Sunday, 9 June 2013

First quantization

Single particle systems

The following exposition is based on Dirac's treatise on quantum mechanics. In the classical mechanics of a particle, there are dynamic variables which are called coordinates (x) and momenta (p). These specify the state of a classical system. The canonical structure (also known as the symplectic structure) of classical mechanics consists of Poisson brackets between these variables, such as {x,p} = 1. All transformations of variables which preserve these brackets are allowed as canonical transformations in classical mechanics. Motion itself is such a canonical transformation.

By contrast, in quantum mechanics, all significant features of a particle are contained in a state $|\psi\rangle$ , called quantum state. Observables are represented by operators acting on a Hilbert space of such quantum states. The (eigen)value of an operator acting on one of its eigenstates represents the value of a measurement on the particle thus represented. For example, the energy is read off by the Hamiltonian operator $(\hat{H})$ acting on a state $|\psi_n\rangle$ , yielding $\hat{H}|\psi_n\rangle=E_n|\psi_n\rangle$ , where $E_n$ is the characteristic energy associated to this $|\psi_n\rangle$ eigenstate.

Any state could be represented as a linear combination of eigenstates of energy; for example, $|\psi\rangle=\sum_{n=0}^{\infty} a_n|\psi_n\rangle$ , where a_n are constant coefficients.

As in classical mechanics, all dynamical operators can be represented by functions of the position and momentum ones, $\hat{X}$ and $\hat{P}$ , respectively. The connection between this representation and the more usual wavefunction representation is given by the eigenstate of the position operator $\hat{X}$ representing a particle at position x, which is denoted by an element $|x\rangle$ in the Hilbert space, and which satisfies $\hat{X}|x\rangle = x|x\rangle$ . Then, $\psi(x)= \langle x|\psi\rangle$ .

Likewise, the eigenstates $|p\rangle$ of the momentum operator $\hat{P}$ specify the momentum representation: $\psi(p)= \langle p|\psi\rangle$ .

The central relation between these operators is a quantum analog of the above Poisson bracket of classical mechanics, the canonical commutation relation,

$[\hat{X},\hat{P}] = \hat{X}\hat{P}-\hat{P}\hat{X} = i\hbar$ .

This relation encodes (and formally leads to) the uncertainty principle, in the form Δx Δp ≥ ħ/2. This algebraic structure may be thus considered as the quantum analog of the canonical structure of classical mechanics.

History of Canonical quantization

Quantum physics first dealt only with the quantization of the motion of particles, leaving the electromagnetic field classical, hence the name quantum mechanics.

Later the electromagnetic field was also quantized, and even the particles themselves were represented through quantized fields, resulting in the development of quantum electrodynamics (QED) and quantum field theory in general. Thus, by convention, the original form of particle quantum mechanics is denoted first quantization, while quantum field theory is formulated in the language of second quantization.

Canonical quantization

In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the extent possible.

Historically, this was not quite Werner Heisenberg's route to obtaining quantum mechanics, but Paul Dirac introduced it in his 1926 doctoral thesis, the "method of classical analogy" for quantization, and detailed it in his classic text. The word canonical arises from the Hamiltonian approach to classical mechanics, in which a system's dynamics is generated via canonical Poisson brackets, a structure which is only partially preserved in canonical quantization.

This method was further used in the context of quantum field theory by Paul Dirac, in his construction of quantum electrodynamics. In the field theory context, it is also called second quantization, in contrast to the semi-classical first quantization for single particles.

Quantum field operators

Defining ${a^{(\dagger)}}_{\nu}$ as a general annihilation(creation) operator that could be either fermionic $({c^{(\dagger)}}_{\nu})$ or bosonic $({b^{(\dagger)}}_{\nu})$ , the real space representation of the operators defines the quantum field operators $\Psi(\bold{r})$ and $\Psi^{\dagger}(\bold{r})$ by

$\Psi(\bold{r})=\sum_{\nu} \psi_{\nu} \left( \bold{r} \right) a_{\nu}$

$\Psi^{\dagger}(\bold{r})=\sum_{\nu} {\psi^*}_{\nu} \left( \bold{r} \right) {a^{\dagger}}_{\nu}$

Second quantization operators, while the coefficients $\psi_{\nu} \left( \bold{r} \right)$ and ${\psi^*}_{\nu} \left( \bold{r} \right)$ are the ordinary first quantization wavefunctions. Loosely speaking, $\Psi^{\dagger}(\bold{r})$ is the sum of all possible ways to add a particle to the system at position r through any of the basis states $\psi_{\nu}\left(\bold{r}\right)$ . Since $\Psi(\bold{r})$ and $\Psi^{\dagger}(\bold{r})$ are second quantization operators defined in every point in space they are called quantum field operators. They obey the following fundamental commutator and anti-commutator,

$\begin{align} \left[\Psi(\bold{r}_1),\Psi^\dagger(\bold{r}_2)\right]=\delta (\bold{r}_1-\bold{r}_2) &\text{ boson fields,}\\ \{\Psi(\bold{r}_1),\Psi^\dagger(\bold{r}_2)\}=\delta (\bold{r}_1-\bold{r}_2)&\text{ fermion fields.} \end{align}$

In homogeneous systems it is often desirable to transform between real space and the momentum representations, hence, the quantum fields operators in Fourier basis yields:

$\Psi(\bold{r})={1\over \sqrt {V}} \sum_{\bold{k}} e^{i\bold{k\cdot r}}a_{\bold{k}}$

$\Psi^{\dagger}(\bold{r})={ 1\over \sqrt{V}} \sum_{\bold{k}} e^{-i\bold{k\cdot r}}{a^{\dagger}}_{\bold{k}}$

Bosons in Second qantizations

Given the occupation number, we introduce the annihilation $b_{\nu_j}$ and creation ${b^{\dagger}}_{\nu_j}$ operators that lowers(raises) the occupation number in the state $| \nu_j \rang$ by 1,

$b_{\nu_j}|\dots,n_{\nu_{j-1}}, n_{\nu_j}, n_{\nu_{j+1}},\dots \rang=\sqrt{n_{\nu_j}}|\dots,n_{\nu_{j-1}}, n_{\nu_j}-1, n_{\nu_{j+1}},\dots \rang$

${b^{\dagger}}_{\nu_j}|\dots,n_{\nu_{j-1}}, n_{\nu_j}, n_{\nu_{j+1}},\dots \rang=\sqrt{n_{\nu_j}+1}|\dots,n_{\nu_{j-1}}, n_{\nu_j}+1, n_{\nu_{j+1}},\dots \rang$

Since bosons are symmetric in the single-particle state index $\nu_j$ we demand that $b_{\nu_j}$ and ${b^{\dagger}}_{\nu_j}$ commute, So, we can obtain the mean properties of these operators:

$\begin{matrix} [{b^{\dagger}}_{\nu_j},{b^{\dagger}}_{\nu_k}] = 0 & [b_{\nu_j},b_{\nu_k}]=0 & [b_{\nu_j},{b^{\dagger}}_{\nu_k}]=\delta_{\nu_j\nu_k}\\ {b^{\dagger}}_{\nu_j}|n_{\nu_j}\rang=\sqrt{n_{\nu_j}+1}|n_{\nu_j}+1 \rang & b_{\nu_j}|n_{\nu_j}\rang=\sqrt{n_{\nu_j}}|n_{\nu_j} -1\rang & b_{\nu_j}|0\rang=0\\ {b^{\dagger}}_{\nu_j}b_{\nu_j}|n_{\nu_j} \rang=n_{\nu_j}|n_{\nu_j} \rang &\left({b^{\dagger}}_{\nu_j}\right)^{n_{\nu_j}}|0 \rang=\sqrt{(n_{\nu_j})!}|n_{\nu_j} \rang & n_{\nu_j}=0,1,2,\dots\\ \end{matrix}$

and therefore identify the first and second quantized states,

$\hat{S}_+|\psi_{n_{\nu_1}}(\bold{r}_1)\rang|\psi_{n_{\nu_2}}(\bold{r}_2)\rang\dots |\psi_{n_{\nu_1}}(\bold{r}_N)\rang= {b^{\dagger}}_{n_{\nu_1}}{b^{\dagger}}_{n_{\nu_2}}\dots{b^{\dagger}}_{n_{\nu_N}}|0\rang$

with $\hat{S}$ the symmetrization operator. Here, both contain N-particle state-kets completely symmetric in the single-particle state index $\psi_{\nu_j}$ . Because the creation and annihilation operators of the quantum harmonic oscillator obey these properties, one can classify the field associated to it as bosonic.

Creation and annihilation operators of Second qantization

The creation and annihilation operators are the way to connect the first and second quantizations. It is fundamental for the many-body theory that every operator can be expressed in terms of annihilation and creation operators. Originally constructed in the context of the quantum harmonic oscillator, these operators are the most general form to describe quantum fields. Depending on the nature of the fields we can use two different approaches:

Second quantization

Second quantization is a powerful procedure used in quantum field theory for describing the many-particle systems by quantizing the fields using a basis that describes the number of particles occupying each state in a complete set of single-particle states. This differs from the first quantization, which uses the single-particle states as basis.

The starting point of this formalism is the notion of indistinguishability of particles that bring us to use determinants of single-particle states as a basis of the Hilbert space of N-particles states Quantum theory can be formulated in terms of occupation numbers (amount of particles occupying one determined energy state) of these single-particle states. The formalism was introduced in 1927 by Dirac.

The occupation number representation

Consider an ordered and complete single-particle basis $\left\{| \nu_1 \rang, | \nu_2 \rang, | \nu_3 \rang, ...\right\}$ , where $| \nu_i \rang$ is the set of all states $\nu$ available for the $i$ -th particle. In an N-particle system, only the occupied single-particle states play a role. So it is simpler to formulate a representation where one just counts how many particles there are in each orbital $| \nu \rang$ . This simplification is achieved with the occupation number representation. The basis states for an N-particle system in this representation are obtained simply by listing the occupation numbers of each basis state, $|n_{\nu_1}, n_{\nu_2}, n_{\nu_3},\dots \rang$ , where $\sum_j n_{\nu_j} = N$ The notation means that there are $n_{\nu_j}$ particles in the state $\nu_j$ . It is therefore natural to define the occupation number operator $\hat{n}_{\nu_j}$ which obeys

$\hat{n}_{\nu_j}|n_{\nu_j} \rang=n_{\nu_j}|n_{\nu_j} \rang$

For fermions $n_{\nu_j}$ can be 0 or 1, while for bosons it can be any non negative number

$n_{\nu_j}= \begin{cases} \ 0, 1. &\text{fermions}\\ 0,1,2,... &\text{bosons} \end{cases}$

The space spanned by the occupation number basis is denoted the Fock space.