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diff --git a/content/know/concept/second-quantization/index.pdc b/content/know/concept/second-quantization/index.pdc new file mode 100644 index 0000000..b8d9a18 --- /dev/null +++ b/content/know/concept/second-quantization/index.pdc @@ -0,0 +1,332 @@ +--- +title: "Second quantization" +firstLetter: "S" +publishDate: 2021-02-26 +categories: +- Quantum mechanics +- Physics + +date: 2021-02-26T10:04:16+01:00 +draft: false +markup: pandoc +--- + +# Second quantization + +The **second quantization** is a technique to deal with quantum systems +containing a large and/or variable number of identical particles. +Its exact formulation depends on +whether it is fermions or bosons that are being considered +(see [Pauli exclusion principle](/know/concept/pauli-exclusion-principle/)). + +Regardless of whether the system is fermionic or bosonic, +the idea is to change basis to a set of certain many-particle wave functions, +known as the **Fock states**, which are specific members of a **Fock space**, +a special kind of [Hilbert space](/know/concept/hilbert-space/), +with a well-defined number of particles. + +For a set of $N$ single-particle energy eigenstates +$\psi_n(x)$ and $N$ identical particles $x_n$, the Fock states are +all the wave functions which contain $n$ particles, for $n$ going from $0$ to $N$. + +So for $n = 0$, there is one basis vector with $0$ particles, +for $n = 1$, there are $N$ basis vectors with $1$ particle each, +for $n = 2$, there are $N (N \!-\! 1)$ basis vectors with $2$ particles, +etc. + +In this basis, we define the **particle creation operators** +and **particle annihilation operators**, +which respectively add/remove a particle to/from a given state. +In other words, these operators relate the Fock basis vectors +to one another, and are very useful. + +The point is to express the system's state in such a way that the +fermionic/bosonic constraints are automatically satisfied, and the +formulae look the same regardless of the number of particles. + + +## Fermions + +Fermions need to obey the Pauli exclusion principle, so each state can only +contain one particle. In this case, the Fock states are given by: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + n &= 0: + \qquad \ket{0, 0, 0, ...} + \\ + n &= 1: + \qquad \ket{1, 0, 0, ...} \quad \ket{0, 1, 0, ...} \quad \ket{0, 0, 1, ...} \quad \cdots + \\ + n &= 2: + \qquad \ket{1, 1, 0, ...} \quad \ket{1, 0, 1, ...} \quad \ket{0, 1, 1, ...} \quad \cdots + \end{aligned} + } +\end{aligned}$$ + +The notation $\ket{N_\alpha, N_\beta, ...}$ is shorthand for +the appropriate [Slater determinants](/know/concept/slater-determinant/). +As an example, take $\ket{0, 1, 0, 1, 1}$, +which contains three particles $a$, $b$ and $c$ +in states 2, 4 and 5: + +$$\begin{aligned} + \ket{0, 1, 0, 1, 1} + = \Psi(x_a, x_b, x_c) + = \frac{1}{\sqrt{3!}} \det\! + \begin{bmatrix} + \psi_2(x_a) & \psi_4(x_a) & \psi_5(x_a) \\ + \psi_2(x_b) & \psi_4(x_b) & \psi_5(x_b) \\ + \psi_2(x_c) & \psi_4(x_c) & \psi_5(x_c) + \end{bmatrix} +\end{aligned}$$ + +The creation operator $\hat{c}_\alpha^\dagger$ and annihilation +operator $\hat{c}_\alpha$ are defined to live up to their name: +they create or destroy a particle in the state $\psi_\alpha$: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \hat{c}_\alpha^\dagger \ket{... (N_\alpha\!=\!0) ...} + &= J_\alpha \ket{... (N_\alpha\!=\!1) ...} + \\ + \hat{c}_\alpha \ket{... (N_\alpha\!=\!1) ...} + &= J_\alpha \ket{... (N_\alpha\!=\!0) ...} + \end{aligned} + } +\end{aligned}$$ + +The factor $J_\alpha$ is sometimes known as the **Jordan-Wigner string**, +and is necessary here to enforce the fermionic antisymmetry, +when creating or destroying a particle in the $\alpha$th state: + +$$\begin{aligned} + J_\alpha = (-1)^{\sum_{j < \alpha} N_j} +\end{aligned}$$ + +So, for example, when creating a particle in state 4 +of $\ket{0, 1, 1, 0, 1}$, we get the following: + +$$\begin{aligned} + \hat{c}_4^\dagger \ket{0, 1, 1, 0, 1} + = (-1)^{0 + 1 + 1} \ket{0, 1, 1, 1, 1} +\end{aligned}$$ + +The point of the Jordan-Wigner string +is that the order matters when applying the creation and annihilation operators: + +$$\begin{aligned} + \hat{c}_1^\dagger \hat{c}_2 \ket{0, 1} + &= \hat{c}_1^\dagger \ket{0, 0} + = \ket{1, 0} + \\ + \hat{c}_2 \hat{c}_1^\dagger \ket{0, 1} + &= \hat{c}_2 \ket{1, 1} + = - \ket{1, 0} +\end{aligned}$$ + +In other words, $\hat{c}_1^\dagger \hat{c}_2 = - \hat{c}_2 \hat{c}_1^\dagger$, +meaning that the anticommutator $\{\hat{c}_2, \hat{c}_1^\dagger\} = 0$. +You can verify for youself that +the general anticommutators of these operators are given by: + +$$\begin{aligned} + \boxed{ + \{\hat{c}_\alpha, \hat{c}_\beta\} = \{\hat{c}_\alpha^\dagger, \hat{c}_\beta^\dagger\} = 0 + \qquad \quad + \{\hat{c}_\alpha, \hat{c}_\beta^\dagger\} = \delta_{\alpha\beta} + } +\end{aligned}$$ + +Each single-particle state can only contain 0 or 1 fermions, +so these operators **quench** states that would violate this rule. +Note that these are *scalar* zeros: + +$$\begin{aligned} + \boxed{ + \hat{c}_\alpha^\dagger \ket{... (N_\alpha\!=\!1) ...} = 0 + \qquad \quad + \hat{c}_\alpha \ket{... (N_\alpha\!=\!0) ...} = 0 + } +\end{aligned}$$ + +Finally, as has already been suggested by the notation, they are each other's adjoint: + +$$\begin{aligned} + \matrixel{... (N_\alpha\!=\!1) ...}{\hat{c}_\alpha^\dagger}{... (N_\alpha\!=\!0) ...} + = \matrixel{...(N_\alpha\!=\!0) ...}{\hat{c}_\alpha}{... (N_\alpha\!=\!1) ...} +\end{aligned}$$ + +Let us now use these operators to define the **number operator** $\hat{N}_\alpha$ as follows: + +$$\begin{aligned} + \boxed{ + \hat{N}_\alpha = \hat{c}_\alpha^\dagger \hat{c}_\alpha + } +\end{aligned}$$ + +Its eigenvalue is the number of particles residing in state $\psi_\alpha$ +(look at the hats): + +$$\begin{aligned} + \hat{N}_\alpha \ket{... N_\alpha ...} + = N_\alpha \ket{... N_\alpha ...} +\end{aligned}$$ + + +## Bosons + +Bosons do not need to obey the Pauli exclusion principle, so multiple can occupy a single state. +The Fock states are therefore as follows: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + n &= 0: + \qquad \ket{0, 0, 0, ...} + \\ + n &= 1: + \qquad \ket{1, 0, 0, ...} \quad \ket{0, 1, 0, ...} \quad \ket{0, 0, 1, ...} \quad \cdots + \\ + n &= 2: + \qquad \ket{1, 1, 0, ...} \quad \ket{1, 0, 1, ...} \quad \ket{0, 1, 1, ...} \quad \cdots + \\ + &\qquad\:\:\: + \qquad \ket{2, 0, 0, ...} \quad \ket{0, 2, 0, ...} \quad \ket{0, 0, 2, ...} \quad \cdots + \end{aligned} + } +\end{aligned}$$ + +They must be symmetric under the exchange of two bosons. +To achieve this, the Fock states are represented by Slater *permanents* +rather than determinants. + +The boson creation and annihilation operators $\hat{c}_\alpha^\dagger$ and +$\hat{c}_\alpha$ are straightforward: + +$$\begin{gathered} + \boxed{ + \begin{aligned} + \hat{c}_\alpha^\dagger \ket{... N_\alpha ...} + &= \sqrt{N_\alpha + 1} \: \ket{... (N_\alpha \!+\! 1) ...} + \\ + \hat{c}_\alpha \ket{... N_\alpha ...} + &= \sqrt{N_\alpha} \: \ket{... (N_\alpha \!-\! 1) ...} + \end{aligned} +}\end{gathered}$$ + +Applying the annihilation operator $\hat{c}_\alpha$ when there are zero +particles in $\alpha$ will quench the state: + +$$\begin{aligned} + \boxed{ + \hat{c}_\alpha \ket{... (N_\alpha\!=\!0) ...} = 0 + } +\end{aligned}$$ + +There is no Jordan-Wigner string, and therefore no sign change when commuting. +Consequently, these operators therefore satisfy the following: + +$$\begin{aligned} + \boxed{ + [\hat{c}_\alpha, \hat{c}_\beta] = [\hat{c}_\alpha^\dagger, \hat{c}_\beta^\dagger] = 0 + \qquad + [\hat{c}_\alpha, \hat{c}_\beta^\dagger] = \delta_{\alpha\beta} + } +\end{aligned}$$ + +The constant factors applied by $\hat{c}_\alpha^\dagger$ and $\hat{c}_\alpha$ +ensure that $\hat{N}_\alpha$ keeps the same nice form: + +$$\begin{aligned} + \boxed{ + \hat{N}_\alpha = \hat{c}_\alpha^\dagger \hat{c}_\alpha + } +\end{aligned}$$ + + +## Operators + +Traditionally, an operator $\hat{V}$ simultaneously acting on $N$ indentical particles +is the sum of the individual single-particle operators $\hat{V}_1$ acting on the $n$th particle: + +$$\begin{aligned} + \hat{V} + = \sum_{n = 1}^N \hat{V}_1 +\end{aligned}$$ + +This can be rewritten using the second quantization operators as follows: + +$$\begin{aligned} + \boxed{ + \hat{V} + = \sum_{\alpha, \beta} \matrixel{\alpha}{\hat{V}_1}{\beta} \hat{c}_\alpha^\dagger \hat{c}_\beta + } +\end{aligned}$$ + +Where the matrix element $\matrixel{\alpha}{\hat{V}_1}{\beta}$ is to be +evaluated in the normal way: + +$$\begin{aligned} + \matrixel{\alpha}{\hat{V}_1}{\beta} + = \int \psi_\alpha^*(\vec{r}) \: \hat{V}_1(\vec{r}) \: \psi_\beta(\vec{r}) \dd{\vec{r}} +\end{aligned}$$ + +Similarly, given some two-particle operator $\hat{V}$ in first-quantized form: + +$$\begin{aligned} + \hat{V} + = \sum_{n \neq m} v(\vec{r}_n, \vec{r}_m) +\end{aligned}$$ + +We can rewrite this in second-quantized form as follows. +Note the ordering of the subscripts: + +$$\begin{aligned} + \boxed{ + \hat{V} + = \sum_{\alpha, \beta, \gamma, \delta} + v_{\alpha \beta \gamma \delta} \hat{c}_\alpha^\dagger \hat{c}_\beta^\dagger \hat{c}_\delta \hat{c}_\gamma + } +\end{aligned}$$ + +Where the constant $v_{\alpha \beta \gamma \delta}$ is defined from the +single-particle wave functions: + +$$\begin{aligned} + v_{\alpha \beta \gamma \delta} + = \iint \psi_\alpha^*(\vec{r}_1) \: \psi_\beta^*(\vec{r}_2) + \: v(\vec{r}_1, \vec{r}_2) \: \psi_\gamma(\vec{r}_1) + \: \psi_\delta(\vec{r}_2) \dd{\vec{r}_1} \dd{\vec{r}_2} +\end{aligned}$$ + +Finally, in the second quantization, changing basis is done in the usual way: + +$$\begin{aligned} + \hat{c}_b^\dagger \ket{0} + = \ket{b} + = \sum_{\alpha} \ket{\alpha} \braket{\alpha}{b} + = \sum_{\alpha} \braket{\alpha}{b} \hat{c}_\alpha^\dagger \ket{0} +\end{aligned}$$ + +Where $\alpha$ and $b$ need not be in the same basis. +With this, we can define the **field operators**, +which create or destroy a particle at a given position $\vec{r}$: + +$$\begin{aligned} + \boxed{ + \hat{\psi}^\dagger(\vec{r}) + = \sum_{\alpha} \braket{\alpha}{\vec{r}} \hat{c}_\alpha^\dagger + \qquad \quad + \hat{\psi}(\vec{r}) + = \sum_{\alpha} \braket{\vec{r}}{\alpha} \hat{c}_\alpha + } +\end{aligned}$$ + + +## References +1. L.E. Ballentine, + *Quantum mechanics: a modern development*, 2nd edition, + World Scientific. |