§1.1 Second Quantization in Quantum Many-Particle Theory
1. Introduction
First quantization in physics is the process of converting a classical system into a quantum system by assigning a wave function to each particle, which describes its behavior at the quantum level. This wave function is determined by solving the Schrödinger equation. But for N-body system, if we construct the N-body wave function, it is impractical to find a direct solution of the Schrödinger equation. So we must explore other techniques, we shall rely on second quantization, quantum field theory, and the use of Green’s functions.
The second quantization merely reformulates the original Schrödinger equation, so it doesn’t mean that we quantize the system twice. Second quantization, simply put, is a method of treating identical particles in a symmetrized Hilbert space using creation and annihilation operators.
2. The difficulty of the first quantization
For a general $N$-body system, the Hamiltonian takes the form
\[\begin{equation} \hat{H} = \sum_{k=1}^{N} \hat{T}(x_k) + \frac{1}{2} \sum_{\substack{k, l=1\\[.2cm] k\neq l}}^{N} \hat{V}(x_k,x_l), \end{equation}\]where $x_k$ denotes the coordinates of the $k$th particle, including all freedoms. The time-independent Schrödinger equation for the N-body wave function is
\[i\hbar \pdv{t} \Psi (x_1\cdots x_N,t) = \hat{H} \Psi(x_1\cdots x_N,t),\]together with an appropriate set of boundary conditions. The space of the $N$-body wave function $\Psi$ is just direct product of $N$ Hilbert spaces,
\[\begin{equation} \mathcal{H}^{N} = \underbrace{\mathcal{H}\otimes\cdots\otimes \mathcal{H}}_{\text{N-copies}}. \end{equation}\]So $\Psi$ can be expanded in terms of a complete set of time-independent single-particle wave functions $\qty{\psi_{E_k}}$ that incorporate the boundary conditions,
\[\begin{equation} \Psi (x_1\cdots x_N,t) = \sum_{E_1\cdots E_N} C(E_1\cdots E_N,t) \psi_{E_1}(x_1)\cdots \psi_{E_N}(x_N), \label{equ:N-body wave function} \end{equation}\]where $E_k$ represents a complete set of single-particle quantum numbers, note that they do not necessarily represent energy. And these single-particle wave functions $\qty{\psi_{E_k}}$ are orthonormal complete. Note that all of the time dependence of the wave function are included in the coefficients $C(E_1 \cdots E_N,t)$.
In physics, we are often interested in systems consisting of many identical particles which means that all intrinsic physical properties of the particles are the same, then the quantum indistinguishabilitiy implies that
\[\begin{equation} |\Psi(x_1\cdots x_i \cdots x_j \cdots x_N,t)|^2 = |\Psi(x_1\cdots x_j\cdots x_i\cdots x_N,t)|^2. \end{equation}\]In $3$ spatial dimensions, this can be shown to lead to only two different possibilities under exchange of the coordinates of two particles:
\[\begin{align} &\Psi \Rightarrow + \Psi, \quad \text{i.e. the wave function is symmetric, or}\\[.2cm] &\Psi \Rightarrow -\Psi, \quad \text{i.e. the wave function is anti-symmetric.} \end{align}\]Particles with a symmetric wave function are called bosons, particles with an antisymmetric wave function are called fermions.
For particles restricted to move in 2 spatial dimensions there also turns out to be more exotic possibilities than just bosons and fermions, namely so- called anyons.
Note that many textbooks tell us that if we exchange two particles of a system twice, the wave function should be equivalent to no exchange at all, so that if $\eta$ is the phase factor for one exchange, two exchanges give $\eta^2 =1$, which thus only gives the solutions $\eta=\pm 1$ corresponding to bosons and fermions.
That approach is indeed wrong, because exchanging two particles is not just permuting coordinates for the wave function mathematically, we must consider it physically by moving them in continuous paths around each other.
The expansion coefficients reflect the identity of particles,
\[\begin{align} &C \Rightarrow + C, \quad \text{i.e. Bosons, or}\\[.2cm] &C \Rightarrow -C, \quad \text{i.e. Fermions.} \end{align}\]If we substitute $\eqref{equ:N-body wave function}$ into the Schrödinger equation, we can derive a differential equation for these coefficients
\[\begin{align*} &i\hbar \pdv{t} C(E_1\cdots E_N,t) \\[.2cm] ={}& \sum_{k=1}^{N}\sum_{W} \int\dd{x_k}\psi_{E_k}^{\dagger}(x_k) T(x_k)\psi_{W}(x_k) C(E_1 \cdots E_{k-1}W E_k \cdots E_N,t)\\[.2cm] +{}\ &\frac{1}{2}\sum_{\substack{k,l=1\\[.2cm]k\neq l}}^N \sum_{W}\sum_{W'}\iint\dd{x_k}\dd{x_l}\psi_{E_k}^{\dagger}(x_k) \psi_{E_l}^{\dagger}(x_l) V(x_k,x_l)\psi_W(x_k)\psi_{W'}(x_l)\\[.2cm] \times \ & C(E_1 \cdots E_{k-1} W E_{k+1} \cdots E_{l-1}W' E_{l+1}\cdots E_N,t). \end{align*}\]This equation is extremely complex, and seeking a solution for it is impractical, which essentially encapsulates the challenge of the first quantization when dealing with $N$-body problems.
We don’t need to expend too much effort on it. We only need to know that it’s very, very, very difficult.
3. Second quantization
3.1. Bosons
For bosons, out of the given set of quantum numbers $E_1,\cdots,E_N$, suppose that the state $1$ occurs $n_1$ times, the state 2 occurs $n_2$ times, and so on. We can defined a special wave function
\[\begin{equation} \Phi_{n_1n_2\cdots n_{\infty}}(x_1\cdots x_N) \equiv \sqrt{\frac{n_1!n_2!\cdots n_{\infty}!}{N!}} \sum_{\substack{E_1\cdots E_N\\[.2cm](n_1 \cdots n_\infty)}}\psi_{E_1}(x_1)\cdots \psi_{E_N}(x_N). \end{equation}\]The most general $N$-body wave function can be expanded in terms of these completely symmetrized wave functions $\Phi_{n_1\cdots n_\infty}(x_1,\cdots,x_N)$
\[\begin{equation} \Psi(x_1\cdots x_N,t) = \sum_{\substack{n_1n_2\cdots n_{\infty}\\[.2cm] \sum_i n_i = N}} f(n_1n_2\cdots n_{\infty},t) \Phi_{n_1n_2\cdots n_{\infty}}(x_1 x_2\cdots x_N). \end{equation}\]This because ${\Phi}$ forms a complete orthonormal basis which means
\[\begin{gather} \Phi(\cdots x_i \cdots x_j\cdots) = \Phi(\cdots x_j \cdots x_i \cdots),\\[.2cm] \int \dd{x_1}\cdots\dd{x_N} \Phi_{n_1'\cdots n_{\infty}'}^{\dagger}(x_1\cdots x_N) \Phi_{n_1\cdots n_{\infty}}(x_1\cdots x_N) = \delta_{n_1'n_1}\cdots \delta_{n_\infty' n_\infty}, \end{gather}\]These properties are easily proven and will be omitted here.
Considering a bosonic system where two particles are in state $1$ and one particle is in state $2$, its wave function can be written as
\[\Phi_{210\cdots 0}(x_1x_2x_3) = \frac{1}{\sqrt{3}}[\psi_1(x_1)\psi_1(x_2)\psi_2(x_3)+\psi_1(x_1)\psi_2(x_2)\psi_1(x_3) + \psi_2(x_1)\psi_1(x_2)\psi_1(x_3)].\]3.2. Fermions
For fermions, the occupation number $n_i$ must be either zero or one because of the Pauli exclusion principle. In the context of considering the antisymmetry of $\Psi$, we can write normalized basis wave functions
\[\begin{equation} \Phi_{n_1\cdots n_{\infty}}(x_1\cdots x_N) = \sqrt{\frac{n_1! n_2!\cdots n_{\infty}!}{N!}} \mdet{\psi_{E_1}(x_1)& \cdots & \psi_{E_1}(x_N)\\[.2cm] \vdots& &\vdots \\[.2cm] \psi_{E_N}(x_1) & \cdots & \psi_{E_N}(x_N)}. \end{equation}\]These functions form a complete set of orthonormal antisymmetric time-independent many-particle wave functions and are usually referred to as the Slater determinant, then
\[\begin{gather} \Phi(\cdots x_i \cdots x_j\cdots) = -\Phi(\cdots x_j \cdots x_i \cdots),\\[.2cm] \int \dd{x_1}\cdots\dd{x_N} \Phi_{n_1'\cdots n_{\infty}'}^{\dagger}(x_1\cdots x_N) \Phi_{n_1\cdots n_{\infty}}(x_1\cdots x_N) = \delta_{n_1'n_1}\cdots \delta_{n_\infty' n_\infty}. \end{gather}\]The $N$-body wave function $\Psi$ can be expanded as
\[\begin{equation} \Psi(x_1\cdots x_N,t) = \sum_{n_1\cdots n_\infty=0}^{1} f(n_1\cdots n_{\infty},t )\Phi_{n_1\cdots n_{\infty}}(x_1\cdots x_N). \end{equation}\]Considering a fermionic system where one particle is in state $1$ and another particle is in state $2$, the wave function of the system can be expressed as:
\[\Phi_{110\cdots 0}(x_1x_2) = \frac{1}{\sqrt{2}}[\psi_1(x_1)\psi_2(x_2) - \psi_2(x_1)\psi_1(x_2)].\]3.3. Occupation number formalism
Instead of using the previous results, we typically address many-particle problem in occupation number formalism, as determining the coefficients in those cases is impractical. We introduce the time-independent abstract state vectors
\[\ket{n_1 n_2 \cdots n_{\infty}} = \ket{n_1}\otimes\ket{n_2}\otimes\cdots \otimes \ket{n_{\infty}},\]where the notation means that there are $n_1$ particles in the eigenstate $1$, $n_2$ particles in the eigenstate $2$, and so on. We want this basis to be complete and orthonormal, which requires that these states satisfy the conditions
\[\begin{align} &\braket{n_1'n_2'\cdots n_\infty'}{n_1n_2\cdots n_{\infty}} = \delta_{n_1'n_1}\delta_{n_2'n_2}\cdots \delta_{n_{\infty}'n_{\infty}}, \quad \text{orthogonality},\\[.2cm] &\sum_{n_1n_2\cdots n_{\infty}}\ket{n_1n_2\cdots n_{\infty}}\bra{n_1n_2\cdots n_{\infty}} = 1, \quad \text{completeness}. \end{align}\]The general $N$-body state can also be expanded in terms of these abstract state.
3.3.1. Bosons
We can introduce time-independent creation and annihilation operators $\hat{b}_k^\dagger,\hat{b}_k$ which satisfy the following commutation relations
\[\begin{equation} [\hat{b}_k,\hat{b}_{k'}^\dagger] = \delta_{kk'}, \quad [\hat{b}_k,\hat{b}_k'] = [\hat{b}_{k}^\dagger,\hat{b}_{k'}^{\dagger}] =0. \end{equation}\]This leads to all the properties of bosons, in particular,
\[\begin{align} \hat{b}_k \ket{n_1\cdots n_k\cdots} &= \sqrt{n_k}\ket{n_1\cdots n_k-1\cdots},\\[.2cm] \hat{b}_k^{\dagger}\ket{n_1\cdots n_k\cdots} &=\sqrt{n_k+1}\ket{n_1\cdots n_k+1\cdots}. \end{align}\]Any given state can be generated from the vacuum state $\ket{0}\equiv \ket{0\cdots n_k=0\cdots}$
\[\begin{equation} \ket{n_1\cdots n_k} = \frac{(\hat{b}_1^\dagger)^{n_1}}{\sqrt{n_1!}}\cdots \frac{(\hat{b}_k^\dagger)^{n_k}}{\sqrt{n_k!}}\cdots \ket{0}. \end{equation}\]3.3.2. Fermions
Similarly, we introduce time-independent creation and annihilation operators $\hat{c}_k,\hat{c}_k^\dagger$, but they obey the following anticommutation relations
\[\begin{equation} \{\hat{c}_k,\hat{c}_k'^\dagger\} =\delta_{k,k'},\quad \{\hat{c}_k,\hat{c}_{k'}\} = \{\hat{c}_k^\dagger,\hat{c}_{k'}^\dagger\} = 0. \end{equation}\]The occupation of any state $k$ can only be $n_k=0$ or $n_k=1$. With definition
\[\begin{equation} \ket{n_1\cdots n_k \cdots} = (\hat{c}_1^\dagger)^{n_1}\cdots (\hat{c}_k^\dagger)^{n_k}\cdots \ket{0}, \end{equation}\]we can derive that
\[\begin{equation} \hat{c}_k\ket{n_1\cdots n_k \cdots} = \begin{cases} (-1)^{S_k}\ket{n_1\cdots n_{k-1}\cdots}, \quad &n_k=1,\\[.2cm] 0, &n_k=0, \end{cases} \end{equation}\]and
\[\begin{equation} \hat{c}_k^\dagger \ket{n_1 \cdots n_k \cdots} = \begin{cases} (-1)^{S_k} \ket{n_1\cdots n_{k+1}\cdots}, \quad &n_k=0,\\[.2cm] 0, & n_k=1 , \end{cases} \end{equation}\]where the phase factor $S_k$ is defined by $S_{k} = n_1+n_2 +\cdots n_{k-1}$.
4. Field operators
It is often convenient to form the linear combinations of the creation and destruction operators (denoted \(\hat{a}_{k\alpha}^\dagger\) and \(\hat{a}_{k\alpha}\) for generality),
where $\alpha$ is an index used to distinguish different spin component. \(\hat{\psi}_{\alpha}\) and \(\hat{\psi}^\dagger_{\alpha}\) are called field operators that satisfy the following commutation or anticommutation relations
\[\begin{equation} [\hat{\psi}_{\alpha}(\vb{x}),\hat{\psi}_{\beta}^{\dagger}(\vb{x}')]_{\mp} = \delta_{\alpha\beta} \delta(\vb{x}-\vb{x}'), \quad [\hat{\psi}_{\alpha}(\vb{x}),\hat{\psi}_{\beta}(\vb{x}')]_{\mp} = [\hat{\psi}_{\alpha}^\dagger(\vb{x}),\hat{\psi}_{\beta}^\dagger(\vb{x}')]_{\mp} = 0. \end{equation}\]where the upper (lower) sign refers to bosons (fermions).
In the next section, we will discuss the second quantization representation of some observables (one- and two- particle operators).
5. References
- Fetter, A. L., Walecka, J. D., Quantum Theory of Many-Particle Systems. Courier Corporation. (2002)
- Mahan, G. D., Many-Particle Physics. Springer Science+Business Media, LLC. (2000).
- Altland, A., Simons, B. D., Condensed Matter Field Theory. Cambridge University Press. (2010)
- 金彪. 量子多体理论. 2024年春