Identical particles ↩

shorthands: {}
aliases: [Indistinguishable particles]
created: 2022-01-06 18:11:56
modified: 2022-11-04 14:59:22

One-particle wave function with spin

The one-particle wave function in coordinate-spin representation:

Where is the spatial part of the wavefunction and is the spin part. The Hilbert space is the tensor product of space of square integrable functions and is a dimensional complex Hilbert space where is the spin of the considered particle.

Wave function of N identical particles with spin

The wave function of identical particles is a member of the tensor product of the one-particle Hilbert space with itself times:

Exchange operator

Other name: Permutation operator
The exchange operator exchanges two particles:

Properties of the exchange operator

• Hermitian:

• Unitary:

• Eigenvalues:

Principle of indistinguishability

The result of the measurement is invariant under the exchange of two identical particles, i.e. for any Hermitian -particle operator :

Take , where . In this case

Which can also be written as

In order to satisfy the above equation for any , a sufficient condition can be written as

which implies

This means that any is an eigenfunction of :

Where we already know the possible values for the eigenvalue of the exchange operator:

Now based on the value of , we can classify particles into two group, fermions and bosons:

Invariance of the Hamiltonian operator

Applying the exchange operator to the Schrödinger equation:

Where is the -particle Hamiltonian operator, we obtain

On the other hand, also solves the Schrödinger equation:

Comparing the above two equations, we get:

Thus, we conclude that the Hamiltonian operator commutes with the exchange operator:

What is the meaning of the operator ?

That is

So

Therefore the Hamiltonian operator of identical particles should be invariant under the exchange of two particles.

Consequently, the symmetry of the -particle wavefunction is a conserved quantity (see Quantum mechanical time derivative):

Which means that fermionic particles remain fermionic under the time evolution dictated by the Schrödinger equation and vice versa.

Electrons

Electrons have spin-1/2, so they are fermionic. Therefore, the many-electron wavefunction is antisymmetric against the exchange of two electrons (the eigenvalue is ).

Construction of the antisymmetric wave functions

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First we show the construction of the two-electron wave function and then we generalize it for electrons.

Two electrons

Let's choose two one-electron basis functions (these represent the possible states of the electrons) and .
The tensor product vectors that can be generated from these basis functions are

Where is the outer product.
Or, by dropping the notation :

The general form of the two-particle wave functions is

After the exchange of the two particles we obtain,

while using that the wavefunction is antisymmetric,

Matching the coefficients from the two last equations yields,

Where is normalized to . This can be formally written in a determinant form:

Generalization for electrons

Generalization for orthonormal functions:

Where is the permutation of the elements and is the parity of the given permutation and the summation goes through all possible permutations. This is exactly the expansion of a determinant, called the Slater determinant:

Pauli exclusion principle

In the -fermion wave function, constructed as an antisymmetric linear combination of the outer product of one-particle wave functions, each one-particle wave function appears only once. (Other words: there can not be more than one fermion in the same one-particle state.)

The general fermionic wavefunction: if is a complete orthonormal set,

Symmetric wave functions

The bosonic wavefunction is symmetric for the interchange of particles. In the two-boson case, the following combinations are symmetric:

Consequently, the bosons do not respect the Pauli exclusion principle, so there can be more particles in the same one-particle state, or even all of them.

General construction: if is a complete orthonormal set,

Where contains the permutations, between different one-particle states, because the permutation of identical states does not give a new -particle wavefunction. stands for the number of identical one-particle states.
The general boson wavefunction is the linear combination of such symmetrized wave functions:

Occupation number representation

On the tensor product space of a complete set of one-particle states (called Fock space) the basis functions of the symmetric and antisymmetric particle states can be uniquely determined by the numbers specifying how many times the one particle states occur, called the occupation numbers,

The sum of the occupation numbers is necessarily :

Energy if the occupied states are eigenstates

If the one-particle wave functions are the eigenvectors of the one-particle Hamiltonian,

the state is an eigenfunction of the non-interacting particle Hamiltonian:

With the energy:

This follows from: