Agenda for week 2: Many-body quantum mechanics

Learning goals

• Fermions and bosons and antisymmetric/symmetric many-body wave functions
• Fock space as a new Hilbert space for many-particle systems
• Creation and annihilation operators acting on the Fock space vectors, along with their commutation relations
• Understanding how creation/annihilation operators change with transformations in the 1-particle basis
• Understanding the basic rules for calculating matrix elements; Wick's theorem

Reading assignment:

Notes for week 2: Many particles in quantum mechanics

More details:

• Tuominen, Sec 7.1, 7.3-7.4 quite well covers the material of this week.
• Sakurai: Sec. 6.1, 6.2; Sakurai&Napolitano: Secs. 7.1, 7.2 and 7.5 cover the basics, but not everything.

Preliminary exercises,

Do these during/after reading the assignment work. Will be discussed in class March 16th.

1. Consider three single-particle states with wave functions and energies (). Find all possible three-particle (a) bosonic and (b) fermionic wavefunctions along with their energies. Express them also in occupation number representation. Hint: there may be many of them, but it is enough to show the different types of wavefunctions.

2. Consider the fermionic creation and annihilation operators and and the bosonic creation and annihilation operators and with indexing the corresponding single-particle state.

   1. Express the fermionic occupation-number state in terms of the creation operators and the vacuum state . Here the first item refers to , the second to , and so on, and the states with are assumed to be unoccupied.

   2. Express the bosonic occupation-number state in terms of the creation operators and the vacuum state . Here the first item refers to , the second to , and so on, and the states with are assumed to be unoccupied.

   3. Calculate the commutators

3. Using Wick's theorem, calculate the following matrix elements

   1. 

   2. 

   3. 

   Here . Consider both fermionic and bosonic operators .

Homework exercises, second week

Will be discussed in the tutorial session on Thursday March 18th. Return a scanned pdf with your solution by Friday March 19th, at 9 pm using the form below. Then check and grade your solution with the help of the model solutions and resubmit your graded solutions by Monday March 22th at 2 pm.

Notes for week 2: Many particles in quantum mechanics

The purpose of this chapter is to extend the single-particle quantum mechanics discussed in earlier quantum mechanics courses to that of many indistinguishable particles occupying the states obtained after solving the single-particle problem. At first sight this step seems quite simple, although first requiring us to write long sequences of wave functions. This step is only intermediate, as we soon find it to be much more convenient to use a simplified notation of the occupation number representation. This representation brings forth a new complete vector space, Fock space. However, it is the indistinguishability of the particles that brings forth new physics, because it requires us to consider a new symmetry, that of permuting the particles between the single-particle states. In quantum systems this has somewhat unexpected consequences: most solutions to the Schrödinger equation cannot occur in the nature! Rather, the only physically allowed solutions are those which are either completely symmetric upon permutations (bosons) or completely antisymmetric (fermions). This nature of the particles can be expressed through the algebra of the possible operators mapping bosonic or fermionic Fock space vectors to each other, called the creation and annihilation operators.

We start first from analysing these different possibilities for the many-body systems. Then we dwell a little more on calculating expectation values of those operators and develop a tool called Wick's theorem for this. This will be the focus of the first week.

In the following two weeks, we will move to more concrete applications. The most important of these is the Fermi gas. We will first study the special case of a non-interacting Fermi system (gas), which is a useful starting point for even studying interacting systems. In fact, in general the full interacting problem of a large number of particles turns out impossible to solve, which is why we need to develop approximative techniques to deal with this problem. The main technique we learn is the Hartree-Fock method, which tries to pose the interacting fermion system problem as the closest analogue noninteracting problem, but with a new definition of the fermions. The case of quantum effects in bosonic systems is deferred to the following chapter.

A reading hint for this chapter: what we do here is to overlay the many-body description on top of the single-body quantum mechanics. The latter is often denoted simply by some indices of the states. So, when reading the material, try to distinguish which part of the description is relevant for the single-particle problem, and which part for the many-particle one. In both cases we may solve a Schrödinger equation, but the details of how it is written is quite different for the two cases.

N non-identical particles

In the case of non-identical particles, the many-body quantum mechanics is a straigthforward generalization of the one-particle case:

• Number of degrees of freedom extends from a single particle to particles
• Observables and the related operators denote the operators separately for each particle, i.e., and so on.
• Suppose that the single-particle states have the basis , where are the quantum numbers of the state.
• The -particle states then have a basis , which denotes a state where the th particle occupies a state with quantum numbers . Note that the above (tensor) product between single particle states does not commute, and we cannot reorder the terms. Also note the notation here, as we will later introduce a notation for bosonic and fermionic states, which will in general be a different state than the above tensor product.

For example, the position-spin basis vectors for 2 particles are where is the position of the th particle and is the z-component of its spin. - The completeness relation includes sums and integrals over all particles. The single-particle completeness relation in position basis generalizes to where on the second line, we used a short-hand for the position-spin basis vectors.

• The inner product between two -particle states and is a product of the inner products between the constituent single-particle states:
• The wave functions are written for particles as The latter short-hand notation is used also in the rest of the chapter.
• The dynamics is given by the -particle Hamilton operator , which has a position representation More generally, the Hamiltonian could be non-local in space and non-diagonal in spin. In that case we would need to write it as a matrix The matrix form of a local spin-diagonal Hamiltonian is non-zero only for the diagonal matrix elements

These steps of course increase the complication of the theory, but do not give anything qualitatively new to the description. On the other hand, when the particles are indistinguishable, we need to include this property in the description. To understand how this can be done, let us make a small detour to group theory and representations of symmetry groups in quantum mechanics.

Symmetry and groups in quantum mechanics

This should be familiar to you from the discussion of angular momentum in part A of the course. But you can refresh your memory here.

N identical particles

If the particles are identical, there is a new symmetry in the system: where is an arbitrary permutation of .

For example, the Hamilton operator could be of the form where every particle has the same mass , feels the same potential and all pairs of particles interact via the same potential . This is in position basis, so .

We denote a permutation with the symbol A permutation is a bijective function from the set to itself, such that , , ... . All possible permutations of form a so-called symmetric group (or permutation group) .

Let us define an operator , which represents the permutation . Its action on the many-particle wavefunction is The action (of the inverse permutation) on the state vectors is which can be verified by comparing the wavefunctions and . The effect of the permutation operator is to assign the quantum numbers of particle to particle .

We apply this operator on both sides of the Schrödinger equation and get where on the right hand side we inserted the identity operator between and . This can be written as where is defined as To see how the permutation operator acts on the Hamiltonian, let us consider the matrix element For a Hamiltonian describing identical particles, it follows that

We find that , i.e. and have common eigenstates. As we found out above, this implies that the eigenstates of should transform under permutations according to the irreducible representations of : where is the matrix element of and the eigenstates of which have the same energy as , in the representation of defined by the operators .

Representations of the permutation group: Bosons and fermions

What kind of matrix representation does the permutation group have? It turns out that has exactly two 1-dimensional representations:

1. The "trivial" representation, where the wavefunction is fully symmetric under any permutation of . Particles, whose wavefunction transforms according to this representation, are called bosons.

2. Wavefunction can change sign under an exchange of partiles: Here

   • if is even, i.e. if is obtained from by performing an even number of exchanges (An exchange swaps the places of two elements, e.g. )
   • if is odd.

   In this case, the wavefunction is fully antisymmetric under an exchange of any two particles . Particles, whose wavefunction transforms according to this representation of , are called fermions.

Counting states in statistical mechanics You might remember from the statistical mechanics course the so called "correct Boltzmann counting": when calculating the partition function for a system of identical particles one sums over the states of the particles, but divides by : Remember also that this itself is still an approximation that is only valid at high temperature. It does not treat correctly the possibility that two or more of the particles could actually be in the same state if they are bosons but not if they are fermions, which starts to be important at lower temperatures.

Now you understand the origin of this : if you have identical particles to be distributed to different states, you have a total of possible ways to do this. The energy for each of these permutations is the same, and the sum over states in the above equation includes all of them as separate states. However, only one (one linear combination) of these states is fully symmetric (antisymmetric) in all permutations of particles, and thus a possible state of the system. The "correct Boltzmann counting" removes the effect of the other states that are not allowed by the particle exchange symmetry.

Paraparticles In addition, (for ), has other unitary and irreducible higher-dimensional representations. Particles whose wavefunctions would transform according to these would be called "para-particles". However, these particles have not been detected.

However, they could occur as low-energy excitations in some condensed-matter systems, especially in cases where the particles are confined into one or two dimensions. In fact, the search of parafermions is at present an active field of study (see for example this review paper).

Spin-statistics theorem

There is a connection between the spin of an particle and the permutation symmetry of the many-body wavefunction, known as the spin-statistics theorem:

Particles, whose spin is an integer number () are bosons; they obey the Bose-Einstein statistics.

Particles, whose spin is a half-integer number () are fermions; they obey the Fermi-Dirac statistics.

The proof of this theorem requires the tools of relativistic QFT.

Noninteracting identical particles

Let us suppose we can write the Hamilton operator of a nonrelativistic system of identical particles as where and are symmetric functions of their arguments and is a 1-particle operator.

Particles are independent (non-interacting) if the mutual interactions are so weak that we can approximate , or if the mutual interactions can be approximated by a "mean field".

The Hamilton operator for identical, non-interacting, particles is Now suppose that i.e. that the 1-particle eigenvalues and eigenstates of are known, and is a set of suitable quantum numbers that describe the system; e.g. .

Suppose also that the eigenstates are normalized, i.e. where .

Since we now have noninteracting particles, the solution of the Schrödinger equation separates into a product of single-particle wavefunctions, i.e. . Substituting this ansatz to the SE, we find so that the energy is obtained as a sum of single-particle energies, i.e. . Since is the same for any permutation , the degree of degeneracy of this energy is : any one of these permutations is a solution of the Schrödinger equation with the same . Howevern, none of them is by itself a valid wavefunction for the system.

Bosons: fully symmetric wave function

The bosonic wave function is of the form where is a normalization factor and is the number of particles in state , i.e., the number of times the state appears in the product. Obviously .

The above is in position basis. In Dirac notation the symmetrized state vector corresponding to the above wavefunction is

The examples above show that labelling the -particle states as becomes soon tedious. Therefore, it is easier to use the Occupation number representation in the Fock space: where how many times the one-particle state appears in . This number for bosons.

Note that the number of particles is and the eigenenergy of the many-particle state can be found by .

Fermions: fully antisymmetric wave function

Antisymmetrizing the product for an arbitrary exchange gives This kind of wavefunction is known as Slater determinant. The corresponding Hilbert space vector in Dirac notation is The normalization constant guarantees again .

Unlike in the bosonic case, here the order of 's does matter. and represent the same state, but with a phase difference of :

Recall here the definition of a determinant of an matrix M:

From this definition it follows that if two rows are identical, i.e. the wavefunction vanishes, , if two fermions share the same state ( for some pair ).

This property of the fermions is commonly stated as Pauli exclusion principle
Two identical fermions cannot have identical quantum numbers, i.e. two identical fermions do not appear in the same quantum state.

also if two columns are identical, in other words if two identical fermions in different 1-particle states have the same coordinates , e.g. and

Even without interactions there is an effective repulsion force between identical fermions, (), known as the Fermi repulsion

Also in the identical-fermion systems, we use the occupation number representation in the Fock space: where or for fermions. The total number of particles and the energy are

Examples of antisymmetric wavefunctions of identical fermions

Ground state of helium

A crude model for helium is given by the Hamiltonian where the 1-particle Hamiltonians are where , or less if we use a variational approach (see QMI). The single-particle Hamiltonians are familiar from the hydrogen atom.

1-particle solutions are known: The spatial part of the many-particle wavefunction consists of , and the energy is . There is no spin in , so the different spin states are degenerate.

Consider then the lowest possible energy (the ground state) of the system, and set . We see that the spatial part is symmetric in the exchange . The full wavefunction on the other hand is antisymmetric. Thus the spin-part of the wavefunction must be antisymmetric. As we find out in the box below, only the singlet state qualifies.

We denote the singlet state by where .

The ground state wavefunction is thus which is a Slater determinant, and and are single-particle states with quantum numbers , and .

Now you should be able to go back and do Question 1 in the preliminary exercises

Fock space

In many-particle systems we might want to consider cases where the particle number can change --- such as creation and annihilation of particles in relativistic quantum field theory, particle-hole creation or phonons in condensed-matter physics, or quantization of electromagnetic field (with a different number of photons), one should consider all -particle sectors simultaneously.

This means that we should generalize the above considerations and introduce the Fock space : where denotes the -particle Hilbert space.

In the occupation number representation these spaces can be specified as follows

• : consists of the single vector , i.e., the vacuum state, with no particles. This vacuum state is normalized: .

• : consists of an infinite number of vectors , , , and so on. In other words, there is one particle on one of the possible one-particle eigenstates, and no particles on the others.

• : consists of the vectors where the states of the type in the second line is possible only for bosons.

In the case of we hence have the states of the form

For a generic Fock space vector , we can however lift the requirement of the number of particles, since it can vary.

We can choose the basis for the one-particle states so that it is orthonormal and complete, i.e., Similarly we can write for the Fock states In addition, the states corresponding to a different total number of particles are orthogonal, This means that an inner product of Fock space states and , where , becomes

The occupation number states of the form hence form an orthonormal basis for the Fock space .

Bosonic system

Let us now discuss the operators acting on the Fock space vectors. In fact, it is enough to define two types of operators: the annihilation operator removing a particle from a certain one-particle state, and its conjugate operator, creation operator, that creates a particle in a given one-particle state.

Bosonic annihilation operator
In other words, removes one particle from state . There are as many such operators as there are one-particle states considered in the problem.

The above then also implies that since there is nothing to be annihilated in a vacuum state.

Note that you have encountered this type of an operator already before: in the discussion of the harmonic oscillator, where the states correspond to its excitation states, and the annihilation operator thus couples the state with state .

In quantum mechanics, operators are typically defined via their matrix elements. From the above definition, the matrix element of is

This is a complex number (or in fact a real number in this case). Let us take its complex conjugate. We get This defines the Bosonic creation operator
which adds the occupation of state by one.

Let us check what happens if we first create a particle in state and then annihilate one from state .

We find that the bosonic operators satisfy the Bosonic commutation relations and The first of these is proven in the above collapsible region. The second is straightforward to show with a similar manner.

Note that we can now start from the vacuum and generate all states by operating with the creation operator multiple times: Continuing this, we find that a given Fock state specified by the set of occupation numbers can be defined by

During the derivation of the bosonic commutation relations, we also found that the Fock states are eigenstates of the operator with the eigenvalue . In other words, this operator counts the number of particles in the one-particle state : Particle number operator The total particle number operator is then

Fermionic system

The state vector in a fermionic system can be presented in terms of the Slater determinant, so that the occupation number representation (the last line) contains all those states involved in the Slater determinant.

The state vector has the property that it changes sign if the indices of any pair of states is changed, i.e., if for example . This means that we have to fix the order in which the one-particle states or their occupation number appear. The precise way of fixing the order does not matter, as long as it stays consistent throughout the description.

Let us define the Fermionic creation operator and The latter property is needed to make sure that we cannot create two fermions in the same single-particle state. It also implies

Let us create now two particles in different single-particle states and so that is further right from : Note that this uses the fact that in the first sum . We can do it also in the opposite order: We hence get an extra "1" in the exponent, because the latter expression assumes . Because of that, the latter result is times the previous one.

Adding these results together hence yields If either of the states is already occupied, the result is zero no matter which order these operators operate. The result is hence valid for an arbitrary Fock state. Since we required also that , we can write Here we defined the anticommutator between two operators and : .

Correspondingly, we can define the Fermion annihilation operator In the same manner as for the creation operators, we can get the anticommutation relations for and : Alternatively, this could be obtained by taking the conjugate of the matrix elements of .

What about the anticommutator ? Let us do it explicitely. First, with : and In other words, we get the same anticommutation relation as above, and it is easy to see (for example by taking the conjugate matrix element) that this holds also for On the other hand, if , we get We thus found the Fermion number operator for which .

On the other hand, Since the operator connects the number states to themselves (i.e., it has only diagonal matrix elements), and has the eigenvalue , we can identify it with Putting this together with the definition of the fermion number operator, we get the remaining Fermion anticommutation relations

Let us then create the fermionic state from the vacuum state by operating multiple times with the creation operators of the different single-particle states. Consider for example the state , i.e., where all states above the 4th state are empty. To get there, we start from the rightmost occupied single-particle state: This we can generalize to an arbitrary fermion state Constructing a fermion state from vacuum with . The order of the operators is important!

Analogous to the bosonic state, the total fermion number operator is Notice how the fermionic nature of the particles (states) lies in the definition of the creation and annihilation operators, or equivalently in their algebra, originating from their anticommutation relations!

Now you should be able to go back and do Question 2 in the preliminary exercises

Second quantization and field operators

Consider a change of basis for 1-particle states from one basis to another basis . For fermions, we define and .

Now, which is equivalent to Therefore, we have This and the following holds similarly for bosonic operators.

Take then with for brevity and interpret as an operator which creates a fermion at with .

We now denote and say that is a field operator.

Correspondingly, for the annihilation operator is a field operator that destroys a fermion at with .

This looks as if we have replaced the wave function by an operator, by interpreting the coefficients of the basis-WF 's as operators . Such a procedure is of fundamental importance in e.g. relativistic QM and the theory of open many-particle Q-systems. This procedure is called the second quantization, in reference to "the first quantization" where one relates physical observables to Hermitian operators.

Notice, however, that now, for nonrelativistic theory, we have not done anything new but basically just rewritten and its Hermitian conjugate.

Using the orthogonality of 's, we obtain

The anticommutation rules remain as before which is expected; the same rules hold in the new basis.

For field operators, we get similarly (using ) which leads to: Anticommutation rules for fermionic field operators

For bosonic operators, just replace , i.e., use the commutation rules .

Now, we can write the many-particle states with the help of the field operators. We again take the examples from fermionic systems.

For a 1-particle state, we have simply and for 2-particle states This idea generalizes to N-particle states as

Wick's theorem

Often one needs to compute matrix elements in the vacuum, vacuum expectation values, of an operator ; such as above. In general, can be quite complicated (long product), and the calculation becomes tedious. Wick's theorem helps to cope with this in a systematic manner.

First we need to define some notation. Let (fermions) or (bosons). We denote a normal-ordered product of operators as and define it as a product where all the creation operators (or ) have been brought to the left and the annihilation operators () to the right. If 's are fermionic, the product is multiplied with , where is the number of exchanges needed for bringing the to the normal order.

Examples

• (bosonic)
• (fermionic)
• Let and , where 's are complex numbers. Then
• For bosonic operators and we get no sign-changes:

Properties of the normal ordered product

Under normal ordering, we can change the order of the operators:

The vacuum expectation value of the normal ordered product is zero: since . (and ).

Contractions

Next we define the contraction of two operators as As an expectation value, the contraction is not an operator anymore, but a complex number.

Table of contractions

Next we define the contracted normal-ordered product: where in the product means that is absent. is the number of exchanges necessary to change from to .

An example:

For an operator , Wick's theorem tells us that it can be written as

Which can be checked by using the commutation relations. The general proof of Wick's theorem can be done by mathematical induction.

The Wick's theorem and (its generalizations in field theory) is used when calculating expectation values. For a vacuum expectation value all the terms except those in which all the operators are contracted, disappear thanks to and . For an odd number of operators , the vacuum expectation value is thus always 0.

Let us calculate an earlier example using Wick's theorem: Wick's theorem removed the need to commute the operators by hand. When using the theorem, one typically does not want to write all the possible contractions, but only those which are not obviously zero. One can immediately discard e.g. the terms containing a contraction of two annihilation or two creation operators, and the terms which contain a contraction in the wrong order (see the table of contractions above). This reduces the number of terms that need to be evaluated drastically.

The sign of a single term in the contraction expansions can be determined from the number of intersections of contraction lines (first line of the above equation). For odd number of crossings we get a minus sign (e.g. first term above) and for even number a plus sign (the second term).

Now you should be able to go back and do Question 3 in the preliminary exercises