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diff --git a/source/know/concept/second-quantization/index.md b/source/know/concept/second-quantization/index.md index 975921c..e446557 100644 --- a/source/know/concept/second-quantization/index.md +++ b/source/know/concept/second-quantization/index.md @@ -20,13 +20,13 @@ 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 $N$. +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, +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** @@ -60,10 +60,10 @@ $$\begin{aligned} } \end{aligned}$$ -The notation $\Ket{N_\alpha, N_\beta, ...}$ is shorthand for +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 $c$ +As an example, take $$\Ket{0, 1, 0, 1, 1}$$, +which contains three particles $$a$$, $$b$$ and $$c$$ in states 2, 4 and 5: $$\begin{aligned} @@ -77,9 +77,9 @@ $$\begin{aligned} \end{bmatrix} \end{aligned}$$ -The creation operator $\hat{c}_\alpha^\dagger$ and annihilation -operator $\hat{c}_\alpha$ are defined to live up to their name: -they create or destroy a particle in the state $\psi_\alpha$: +The creation operator $$\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{ @@ -93,16 +93,16 @@ $$\begin{aligned} } \end{aligned}$$ -The factor $J_\alpha$ is sometimes known as the **Jordan-Wigner string**, +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: +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: +of $$\Ket{0, 1, 1, 0, 1}$$, we get the following: $$\begin{aligned} \hat{c}_4^\dagger \Ket{0, 1, 1, 0, 1} @@ -122,8 +122,8 @@ $$\begin{aligned} = - \Ket{1, 0} \end{aligned}$$ -In other words, $\hat{c}_1^\dagger \hat{c}_2 = - \hat{c}_2 \hat{c}_1^\dagger$, -meaning that the anticommutator $\{\hat{c}_2, \hat{c}_1^\dagger\} = 0$. +In other words, $$\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$$. You can verify for youself that the general anticommutators of these operators are given by: @@ -154,7 +154,7 @@ $$\begin{aligned} = \matrixel{...(N_\alpha\!=\!0) ...}{\hat{c}_\alpha}{... (N_\alpha\!=\!1) ...} \end{aligned}$$ -Let us now use these operators to define the **number operator** $\hat{N}_\alpha$ as follows: +Let us now use these operators to define the **number operator** $$\hat{N}_\alpha$$ as follows: $$\begin{aligned} \boxed{ @@ -162,7 +162,7 @@ $$\begin{aligned} } \end{aligned}$$ -Its eigenvalue is the number of particles residing in state $\psi_\alpha$ +Its eigenvalue is the number of particles residing in state $$\psi_\alpha$$ (look at the hats): $$\begin{aligned} @@ -198,8 +198,8 @@ They must be symmetric under the exchange of two bosons. To achieve this, the Fock states are represented by Slater *permanents* rather than determinants. -The boson creation and annihilation operators $\hat{c}_\alpha^\dagger$ and -$\hat{c}_\alpha$ are straightforward: +The boson creation and annihilation operators $$\hat{c}_\alpha^\dagger$$ and +$$\hat{c}_\alpha$$ are straightforward: $$\begin{gathered} \boxed{ @@ -212,8 +212,8 @@ $$\begin{gathered} \end{aligned} }\end{gathered}$$ -Applying the annihilation operator $\hat{c}_\alpha$ when there are zero -particles in $\alpha$ will quench the state: +Applying the annihilation operator $$\hat{c}_\alpha$$ when there are zero +particles in $$\alpha$$ will quench the state: $$\begin{aligned} \boxed{ @@ -232,8 +232,8 @@ $$\begin{aligned} } \end{aligned}$$ -The constant factors applied by $\hat{c}_\alpha^\dagger$ and $\hat{c}_\alpha$ -ensure that $\hat{N}_\alpha$ keeps the same nice form: +The constant factors applied by $$\hat{c}_\alpha^\dagger$$ and $$\hat{c}_\alpha$$ +ensure that $$\hat{N}_\alpha$$ keeps the same nice form: $$\begin{aligned} \boxed{ @@ -244,8 +244,8 @@ $$\begin{aligned} ## Operators -Traditionally, an operator $\hat{V}$ simultaneously acting on $N$ indentical particles -is the sum of the individual single-particle operators $\hat{V}_1$ acting on the $n$th particle: +Traditionally, an operator $$\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} @@ -261,7 +261,7 @@ $$\begin{aligned} } \end{aligned}$$ -Where the matrix element $\matrixel{\alpha}{\hat{V}_1}{\beta}$ is to be +Where the matrix element $$\matrixel{\alpha}{\hat{V}_1}{\beta}$$ is to be evaluated in the normal way: $$\begin{aligned} @@ -269,7 +269,7 @@ $$\begin{aligned} = \int \psi_\alpha^*(\vec{r}) \: \hat{V}_1(\vec{r}) \: \psi_\beta(\vec{r}) \dd{\vec{r}} \end{aligned}$$ -Similarly, given some two-particle operator $\hat{V}$ in first-quantized form: +Similarly, given some two-particle operator $$\hat{V}$$ in first-quantized form: $$\begin{aligned} \hat{V} @@ -287,7 +287,7 @@ $$\begin{aligned} } \end{aligned}$$ -Where the constant $v_{\alpha \beta \gamma \delta}$ is defined from the +Where the constant $$v_{\alpha \beta \gamma \delta}$$ is defined from the single-particle wave functions: $$\begin{aligned} @@ -306,9 +306,9 @@ $$\begin{aligned} = \sum_{\alpha} \Inprod{\alpha}{b} \hat{c}_\alpha^\dagger \Ket{0} \end{aligned}$$ -Where $\alpha$ and $b$ need not be in the same basis. +Where $$\alpha$$ and $$b$$ need not be in the same basis. With this, we can define the **field operators**, -which create or destroy a particle at a given position $\vec{r}$: +which create or destroy a particle at a given position $$\vec{r}$$: $$\begin{aligned} \boxed{ |