the wavefunction for a field where r stands for the set of coordinates of a point. The dynamics of the field are given by the time-dependent Schrödinger equation
where ħ is Planck's constant divided by 2π.
Consider ψ(r) expressed as a linear combination of a set of function {φ_{n}(r); n=0, 1, …, M}; i.e.,
The set of functions to be used are the solutions to the time-independent Schrödinger equation
The solutions are orthogonal and can therefore be made into an orthonormal set; i.e.,
where δ_{nm}=1 if n=m and is zero otherwise.
When the linear combination is put into the time-dependent equation the result is
This equation can be multiplied by φ_{m}(r) and integrated over space. Because of the orthonorality of the set of φ_{n}(r) functions the result is
For particles the expression −(ħ²/(2m))∇²ψ + V(r)ψ is the Hamiltonian function; for a field it is the density of the Hamitonian function. The Hamiltonian for a field is given by its expected value; i.e.,
where ψ* is the complex conjugate of ψ.
The quantity E(r)ψ can replace −(ħ²/(2m))∇² + V(r))ψ(r) in the integral to give
If ψ is replaced by the linear combination of the φ_{n} functions and the integration carried out the result is
This Hamiltonian is in the nature of a sum of the Hamiltonians of a set of harmonic oscillators. Consider the harmonic oscillator for a particular value of n. It is an oscillator with a frequency of E_{n}/ħ.
Let B_{n−} denote the operator corresponding to b_{n} and B_{n+} the operator corresponding to b_{n}*. B_{n+} is the operator adjoint to B_{n−}. The reason for this notation is that B_{n+} represents the creation of an additional particle in the field and B_{n−} the annilhilation of a particle in the field.
The Hamiltonian operator is then
Let the bracket symbol [P, Q] denoted the commutator of two operations, P and Q; i.e.,
The set of operators B_{n−} and B_{n+} are required to satisfy the following canonical quantification conditions
where 0^ denotes the zero operation; i.e., the operation that generates the zero function.
From these it follows that
These conditions along with the previously derived
make the results consistent with the Heisenberg equation for the dynamics of the operators
Consider the operator B_{n+}B_{n−}. It is so significant for the analysis that it is given a special name and symbol. It is called the number operator and is denoted as N_{n}
Let L be an eigenfunction of N_{n} and λ its eigenvalue; i.e.,
In the webpage Operator Relations it is shown that (B_{n−})L is also an eigenfunctions of N_{n} but with an eigenvalue which is 1 less than that of L. From this it follows that λ must be an integer and hence the eigenvalues of N_{n} are the nonnegative integers 0, 1, 2, 3, …. If n is the eigenvalue of N_{n} then the eigenvalue of (B_{n−})L is (n−1). This is why B_{n−} is the annilhilator operator; it diminishes the number of particles in the field by one.
Similarly it can be shown that (B_{n+})L is an eigenfunction of N_{n} but with an eigenvalue that is one unit more than that of L. Thus B_{n+} is a creation operator that creates an additional particle for a field. This is the case of bosons.
For the case of fermions it is the anticommutator of two operations that must be considered. The anticommutator of two operations, P and Q, is defined as
The operator corresponding to b_{n} will be denoted as β_{n−} and that of b_{n}* as β_{n+}. The canonical quantification conditions to be satisfied by β_{n−} and β_{n} are
Note that
where 1^ is the identity operator.
Now consider
But (β_{n+}β_{n+}) and (β_{n−}β_{n−}) are both zero. Thus
Let L be an eigenfunction of (β_{n+}β_{n−}) and λ its eigenvalue. Then
This means that
Thus a fermion state can have at most one particle. This is the Pauli exclusion principle.
HOME PAGE OF Thayer Watkins |