5.9: Appendix I- Second Quantization

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\( \newcommand\msa{m\ns_\ssr{A}}\)
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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 \(\Hh\ns_i\).