Quantum mechanical states
Let us start by discussing quantum mechanical states, by taking the introduction by Dickhoff and Van Neck (Dickhoff and Van Neck 2008). It is an axiom of quantum mechanics that every single-particle quantum mechanical state \(\require{physics} \ket{\psi}\) is an element of an abstract one-particle Hilbert space10 \(\mathscr{H}_1\): \[\begin{equation} \ket{\psi} \in \mathscr{H}_1 % \thinspace . \end{equation}\] If we have a basis \(\set{\ket{\mu}, \ket{\nu}, \ldots}\) for this 1-particle Hilbert space that is orthonormal \[\begin{equation} \braket{\mu}{\nu} = \delta_{\mu \nu} \end{equation}\] and complete \[\begin{equation} \sum_\mu \dyad{\mu} = 1 % \thinspace , \end{equation}\] we can write every single-particle state \(\ket{\psi}\) as a linear expansion in this basis: \[\begin{align} \ket{\psi} % &= \sum_\mu \ket{\mu} \braket{\mu}{\psi} \\ % &= \sum_\mu \psi_\mu \ket{\mu} \thinspace , \end{align}\] in which \(\psi_\mu\) is the coefficient of \(\ket{\psi}\) for the basis vector \(\ket{\mu}\).
One particular basis which will be extremely important to the development of this chapter in first quantization is the so-called position basis \(\set{\ket{\vb{r}}}\), which is composed of the (continuous) set of eigenfunctions of the position operator \(\hat{\vb{r}}\): \[\begin{equation} \hat{\vb{r}} \ket{\vb{r}} = \vb{r} \ket{\vb{r}} % \thinspace , \end{equation}\] which is orthonormal \[\begin{equation} \braket{\vb{r}'}{\vb{r}} = \delta(\vb{r} - \vb{r}') \end{equation}\] and complete \[\begin{equation} \int \dd{\vb{r}} \dyad{\vb{r}} = 1 \end{equation}\] because the position operator \(\hat{\vb{r}}\) is Hermitian. (Acke 2017) In this position basis, we can expand the single-particle wave vector \(\ket{\psi}\) as: \[\begin{align} \ket{\psi} % &= \int \dd{\vb{r}} \ket{\vb{r}} \braket{\vb{r}}{\psi} \\ % &= \int \dd{\vb{r}} \psi(\vb{r}) \ket{\vb{r}} % \thinspace , \end{align}\] in which \(\psi(\vb{r})\) is called the wave function of the one-particle system. The meaning of a wave function is thus that it is the `expansion coefficient’ of the abstract state vector \(\ket{\psi}\) in the position basis \(\set{\ket{\vb{r}}}\): \[\begin{equation} \psi(\vb{r}) = \braket{\vb{r}}{\psi} \thinspace . \end{equation}\]
Up until now, we have only discussed one particle. For \(N\) particles, we start by taking \(N\) copies of the \(1\)-particle Hilbert space \(\mathscr{H}_1\): \[\begin{equation} \mathscr{H}_N = \bigotimes_{i=1}^N \mathscr{H}_1 \end{equation}\] and subsequently introduce an \(N\)-particle basis vector as \[\begin{equation} \label{eq:N-particle_basis_vector} |\mu_1 \cdots \mu_N) = \ket{\mu_1} \cdots \ket{\mu_N} \thinspace , \end{equation}\] in which the product on the right-hand side of the equation is to be seen as a tensor product. The basis composed of these \(N\)-particle basis vectors is orthonormal: \[\begin{align} (\mu_1 \cdots \mu_N | \nu_1 \cdots \nu_N) &= \braket{\mu_1}{\nu_1} \cdots \braket{\mu_N}{\nu_N} \\ &= \delta_{\mu_1 \nu_1} \cdots \delta_{\mu_N \nu_N} \end{align}\] and complete: \[\begin{equation} \sum_{\mu_1 \cdots \mu_N} |\mu_1 \cdots \mu_N) (\mu_1 \cdots \mu_N| = 1 \thinspace . \end{equation}\]
However, for electrons (which are fermions), the basis with elements defined in equation \(\eqref{eq:N-particle_basis_vector}\) does not have the correct symmetry: it is not antisymmetric by exchange of two electron labels. We therefore introduce the antisymmetrized \(N\)-particle basis vectors as \[\begin{equation} \label{eq:antisymmetrized_N-partcile_basis_vector} \ket{\mu_1 \cdots \mu_N} = % \frac{1}{\sqrt{N!}} % \sum_{P \in S_N} (-1)^P |\mu_{P_1} \cdots \mu_{P_N}) % \thinspace . \end{equation}\] The basis that is composed of these elements is orthonormal: \[\begin{equation} \braket{\mu_1 \cdots \mu_N}{\nu_1 \cdots \nu_N} % = \delta_{\mu_1 \nu_1} \cdots \delta_{\mu_N \nu_N} \end{equation}\] and complete: \[\begin{equation} \label{eq:antisymmetrized_completeness} \sum_{\mu_1 \cdots \mu_N} \dyad{\mu_1 \cdots \mu_N} = 1 % \thinspace . \end{equation}\]
What is then an \(N\)-particle wave function \(\Psi(\vb{r}_1, \ldots, \vb{r}_N)\)? We state that it is the projection of the \(N\)-particle abstract state vector \(\ket{\Psi}\) onto a (non-antisymmetrized) \(N\)-particle position basis vector (Acke 2017): \[\begin{equation} \Psi(\vb{r}_1, \ldots, \vb{r}_N) % = (\vb{r}_1 \cdots \vb{r}_N |\Psi \rangle % \thinspace . \end{equation}\] In quantum chemistry, we introduce a single-particle basis \(\set{ \ket{\phi_p} }\), in which the individual single-particle basis functions are called orbitals, such that after invoking the antisymmetrized completeness relation \(\eqref{eq:antisymmetrized_completeness}\) and plugging in \(\eqref{eq:antisymmetrized_N-partcile_basis_vector}\), the wave function \(\Psi(\vb{r}_1, \ldots, \vb{r}_N)\) can be written as \[\begin{equation} \Psi(\vb{r}_1, \ldots, \vb{r}_N) % = % \frac{1}{\sqrt{N!}} % \sum_{\phi_1 \cdots \phi_N} % \Psi_{\phi_1 \cdots \phi_N} % \qty[ % \frac{1}{\sqrt{N!}} % \sum_{P \in S_N} (-1)^P % (\vb{r}_1 \cdots \vb{r}_N % | \phi_{P_1} \cdots \phi_{P_N}) % ] % \thinspace . \end{equation}\] In this expression, we now recognize the Slater determinant \[\begin{align} \Phi(\vb{r}_1, \ldots, \vb{r}_N) % &= % \frac{1}{\sqrt{N!}} % \sum_{\phi_1 \cdots \phi_N} % \Psi_{\phi_1 \cdots \phi_N} % \qty[ % \frac{1}{\sqrt{N!}} % \sum_{P \in S_N} (-1)^P % (\vb{r}_1 \cdots \vb{r}_N % | \phi_{P_1} \cdots \phi_{P_N}) % ] \\ &= % \frac{1}{\sqrt{N!}} % \begin{vmatrix} % \phi_1(\vb{r}_1) & \cdots & \phi_N(\vb{r}_N) \\ \vdots & \ddots & \vdots \\ \phi_1(\vb{r}_1) & \cdots & \phi_N(\vb{r}_N) % \end{vmatrix} \end{align}\] such that the wave function \(\Psi(\vb{r}_1, \ldots, \vb{r}_N)\) is then written as a linear combination of Slater determinants: \[\begin{equation} \Psi(\vb{r}_1, \ldots, \vb{r}_N) % = \sum_{\phi_1 \cdots \phi_N} % \Psi_{\phi_1 \cdots \phi_N} % \thinspace % \Phi_{\phi_1 \cdots \phi_N}(\vb{r}_1, \ldots, \vb{r}_N) % \thinspace , \end{equation}\] where \(\Psi_{\phi_1 \cdots \phi_N}\) finally represents the expansion coefficient inside the antisymmetrized \(N\)-particle basis.
We should note that this treatment of quantum mechanical states is valid for any form of the wave functions, i.e. whether they have one, two, or four components. One-component wave functions will subsequently be called scalar wave functions: \[\begin{equation} \psi(\vb{r}) : \mathbb{R}^3 \rightarrow \mathbb{C} \thinspace . \end{equation}\] Two-component wave functions \[\begin{equation} \psi(\vb{r}) : \mathbb{R}^3 \rightarrow \mathbb{C}^2 \end{equation}\] will then be denoted by spinors and full four-component wave functions \[\begin{equation} \psi(\vb{r}) : \mathbb{R}^3 \rightarrow \mathbb{C}^4 \end{equation}\] will be referred to as bispinors.
References
A mathematical Hilbert space is an inner product space that is furthermore complete in the sense that every Cauchy sequence converges with respect to the norm defined by the inner product.↩︎