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# 5.9: Appendix I- Second Quantization


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## Basis States and Creation/Annihilation Operators

Second quantization is a convenient scheme to label basis states of a many particle quantum system. We are ultimately interested in solutions of the many-body Schrödinger equation, $\HH\RPsi(\Bx\ns_1,\ldots,\Bx\ns_N) = E\,\RPsi(\Bx\ns_1,\ldots,\Bx\ns_N)$ where the Hamiltonian is $\HH=-{\hbar^2\over 2m}\sum_{i=1}^N \bnabla_i^2 + \sum_{j<k}^N V(\Bx\ns_j-\Bx\ns_k)\quad.$ To the coordinate labels $$\{\Bx\ns_1,\ldots\Bx\ns_N\}$$ we may also append labels for internal degrees of freedom, such as spin polarization, denoted $$\{\zeta\ns_1,\ldots,\zeta\ns_N\}$$. Since $$\big[\HH,\sigma\big]=0$$ for all permutations $$\sigma\in S\ns_N$$, the many-body wavefunctions may be chosen to transform according to irreducible representations of the symmetric group $$S\ns_N$$. Thus, for any $$\sigma\in S\ns_N$$, $\RPsi\big(\Bx\ns_{\sigma(1)},\ldots,\Bx\ns_{\sigma(N)}\big)=\bigg\{ {1\atop \textsf{sgn}(\sigma)}\bigg\}\,\RPsi(\Bx\ns_1,\ldots,\Bx\ns_N)\quad,$ where the upper choice is for Bose-Einstein statistics and the lower sign for Fermi-Dirac statistics. Here $$\Bx\ns_j$$ may include not only the spatial coordinates of particle $$j$$, but its internal quantum number(s) as well, such as $$\zeta\ns_j$$.

A convenient basis for the many body states is obtained from the single-particle eigenstates $$\big\{\tket{\alpha}\big\}$$ of some single-particle Hamiltonian $$\HH\ns_0$$ , with $$\sbraket{\Bx}{\alpha}=\vphi\ns_\alpha(\Bx)$$ and $$\HH\ns_0\,\tket{\alpha}=\ve\ns_\alpha\,\tket{\alpha}$$. The basis may be taken as orthonormal, $$\braket{\alpha}{\alpha'}=\delta\ns_{\alpha\alpha'}$$. Now define $\RPsi\ns_{\alpha\ns_1,\ldots,\alpha\ns_N}(\Bx\ns_1,\ldots,\Bx\ns_N)={1\over\sqrt{N!\prod_\alpha n\ns_\alpha!}}\sum_{\sigma\in S\ns_N} \bigg\{ {1\atop \textsf{sgn}(\sigma)}\bigg\}\ \vphi\ns_{\alpha\ns_{\sigma(1)}}(\Bx\ns_1) \cdots \vphi\ns_{\alpha\ns_{\sigma(N)}}(\Bx\ns_N)\quad.$ Here $$n\ns_\alpha$$ is the number of times the index $$\alpha$$ appears among the set $$\{\alpha\ns_1,\ldots,\alpha\ns_N\}$$. For BE statistics, $$n\ns_\alpha\in\{0,1,2,\ldots\}$$ , whereas for FD statistics, $$n\ns_\alpha\in\{0,1\}$$ . Note that the above states are normalized22: \begin{aligned} \int\!d^d\!x\ns_1\cdots\!\int\!d^d\!x\ns_N\>\big|\RPsi\ns_{\alpha\ns_1\cdots\alpha\ns_N}(\Bx\ns_1,\ldots,\Bx\ns_N)\big|^2&= {1\over N!\prod_\alpha n\ns_\alpha!}\sum_{\sigma,\mu\in S\ns_N} \bigg\{ {1\atop \textsf{sgn}(\sigma\mu)}\bigg\}\ \prod_{j=1}^N\int\!d^d\!x\ns_j \ \vphi^*_{\alpha\ns_{\sigma(j)}}(\Bx\ns_j)\,\vphi\ns_{\alpha\ns_{\mu(j)}}(\Bx\ns_j) \nonumber \\ &={1\over\prod_\alpha n\ns_\alpha!}\sum_{\sigma\in S\ns_N} \prod_{j=1}^N \delta\ns_{\alpha\ns_j,\alpha\ns_{\sigma(j)}} = 1\quad.\end{aligned} Note that $\begin{split} \sum_{\sigma\in S\ns_N} \vphi\ns_{\alpha\ns_{\sigma(1)}}(\Bx\ns_1) \cdots \vphi\ns_{\alpha\ns_{\sigma(N)}}(\Bx\ns_N) & \equiv \textsf{per}\big\{ \vphi\ns_{\alpha\ns_i}(\Bx\ns_j)\big\} \\ \sum_{\sigma\in S\ns_N} \textsf{sgn}(\sigma)\> \vphi\ns_{\alpha\ns_{\sigma(1)}}(\Bx\ns_1) \cdots \vphi\ns_{\alpha\ns_{\sigma(N)}}(\Bx\ns_N) & \equiv \textsf{det}\big\{ \vphi\ns_{\alpha\ns_i}(\Bx\ns_j)\big\} \quad, \end{split}$ which stand for permanent and determinant, respectively. We may now write $\RPsi\ns_{\alpha\ns_1\cdots\,\alpha\ns_N}(\Bx\ns_1,\ldots,\Bx\ns_N)=\braket{\Bx\ns_1, \cdots, \Bx\ns_N}{\alpha\ns_1\cdots\alpha\ns_N}\quad,$ where $\ket{\alpha\ns_1\cdots\,\alpha\ns_N}={1\over\sqrt{N!\prod_\alpha n\ns_\alpha!}}\,\sum_{\sigma\in S\ns_N} \bigg\{ {1\atop \textsf{sgn}(\sigma)}\bigg\}\ \ket{\alpha\ns_{\sigma(1)}} \otimes \ket{\alpha\ns_{\sigma(2)}} \otimes \cdots \otimes \ket{\alpha\ns_{\sigma(N)}} \quad.$ Note that $$\sket{\alpha\ns_{\sigma(1)}\cdots\alpha\ns_{\sigma(N)}}=(\pm 1)^\sigma\,\sket{\alpha\ns_1\cdots\alpha\ns_N}$$ , where by $$(\pm 1)^\sigma$$ we mean $$1$$ in the case of BE statistics and $$\sgn(\sigma)$$ in the case of FD statistics.

We may express $$\sket{\alpha\ns_1\cdots\alpha\ns_N}$$ as a product of creation operators acting on a vacuum $$\sket{0}$$ in Fock space. For bosons, $\ket{\alpha\ns_1\cdots\,\alpha\ns_N}=\prod_\alpha {(b\yd_\alpha)^{n\ns_\alpha}\over\sqrt{n\ns_\alpha!}}\,\ket{0} \equiv \ket{\{n\ns_\alpha\}} \quad,$ with $\big[b\nd_\alpha\,,\,b\nd_{\beta}\big]=0 \qquad,\qquad \big[b\yd_\alpha\,,\,b\yd_{\beta}\big]=0 \qquad,\qquad \big[b\nd_\alpha\,,\,b\yd_{\beta}\big]=\delta\ns_{\alpha\beta}\quad,$ where $$[\,\bullet\,,\bullet\,]$$ is the commutator. For fermions, $\ket{\alpha\ns_1\cdots\,\alpha\ns_N}=c\yd_{\alpha\ns_1} c\yd_{\alpha\ns_2}\cdots\, c\yd_{\alpha\ns_N}\,\ket{0} \equiv \ket{\{n\ns_\alpha\}} \quad,$ with $\big\{c\nd_\alpha\,,\,c\nd_{\beta}\big\}=0 \qquad,\qquad \big\{c\yd_\alpha\,,\,c\yd_{\beta}\big\}=0 \qquad,\qquad \big\{c\nd_\alpha\,,\,c\yd_{\beta}\big\}=\delta\ns_{\alpha\beta}\quad,$ where $$\{\bullet\,,\bullet\}$$ is the anticommutator.

## Second Quantized Operators

Now consider the action of permutation-symmetric first quantized operators such as $$\HT=-{\hbar^2\over 2m}\sum_{i=1}^N\bnabla_i^2$$ and $$\HV=\sum_{i<j}^N \Hv(\Bx\ns_i-\Bx\ns_j)$$. For a one-body operator such as $$\HT$$, we have \begin{aligned} \expect{\alpha\ns_1\cdots\,\alpha\ns_N}{\HT}{\alpha'_1\cdots\,\alpha'_N} &=\int\!d^d\!x\ns_1\cdots\!\int\!d^d\!x\ns_N\> \Big(\prod_\alpha n\ns_\alpha!\Big)^{-1/2}\Big(\prod_\alpha n'_\alpha!\Big)^{-1/2} \times \\ &\hskip 0.5in \sum_{\sigma\in S\ns_N} (\pm 1)^\sigma\vphi^*_{\alpha\ns_{\sigma(1)}}(\Bx\ns_1)\cdots \vphi^*_{\alpha\ns_{\sigma(N)}}(\Bx\ns_N) \>\sum_{k=1}^N \HT\ns_i \>\vphi\ns_{\alpha'_{\sigma(1)}}(\Bx\ns_1)\cdots \vphi\ns_{\alpha'_{\sigma(N)}}\nonumber\\ &=\sum_{\sigma\in S\ns_N} (\pm 1)^\sigma\, \Big(\prod_\alpha n\ns_\alpha!\,n'_\alpha!\Big)^{-1/2} \sum_{i=1}^N \prod_{j\atop (j\ne i)}\delta\ns_{\alpha\ns_j,\alpha'_{\sigma(j)}} \int\!d^d\!x\ns_1\>\vphi^*_{\alpha\ns_i}(\Bx\ns_1)\,\HT\ns_1\, \vphi\ns_{\alpha'_{\sigma(i)}}(\Bx\ns_1)\quad.\nonumber\end{aligned} One may verify that any permutation-symmetric one-body operator such as $$\HT$$ is faithfully represented by the second quantized expression, $\HT=\sum_{\alpha,\beta}\expect{\alpha}{\HT}{\beta}\,\psi\yd_\alpha\,\psi\nd_\beta\quad,$ where $$\psi\yd_\alpha$$ is $$b\yd_\alpha$$ or $$c\yd_\alpha$$ as the application determines, and $\expect{\alpha}{\HT}{\beta}=\int\!d^d\!x\ns_1\>\vphi^*_\alpha(\Bx\ns_1)\,\HT\ns_1\,\vphi\ns_\beta(\Bx\ns_1)\quad.$ Similarly, two-body operators such as $$\HV$$ are represented as $\HV=\half\sum_{\alpha,\beta,\gamma,\delta} \expect{\alpha\beta}{\HV}{\gamma\delta}\,\psi\yd_\alpha\,\psi\yd_\beta \,\psi\nd_\delta\,\psi\nd_\gamma\quad,$ where $\expect{\alpha\beta}{\HV}{\gamma\delta}=\int\!d^d\!x\ns_1\!\int\!d^d\!x\ns_2\> \vphi^*_\alpha(\Bx\ns_1)\,\vphi^*_\beta(\Bx\ns_2)\, v(\Bx\ns_1-\Bx\ns_2)\,\vphi\ns_\delta(\Bx\ns_2)\,\vphi\ns_\gamma(\Bx\ns_1)\quad.$ The general form for an $$n$$-body operator is then $\HR={1\over n!}\sum_{\alpha\ns_1\cdots\,\alpha\ns_n\atop\beta\ns_1\cdots\,\beta\ns_n} \expect{\alpha\ns_1\cdots\,\alpha\ns_n}{\HR} {\beta\ns_1\cdots\,\beta\ns_n}\,\psi\yd_{\alpha\ns_n}\!\cdots\psi\yd_{\alpha\ns_n}\,\psi\nd_{\beta\ns_n}\!\cdots\psi\ns_{\beta\ns_1}\quad.$

Finally, if the Hamiltonian is noninteracting, consisting solely of one-body operators $$\HH=\sum_{i=1}^N\Hh\ns_i$$, then $\HH=\sum_\alpha \ve\ns_\alpha\,\psi\yd_\alpha\,\psi\nd_\alpha\quad,$ where $$\{\ve\ns_\alpha\}$$ is the spectrum of each single particle Hamiltonian $$\Hh\ns_i$$.

5.9: Appendix I- Second Quantization is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.