**1. Lecture 3: The many body problem **

In this lecture the *many body problem* is introduced in the context of *first* and *second quantisation*. The lecture notes can also be found here in pdf format.

**2. The Schrödinger equation **

We consider particles; in many cases of interest in this course the hamiltonian for the particles takes the form

where is the position of the th particle, is the kinetic energy, and is the potential energy of interaction, *counted once*, between the particles. (It is often convenient to assume that the position variable includes not only the particle’s coordinates in , but also its internal configurations such as spin, etc.) The time-independent Schrödinger equation then reads

together with some appropriate choice of boundary conditions for the wavefunction.

Suppose that , , is a complete set of orthonormalised single-particle wavefunctions. For example, could be the eigenfunctions of the harmonic oscillator. Then any product is a valid many-body wavefunction. Further, these products are *complete*, in that any many-body wavefunction can be expressed as a linear combination of them:

(This is a consequence of the *tensor product rule*. That this should be true follows from general information-theoretic considerations emerging from the assumption that it is possible to perform *tomography* of the entire wavefunction by separately measuring the particles.)

Substituting (3) into the Schrödinger equation (2) yields

We now accommodate the indistinguishability of the particles into the wavefunction by demanding that an exchange of particles cannot lead to an observable consequence. This means that

i.e., the wavefunction must be the same *up to a phase*. Since a further exchange of particles and yields the original wavefunction the phase in order to ensure that is not multiple valued. (In geometries with nontrivial topologies, or in two dimensions the wavefunction may possibly be multiple valued, yielding the possibility of *anyons*, and other exotic particles.) A necessary and sufficient condition to ensure the (anti-)symmetry of the wavefunction is that the coefficients in (3) satisfy

Exercise: prove this.

** 2.1. Bosons **

The particles described by a wavefunction that is completely symmetric under interchange are known as *bosons*. The symmetry of the coefficients allows us to reorder the indices of the coefficient in the summation (3). Suppose the state occurs times, and the state occurs times, etc. Then all such terms with the same numbers have the same coefficient. It is thus convenient to rename the coefficient function

We now define another coefficient function

The condition that the wavefunction is normalised becomes

where the coefficients must satisfy

Exercise: prove this.

We can now use the coefficients to rewrite the original wavefunction in terms of a new convenient complete orthonormal basis

and the summation is over all indices with the given pattern of s, s, etc. Note that the functions are completely symmetrised.

Exercise: prove that the functions form a symmetrised complete orthonormal set.

Here is an example of a function:

Substituting the expansion (13) into the Schrödinger equation (2) leads to

where there is an additional term for all possible sets of occupation numbers so that two particles are removed, multiplied by an overall factor , and then subsequently added back in at different places. This expression appears very complicated, however, there is a way to write it in an equivalent form which is much more compact.

** 2.2. Fermions **

If a minus sign is chosen in (7) then the s are antisymmetric under the exchange of any two particles:

which shows that must be different from or else the coefficient vanishes. This, in turn, shows that the occupation numbers can only be or . Any coefficients with the same states occupied are equal up to a minus sign, allowing us to define a new coefficient

where all of the numbers are first arranged in increasing order.

Exactly as in the bosonic case the many particle wavefunction can be expanded as

and . These *Slater determinants* form a complete set of orthonormal antisymmetric wavefunctions.

We postpone the task of writing out the Schrödinger equation until next section where we’ll develop a much more compact representation.

**3. Many particle hilbert space; creation and annihilation operators **

In this section we introduce a new orthonormal basis for hilbert space describing the *number* of particles in each state. This must be initially understood as an *abstract* construction until such time we can show it is equivalent to the first-quantised treatments of the previous section. The basis we introduce is denoted

which is meant to mean that particles are in the single-particle eigenstate . This basis is demanded to be complete and orthonormal, meaning that

and

** 3.1. Bosons **

In the bosonic case, associated with this occupation number basis, we introduce the *annihilation* and *creation* operators , satisfying the *canonical commutation relations* (CCR) = \delta_{jk},\quad [b_j, b_k] = [b_j^\dag, b_k^\dag] = 0. These operators are taken to act in the standard way on the number basis, e.g.,

etc. The *mode number operators* are defined to be

We now use the occupation number basis to rewrite the Schrödinger equation. Form the following state

where the s are taken to be the expansion coefficients of (13) which satisfy the coupled differential equations (16). This state vector in our abstract hilbert space obeys the differential equation

Relabelling dummy indices, using the properties of CCR, and rewriting the appropriate operations in terms of the annihiliation and creation operators leads us to the consequence that the abstract state vector satisfies the Schrödinger equation

where the operator is given by

These equations restate the Schrödinger equation in *second quantisation*. All the statistics and operators properties are expressed via the CCR. The physical problem is unchanged by the new formulation, and the coefficients express the connection between first and second quantisations. Thus any solution to the Schrödinger equation in first quantisation yields a solution in second quantisation and vice versa.

** 3.2. Fermions **

In the fermionic case we introduce the *annihilation* and *creation* operators , satisfying the *canonical anticommutation relations* (CAR)

The action of these operators on the vacuum state is expressed by

If we now introduce, as before, the abstract state vector

where the s are taken to be the expansion coefficients of (22) obeying the coupled differential equations coming from substitution into the Schrödinger equation, then we find that

where the operator is given by