Proof of the Wave Function Identity, a New Symmetry Constraint for the Three-Electron Quantum Dot in a Magnetic Field and Ad Hoc Generalization to Bound Many-Electron Systems

1. Introduction

Symmetry plays a significant role in physics. A symmetry operation leaves a physical system invariant and facilitates the solution of a quantum-mechanical system as defined by its Hamiltonian. It further constitutes a constraint or ‘sum-rule’ on both the exact and any approximate solution. In this paper, we prove the Wave Function Identity (WFI), a symmetry constraint on the wave function solutions of the Schrödinger [1] and Schrödinger-Pauli [2] equations for the Coulombically interacting $N=3$ electron system bound by an electrostatic field with harmonic potential and in the presence of a uniform magnetic field. The WFI is a recently discovered [3] symmetry constraint on the exact wave function proved for the $N=2$ electron bound systems. Based on the proofs for the $N=2$ and now the $N=3$ electron cases, we propose ad hoc the validity of the WFI for bound $N\ge 2$ electron systems. Further, that the WFI is valid for arbitrary binding potential, electron interaction of the form $\omega \left(\right|\mathbf{r}-{\mathbf{r}}^{\prime }\left|\right)$, and dimensionality. Thus, the WFI is a symmetry requirement of the solutions of the many-electron Schrödinger-Pauli/Schrödinger equations for systems such as the elements of the Periodic Table and multi-electron $2D$ and $3D$ ‘artificial atoms’.

Consider a system of N electrons (of mass m, charge $-e$, and spin moment ${\mu }_s$), with arbitrary electron interaction of the form $\omega \left(\right|\mathbf{r}-{\mathbf{r}}^{\prime }\left|\right)$, in an arbitrary external binding electric field $\mathcal{E}\left(\mathbf{r}\right)$ such that $-e\mathcal{E}\left(\mathbf{r}\right)=-\nabla v\left(\mathbf{r}\right)$, where $v\left(\mathbf{r}\right)$ is a symmetrical scalar potential, and a magnetic field $\mathcal{B}\left(\mathbf{r}\right)=\nabla \times \mathbf{A}\left(\mathbf{r}\right)$ with $\mathbf{A}\left(\mathbf{r}\right)$ the vector potential. In the corresponding Schrödinger-Pauli theory Hamiltonian ${\widehat{H}}_{\mathrm{spin}}$, the interaction of the spin moment ${\mu }_s$ with the magnetic field is explicitly considered. The exact and approximate wave function solutions $\Psi \left(\mathbf{X}\right)$ (with $\mathbf{X}={\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_N$; $\mathbf{x}=\mathbf{r}\zeta$, $\mathbf{r}$ the spatial and $\zeta$ the spin coordinate) of the stationary-state Schrödinger-Pauli equation must satisfy certain physical and mathematical constraints. The Schrödinger equations in the presence and absence of a magnetic field (with spinless Hamiltonian) constitute special cases. The corresponding solutions too must satisfy the same constraints.

The physical and mathematical constraints on the wave function $\Psi \left(\mathbf{X}\right)$ for $N\ge 2$ are the following:

(i)  Wave Function Identity. This symmetry operation is comprised of the interchange of the spatial coordinates of any two electrons whilst keeping their spin coordinates unchanged, followed by the inversion of all electron coordinates. Thus,

$$\Psi ({\mathbf{r}}_1{\zeta }_1,{\mathbf{r}}_2{\zeta }_2,{\mathbf{X}}^{N-2})=\Psi (-{\mathbf{r}}_2{\zeta }_1,-{\mathbf{r}}_1{\zeta }_2,-{\mathbf{X}}^{N-2}),$$

(1)

where ${\mathbf{X}}^{N-2}={\mathbf{r}}_3{\zeta }_3,\dots ,{\mathbf{r}}_N{\zeta }_N$. The application of the permutation operator [4] to the transformed wave function then leads [3] to its parity.

(ii)  Pauli Principle (PP) [5,6,7]. In an interchange of the coordinates of any two electrons (including the spin coordinate), the wave function $\Psi \left(\mathbf{X}\right)$ must be antisymmetric:

$$\Psi ({\mathbf{r}}_1{\zeta }_1,{\mathbf{r}}_2{\zeta }_2,{\mathbf{X}}^{N-2})=-\Psi ({\mathbf{r}}_2{\zeta }_2,{\mathbf{r}}_1{\zeta }_1,{\mathbf{X}}^{N-2}).$$

(2)

(iii)  Parity. The wave function $\Psi \left(\mathbf{X}\right)$ must possess the correct parity.

(iv)  Electron Coalescence Constraints [8,9,10,11,12,13]. The wave function $\Psi \left(\mathbf{X}\right)$ must satisfy the electron-electron and electron-nucleus coalescence constraints. The general form of the wave function $\Psi \left(\mathbf{X}\right)$ (with the spin coordinate suppressed) in D-dimensional space when two charged particles of mass $m_1$ and $m_2$ coalesce is

$$\Psi ({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_N)=\Psi ({\mathbf{r}}_2,{\mathbf{r}}_2,{\mathbf{r}}_3,\dots ,{\mathbf{r}}_N)\left(1+{\displaystyle \frac{2Z_1{Z}_2{\mu }_{12}}{D-1}}s\right)+\mathbf{s}·\mathbf{C}({\mathbf{r}}_2,{\mathbf{r}}_3,\dots ,{\mathbf{r}}_N),$$

(3)

where ${\mu }_{12}$ is the reduced mass, $\mathbf{s}={\mathbf{r}}_1-{\mathbf{r}}_2$ the relative coordinate, and $\mathbf{C}({\mathbf{r}}_2,{\mathbf{r}}_3,\dots ,\mathbf{N})$ an unknown vector. For electron-electron coalescence, $Z_1=-e$, $Z_2=-e$, and ${\mu }_{12}={\displaystyle \frac{1}{2}}m$. Depending on the state, the wave function $\Psi \left(\mathbf{X}\right)$ must satisfy either the node or cusp coalescence constraint. For electron-nucleus coalescence $Z_1=-e$, $Z_2=Z_e$ the nuclear charge, and ${\mu }_{12}\approx m$.

(v)  Parity about each point of Electron-Electron Coalescence [8]. The wave function $\Psi \left(\mathbf{X}\right)$ must possess the correct odd or even parity about each point of electron-electron coalescence.

(vi)  Single-Valued, Smooth, and Quadratically Integrable. The wave function $\Psi \left(\mathbf{X}\right)$ must be single-valued and smooth. It must also be quadratically integrable to be normalizable, and for the determination of properties as expectation values of Hermitian operators.

(vii)  Nodes. Depending on the state of the interacting system, the corresponding wave function $\Psi \left(\mathbf{X}\right)$ must possess the correct number of nodes.

For the 2-dimensional 3-electron harmonically bound system in a uniform magnetic field, it is possible to derive [14] the general form of the exact solution to the Schrödinger-Pauli equation in the Wigner high-electron-correlation regime [15,16,17,18]. This regime is characterized by low energy density, high electron-interaction energy relative to kinetic energy, and a high electron correlation contribution to the kinetic energy.

(Due to advances in semiconductor technology, such an ‘artificial atom’ or quantum dot can be physically created. In a semiconductor quantum dot [19,20,21], the electrons are confined to 2-dimensions in a quantum well within a thin layer of semiconductor such as GaAs sandwiched between two layers of another semiconductor such as AlGaAs. The mass of the electron is the band-effective mass, and its charge is modified by the dielectric constant of the semiconductor. Further, as opposed to natural atoms, the electrons in an‘artificial atom’ are bound by a harmonic potential. There has been recent work on the creation of such quantum dot Wigner crystals [22,23,24,25].)

Employing the exact general form of the wave function for the quartet state [26,27] of a 3-electron quantum dot, we prove the satisfaction of the Pauli Principle, the Wave Function Identity, and consequently prove the parity of a quartet state to be odd. In a quartet state, the spin moments of each electron are the same. Thus, its multiplicity $2S+1=4$ since $S={\displaystyle \frac{3}{2}}$. (A similar proof can be provided for any state, e.g. the ground state for which two electrons have the same spin moment whilst the third has an opposite moment.) The choice of the quartet state in the present work is further governed by the fact that because of constraints on the solutions of the corresponding Schrödinger-Pauli equation as explained in the following section, a closed-form analytical expression for the wave function can be derived. This expression then allows for the easier determination and pictorial representation of the properties of the quartet state wave function.

In Sect II, we provide a brief outline of the derivation of the general solution of the 2-dimensional Schrödinger-Pauli equation in the Wigner regime for the 3-electron harmonically bound system in a uniform magnetic field. Employing the general form of the wave function for a quartet state, we prove in Sect. III the satisfaction of the Pauli Principle, the Wave Function Identity, and that the parity of a quartet state is odd. The details of the proofs are given in Appendices A, B and C, respectively. (The proof of the Pauli Principle is deemed necessary as it is possible to obtain a solution of the Schrödinger-Pauli equation that leads to an energy lower than the ground state but one which does not satisfy the Pauli Principle.) We then derive an exact analytical expression for the quartet state wave function in the Wigner regime, and present plots demonstrating the satisfaction of all symmetry and coalescence constraints. Concluding remarks are made in Sect. IV.

2. Derivation of General Form of Wave Function in Wigner Regime

The 2-dimensional Hamiltonian ${\widehat{H}}_{\mathrm{spin}}$ of the Schrödinger-Pauli equation for 3-electrons, each of mass m, charge $-e$, and spin moment ${\mu }_s$, in a binding electric field $-e\mathcal{E}\left(\mathbf{r}\right)$ with a harmonic scalar potential $v\left(\mathbf{r}\right)={\displaystyle \frac{1}{2}}{k}_0{r}^2={\displaystyle \frac{1}{2}}m{\omega }_0^2{r}^2$, with $k_0,{\omega }_0$ the force constant and frequency, and a magnetic field $\mathcal{B}\left(\mathbf{r}\right)$ with a vector potential $\mathbf{A}\left(\mathbf{r}\right)$, is the sum of the Feynman kinetic $T_F$, electron-interaction $\widehat{W}$, and external potential $\widehat{V}$ operators:

$${\widehat{H}}_{\mathrm{spin}}={\widehat{T}}_F+\widehat{W}+\widehat{V},$$

(4)

$$\begin{array}{ccc}\hfill {\widehat{T}}_F& =& {\displaystyle \frac{1}{2m}}{\displaystyle \sum _{k=1}^3}({\sigma }_k·{\widehat{\mathbf{p}}}_{\mathrm{k},\mathrm{phys}})({\sigma }_k·{\widehat{\mathbf{p}}}_{\mathrm{k},\mathrm{phys}})\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{2m}}{\displaystyle \sum _{k=1}^3}{({\widehat{\mathbf{p}}}_k+{\displaystyle \frac{e}{c}}\mathbf{A}\left({\mathbf{r}}_k\right))}^2+{\displaystyle \frac{g_s{\mu }_B}{2\hslash }}{\displaystyle \sum _{k=1}^3}\mathcal{B}\left({\mathbf{r}}_{\mathbf{k}}\right)·{\mathbf{s}}_k,\hfill \end{array}$$

(6)

$$\widehat{W}={\displaystyle \frac{1}{2}}{\displaystyle \underset{k,\ell =1}{\overset{3}{\sum ^{\prime }}}}{\displaystyle \frac{e^2}{|{\mathbf{r}}_k-{\mathbf{r}}_{\ell }|}},$$

(7)

$$\widehat{V}={\displaystyle \sum _{k=1}^3}v\left({\mathbf{r}}_k\right),$$

(8)

where the electron spin moment ${\mu }_s=-(g_s{\mu }_B/2\hslash )\mathbf{s}$; the spin gyromagnetic ratio $g_s=2$; the Bohr magneton ${\mu }_B=e\hslash /2mc$, with c the velocity of light; the spin angular momentum operator in its matrix representation $\mathbf{s}={\displaystyle \frac{\hslash }{2}}\sigma$, with $\sigma$ the Pauli spin matrix; the physical or kinetic momentum operator ${\widehat{\mathbf{p}}}_{\mathrm{physical}}=\widehat{\mathbf{p}}+{\displaystyle \frac{e}{c}}\mathbf{A}\left(\mathbf{r}\right)$ with the canonical momentum operator $\widehat{\mathbf{p}}=-i\hslash \nabla$.

The Schrödinger-Pauli equation for the 3-electron system is then

$${\widehat{H}}_{\mathrm{spin}}\Psi \left(\mathbf{X}\right)=E\Psi \left(\mathbf{X}\right),$$

(9)

where $\{\Psi (\mathbf{X}),E\}$ are the eigenfunctions and eigenvalues, and $\mathbf{X}={\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3$; $\mathbf{x}=\mathbf{r}\zeta$. In a uniform magnetic field $\mathcal{B}\left(\mathbf{r}\right)=\mathcal{B}{\mathbf{i}}_z$ and symmetric gauge $\mathbf{A}\left(\mathbf{r}\right)={\displaystyle \frac{1}{2}}\mathcal{B}\times \mathbf{r}$, the form of the eigenfunctions is a product of a spatial $\psi \left(\mathcal{R}\right)$ and spin $\chi \left(Z\right)$ component, where $\mathcal{R}=({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3)$ and $Z=({\zeta }_1,{\zeta }_2,{\zeta }_3)$. Thus,

$$\Psi \left(\mathbf{X}\right)=\psi ({\mathbf{r}}_{\mathbf{1}},{\mathbf{r}}_{\mathbf{2}},{\mathbf{r}}_{\mathbf{3}})\chi ({\zeta }_1,{\zeta }_2,{\zeta }_3).$$

(10)

The Hamiltonian ${\widehat{H}}_{\mathrm{spin}}$ of Eq. (4) can be decoupled [14] so as to be written as a sum of three interacting single-particle-pair (or quasi-particle) Hamiltonians. This is possible only in the high-electron-correlation Wigner regime for which the center of mass coordinate $\mathbf{R}$ is set to zero:

$$\mathbf{R}={\displaystyle \frac{1}{3}}{\displaystyle \sum _{k=1}^3}{\mathbf{r}}_k=0.$$

(11)

One makes an orthogonal transformation that leaves the kinetic energy in a uniform magnetic field and the harmonic potential energy invariant. In this orthogonal transformation, the center of mass coordinate is also invariant. Setting $\mathbf{R}=0$ in the transformed Hamiltonian, one obtains the decoupled Hamiltonian (in a.u., $e=\hslash =m=c=1)$ as

$${\widehat{H}}_{\mathrm{spin}}={\displaystyle \frac{1}{2}}{\displaystyle \underset{k,\ell =1}{\overset{3}{\sum ^{\prime }}}}{\widehat{h}}_{k\ell }\left({\mathbf{y}}_{k\ell }\right),$$

(12)

where ${\mathbf{y}}_{k\ell }={\displaystyle \frac{1}{\sqrt{3}}}({\mathbf{r}}_{\ell }-{\mathbf{r}}_k)$, so that $y_{k\ell }=|{\mathbf{y}}_{k\ell }|={\displaystyle \frac{1}{\sqrt{3}}}\left(\right|{\mathbf{r}}_{\ell }-{\mathbf{r}}_k\left|\right)$, and

$$\widehat{h}\left(\mathbf{y}\right)={\displaystyle \frac{1}{2}}{(\widehat{\mathbf{p}}+A\left(\mathbf{y}\right))}^2+\mathcal{B}\left(\mathbf{y}\right)·\mathbf{s}+\widehat{w}\left(\mathbf{y}\right)+v\left(\mathbf{y}\right),$$

(13)

with

$$\widehat{w}\left(\mathbf{y}\right)={\displaystyle \frac{1}{\sqrt{3}\left|\mathbf{y}\right|}};v\left(\mathbf{y}\right)={\displaystyle \frac{1}{2}}{k}_0{y}^2,$$

(14)

and where $\mathbf{y}$ corresponds to ${\mathbf{y}}_{k\ell }$ for any $(k\ell )$th pair: $(k\ell =12,23,31)$.

The corresponding quasi-particle or pair equation for the spatial component ${\Phi }_q\left(\mathbf{y}\right)$ is

$$\widehat{h}\left(\mathbf{y}\right){\Phi }_q\left(\mathbf{y}\right)=ϵ{\Phi }_q\left(\mathbf{y}\right),$$

(15)

where the $\{{\Phi }_q\left(\mathbf{y}\right),ϵ\}$ are the pair-function and eigenvalue of the $(k\ell )$th pair equation, with q representing the quantum numbers of that pair. The pair-function ${\Phi }_q\left(\mathbf{y}\right)$ satisfies the normalization constraint:

$$\int |{\Phi }_q{\left(\mathbf{y}\right)|}^{2}d\mathbf{y}=1.$$

(16)

The general formula of the solution ${\Phi }_q\left(\mathbf{y}\right)$ of the pair-equation Eq. (15) for each $(k\ell )$th pair in polar coordinates $(y,\alpha )$ is

$${\Phi }_q\left(\mathbf{y}\right)={\Phi }_{n,m,p}\left(\mathbf{y}\right)={\displaystyle \frac{e^{im\alpha }}{\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{1}{2}}\sqrt{k_{\mathrm{eff}}}{y}^2}{\varphi }_{m,p}\left(\mathbf{y}\right),$$

(17)

$${\varphi }_{m,p}\left(\mathbf{y}\right)=y^{\left|m\right|}{\varphi }_p\left(\mathbf{y}\right),$$

(18)

$${\varphi }_p\left(\mathbf{y}\right)=1+d_{2}y+d_3{y}^2+\dots +d_p{y}^{p-1},$$

(19)

where $q=n,m,p$ are the quantum numbers; n is the number of nodes of the pair function; m is the angular momentum quantum number $=\pm 1,\pm 2\dots$; p is the number of terms in the polynomial ${\varphi }_p\left(\mathbf{y}\right)$; $d_i$ the coefficients of the polynomial ${\varphi }_p\left(\mathbf{y}\right)$; $\alpha$ is the angle of the relative coordinate vector $\mathbf{s}={\mathbf{r}}_{\ell }-{\mathbf{r}}_k$; $k_{\mathrm{eff}}={\omega }_0^2+{\omega }_L^2$ is the effective force constant; and ${\omega }_L={\displaystyle \frac{B}{2}}$ is the Larmor frequency.

The number of nodes n of a pair-function ${\Phi }_q\left(\mathbf{y}\right)$ depends on the value of $k_{\mathrm{eff}}$ chosen. For the lowest value of $k_{\mathrm{eff}}$, there are zero nodes; for the next higher value of $k_{\mathrm{eff}}$, there is one node, and so on. (The reader is referred to the original literature [14] and to [8] for details of the derivation of the pair-function).

As the interacting 3-electron system is reduced to 3 noninteracting quasi-particle systems, the wave function $\Psi \left(\mathbf{X}\right)$ of Eq. (10) is then the Hartree product of the wave functions of each pair-system:

$$\Psi ({\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3)=\left[{\Phi }_{q_{12}}({\mathbf{r}}_1,{\mathbf{r}}_2){\chi }_{12}({\zeta }_1,{\zeta }_2)\right]\left[{\Phi }_{q_{23}}({\mathbf{r}}_2,{\mathbf{r}}_3){\chi }_{23}({\zeta }_2,{\zeta }_3)\right]\left[{\Phi }_{q_{31}}({\mathbf{r}}_3,{\mathbf{r}}_1){\chi }_{31}({\zeta }_3,{\zeta }_1)\right],$$

(20)

where the ${\chi }_{k\ell }({\zeta }_k,{\zeta }_{\ell })$ are the normalized spin functions of the $(k\ell )$th pair. Further, the effective force constant $k_{\mathrm{eff}}$ is the same for each pair. With the general form of the spatial component ${\Phi }_{q_{k\ell }}({\mathbf{r}}_k,{\mathbf{r}}_{\ell })$ of Eq. (17) and with the spin functions ${\chi }_{k\ell }({\zeta }_k,{\zeta }_{\ell })$ known, the wave function $\Psi ({\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3)$ is the exact solution to the Schrödinger-Pauli equation Eq. (9) for the harmonically bound 3-electron system in the Wigner regime. The total energy E is the sum of the eigenvalues ${ϵ}_{k\ell }$ of each $(k\ell )$th pair-equation:

$$E={ϵ}_{12}+{ϵ}_{23}+{ϵ}_{31}.$$

(21)

A closed-form analytical solution for a given state can be obtained provided the quantum numbers of each pair-function are the same.

The pair-equation Eq. (15) for the 3-electron system is similar to that of the harmonically bound 2-electron system in a magnetic field [8,28,29]. Hence, the individual pair-function components of the 3-electron sysstem, and thus the wave function $\Psi ({\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3)$, can be obtained from the solutions of the 2-electron system by a simple scaling.

3. Symmetry Properties of the Quartet State

This section is comprised of two parts. The first is the analytical proof of the symmetry properties of the Pauli principle, Wave Function Identity (WFI), and odd parity of a quartet state. The details of the proofs are given, respectively, in Appendices A, B, and C employing the general form of the exact 3-electron wave function derived in the previous section.

In the second part, we derive an analytical expression for the wave function of a quartet state in the high-electron-correlation Wigner regime. We then employ this expression to demonstrate pictorially the satisfaction of the symmetry properties of this state. Additionally, these diagrams demonstrate that the quartet state satisfies the node electron-electron coalescence constraint, that the parity about the point of electron coalescence is odd, and that the wave function exhibits a single node.

In Appendix A we employ the general form of the wave function of Eq. (20) for a quartet state to prove the satisfaction of the Pauli principle. In the quartet state, the electrons have the same spin moment. Hence, the spin function components of each pair-function are symmetric in an interchange of any two spin coordinates. It follows that the total spin function is symmetric in the interchange of the spin of any two of the 3 electrons. Therefore, the spatial part of the total wave function must be antisymmetric in an interchange of the spatial coordinates of any two electrons. In the proof we employ the angular momentum of each pair-function to be $m=1$, so that the total angular momentum in the solvable state is $M_L=3m=3$. (Note that in contrast, the total ground state cannot be built up from identical ground-state pair-functions with $m=0$. This would lead to the lowest energy but would not satisfy the Pauli principle.)

Employing the above general form of the quartet state wave function (with $M_L=3$), we prove in Appendix B the satisfaction of the WFI both analytically and by a vector diagram analysis.

Finally, using the WFI symmetry transformed quartet state wave function, we prove in Appendix C its parity to be odd.

To demonstrate pictorially the above symmetry and constraint properties of the quartet state we have derived the spatial component ${\Phi }_q({\mathbf{r}}_k,{\mathbf{r}}_{\ell })$ (with $k\ell =(12,23,31$) of the pair function of the 3-electron wave function $\Psi ({\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3)$ of Eq. (20). The quantum numbers $q=n,m,p$ employed are $n=1,m=1,p=6$. (See Eq. (17) for the definitions of these quantum numbers.) Thus,

$${\Phi }_{1,1,6}({\mathbf{r}}_k,{\mathbf{r}}_{\ell })=N{e}^{i\alpha }{e}^{-{\displaystyle \frac{1}{2}}\sqrt{k_{\mathrm{eff}}}({\displaystyle \frac{1}{\sqrt{3}}}|{\mathbf{r}}_{\ell }-{\mathbf{r}}_k{|)}^2}{\varphi }_{1,6}({\displaystyle \frac{1}{\sqrt{3}}}|{\mathbf{r}}_{\ell }-{\mathbf{r}}_k|),$$

(22)

$$\begin{array}{ccc}\hfill {\varphi }_{1,6}({\displaystyle \frac{1}{\sqrt{3}}}|{\mathbf{r}}_{\ell }-{\mathbf{r}}_k|)& =& \left({\displaystyle \frac{1}{\sqrt{3}}}|{\mathbf{r}}_{\ell }-{\mathbf{r}}_k|\right){\varphi }_6\left({\displaystyle \frac{1}{\sqrt{3}}}|{\mathbf{r}}_{\ell }-{\mathbf{r}}_k|\right),\hfill \\ \hfill {\varphi }_6\left({\displaystyle \frac{1}{\sqrt{3}}}|{\mathbf{r}}_{\ell }-{\mathbf{r}}_k|\right)& =& 1+d_2|{\mathbf{r}}_{\ell }-{\mathbf{r}}_k|+d_3{|{\mathbf{r}}_{\ell }-{\mathbf{r}}_k|}^2\hfill \end{array}$$

(23)

$$\begin{array}{ccc}& +& d_4|{\mathbf{r}}_{\ell }-{\mathbf{r}}_k{|}^3+d_5|{\mathbf{r}}_{\ell }-{\mathbf{r}}_k{|}^4+d_6{|{\mathbf{r}}_{\ell }-{\mathbf{r}}_k|}^5,\hfill \end{array}$$

(24)

where the normalization constant $N=9.83054\times {10}^{-4}$; the angle $\alpha$ is that of the relative coordinate $\mathbf{s}={\mathbf{r}}_{\ell }-{\mathbf{r}}_k$; $\sqrt{k_{\mathrm{eff}}}={\displaystyle \frac{2}{3}}(2/84.0644)=1.58609\times {10}^{-2}$; $d_i=c_i{\left({\displaystyle \frac{2}{3}}\right)}^{i-1}$; $i=2to6$; $c_2={\displaystyle \frac{1}{3}}$; $c_3=2.67971\times {10}^{-2}$; $c_4=-3.28306\times {10}^{-4}$; $c_5=-9.33717\times {10}^{-5}$; $c_6=-2.22143\times {10}^{-6}$.

The spatial part of the quartet state wave function designated as ${\Psi }_Q({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3)$ for the 3-electron quantum dot is comprised of 6 variables $(r_i,{\theta }_i;i=1,2,3)$. To plot this wave function we fix the coordinates of one electron and the angles of the other two electrons. Thus, we present the Real and Imaginary parts of ${\Psi }_Q$ as a function of the radial coordinates of the two electrons.

In Figure 1(a) we plot the Real part of the quartet state wave function $Re{\Psi }_Q(r_1,r_2)$ in (a.u.) for a fixed position of electron $3:r_3=0.5,{\theta }_3={15}^{\circ }$; and fixed angles of electrons 1 and $2:{\theta }_1={30}^{\circ },{\theta }_2={65}^{\circ }$. In Figure 1(b), the coordinates of electrons 1 and 2 are switched so that it is a plot of the $Re{\Psi }_Q(r_2,r_1)$ with ${\theta }_1={65}^{\circ },{\theta }_2={30}^{\circ }$, and for the same fixed position of electron 3. (Note that the coordinate axes of Figure 1(b) have also been switched.) Observe that multiplication of the plot of Figure 1(b) by $(-1)$ reproduces the plot of Figure 1(a). This proves the satisfaction of the Pauli principle for this arbitrary fixed position of one electron and fixed angles of the other two electrons.

Figure 2a,b are the corresponding plots for the Imaginary parts of the quartet state wave function: $Im{\Psi }_Q(r_1,r_2)$. Once again, the Pauli principle is satisfied for this component of the wave function.

Also observe the following in Figure 1(a) and Figure 2(a). (i) The structure of the $Re{\Psi }_Q(r_1,r_2)$ and $Im{\Psi }_Q(r_1,r_2)$ are distinctly different; (ii) Both the $Re{\Psi }_Q(r_1,r_2)$ and $Im{\Psi }_Q(r_1,r_2)$ exhibit a single node; (iii) At the point of electron-electron coalescence at the origin $(r_1=r_2=0)$, both the $Re{\Psi }_Q(r_1,r_2)$ and $Im{\Psi }_Q(r_1,r_2)$ satisfy the node electron-electron coalescence constraint. (There are, of course, an infinite number of such nodes when any two of the three electrons coalesce.)

In Figure 3 and Figure 4, we show the satisfaction of the WFI symmetry property. Figure 3(a) is a plot of the $Re{\Psi }_Q(r_1,r_2)$ for fixed position of electron 3 at $r_3=0.6,{\theta }_3={20}^{\circ }$; and fixed angles of electrons 1 and 2 at ${\theta }_1={10}^{\circ },{\theta }_2={70}^{\circ }$. In Figure 3(b), the WFI symmetry constraint of interchanging the positions of two electrons followed by an inversion of all three electrons is applied to the $Re{\Psi }_Q(r_1,r_2)$. Thus, what is plotted in Figure 3(b) is $Re{\Psi }_Q(-r_2,-r_1)$ with the position of electron 3 at $r_3=0.6,{\theta }_3={200}^{\circ }$; and the angles of electrons 1 and 2 fixed at ${\theta }_1={250}^{\circ },{\theta }_2={190}^{\circ }$. The plots of Figs. 3(a) and 3(b) are the same, thus demonstrating the satisfaction of the WFI by the Real part of the quartet state wave function. Figure 4(a) and 4(b) are plots of the $Im{\Psi }_Q(r_1,r_2)$ demonstrating the satisfaction of the WFI for this component of the quartet state wave function.

Finally, in Figure 5 we demonstrate that the parity of the quartet state is odd. In Figure 4(a) we plot $Re{\Psi }_Q(r_1,r_2)$ for electron 3 fixed at $r_3=15,{\theta }_3=0^{\circ }$; and the angles of electrons 1 and 2 fixed at ${\theta }_1={\theta }_2=0^{\circ }$. Figure 5(b) is a plot of $Re{\Psi }_Q(r_1,r_2)$ with electron 3 fixed at $r_3=15,{\theta }_3={180}^{\circ }$; and the angles of electrons 1 and 2 fixed at ${\theta }_1={\theta }_2={180}^{\circ }$. A comparison of panels (a) and (b) show the parity to be odd.

4. Concluding Remarks

The principal conclusion of the present work is the ad hoc proposal that for N-electron systems with arbitrary symmetric binding potential, arbitrary interaction of electrons of the form $\omega \left(\right|(\mathbf{r}-{\mathbf{r}}^{\prime }|)$, and arbitrary dimensionality, the wave function solutions to the Schrödinger-Pauli/Schrödinger equations satisfy the symmetry property of the Wave Function Identity (WFI). This symmetry operation is comprised of the interchange of the spatial coordinates of any two electrons whilst keeping their spin coordinates unchanged, followed by an inversion of all the electron coordinates. This conclusion is arrived at based on our prior work of the satisfaction of the WFI by the exact wave function for the $N=2$ electron case, and the present proof for the exact solution of the Schrödinger-Pauli equation of the 2-dimensional Coulombically interacting $N=3$ electron ‘artificial’ atom or semiconductor quantum dot in a uniform magnetic field. For the 3-electron quantum dot, it is possible to obtain the general analytical form of the exact solutions in the high-electron-correlation Wigner regime. The satisfaction of the WFI is proved employing the general form of the wave function of a quartet state. The satisfaction of the Pauli principle and the odd parity of the quartet state wave function is also proved analytically. Additionally, we demonstrate these symmetry properties pictorially by deriving a specific closed-form exact analytical expression for the quartet state (with total angular momentum $M_L=3)$ in the Wigner regime. These Figures further exhibit the following properties of a quartet state: it satisfies the node electron-electron coalescence constraint; the parity about each point of electron-electron coalescence is odd; and the state possesses a single node. Finally, we conclude by reiterating that the WFI symmetry requirement for a bound many-electron system is founded on exact analytical correlated wave function solutions of the Schrödinger-Pauli/Schrödinger equations for the 2- and 3-electron systems.