Theoretical Proof of and Proposed Experimental Search for the Ground Triplet State of a Wigner-Regime Two-Electron ‘Artificial Atom’ in a Magnetic Field

Abstract

It is experimentally established that there is no ground triplet state of the natural $He$ atom. There is also no exact analytical solution to the Schrödinger equation corresponding to this state. For a two-dimensional two-electron ‘artificial atom’ or a semiconductor quantum dot in a magnetic field, as described by the Schrödinger–Pauli equation, we provide theoretical proof of the existence of a ground triplet state by deriving an exact analytical correlated wave function solution to the equation. The state exists in the Wigner high-electron-correlation regime. We further explain that the solution satisfies all requisite symmetry and electron coalescence constraints of a triplet state. Since, due to technological advances, such a Wigner crystal quantum dot can be created, we propose an experimental search for the theoretically predicted ground triplet-state spectral line. We note that there exists an analytical solution to the Schrödinger–Pauli equation for a ground singlet state in the Wigner regime for the same value of the magnetic field. The significance to quantum mechanics of the probable experimental observation of the ground triplet state for an ‘artificial atom’ is discussed.

1. Introduction of and Rationale for Proposed Experiment

It is an experimental fact that there does not exist a ground triplet state of the natural He atom. The absence of the triplet state associated with the singlet ground state in the He atom spectrum led Pauli to the discovery of the Pauli exclusion principle (PEP) [1]. According to the PEP, no two electrons in a many-electron system can occupy the same quantum state. The PEP is a description of the individual electron, with the quantum numbers specifying the occupation of one-particle states. With the advent of the papers by Schrödinger [2] the following year, Dirac [3] and Heisenberg [4], to account for the indistinguishability of electrons, postulated that the many-interacting-electron wave function solutions to the Schrödinger equation must be antisymmetric in an interchange of the coordinates of any two electrons (including the spin coordinate)—the eponymous Pauli principle (PP). Orbital-based theories obeying the PEP confirm that such a ground triplet state for the He atom cannot exist. There is also no analytical or numerical correlated wave function solution to the Schrödinger equation for this state.

In this paper, we present rigorous theoretical proof of the existence of a ground triplet state of a two-dimensional two-electron ‘artificial atom’ or semiconductor quantum dot [5,6,7] in a magnetic field. We derive an exact analytical correlated wave function for this state. The quantum dot additionally possesses the property of being in the low-electron-density high-electron-correlation Wigner regime [8,9]. Based on our understanding of the natural $He$ atom, one would expect that there exists solely a ground singlet state for any two-electron system. However, that is theoretically not the case. We therefore propose the creation of a Wigner-regime two-electron quantum dot and, subsequently, a search for the spectral line of the ground triplet state. A more detailed rationale, experimental and theoretical, for the proposed experiment follows.

(A) As a result of the development of semiconductor technology, it has been possible to create ‘artificial atoms’ or quantum dots [5,6,7] that possess properties similar to those of natural atoms. The motion of the electrons is confined to two dimensions within a quantum well in a thin layer of a semiconductor, such as GaAs sandwiched between two layers of another AlGaAs semiconductor. The motion can be further restricted by electric and magnetic fields. The charge of the electron is modified by the dielectric constant of the semiconductor, and its mass is the band effective mass. For GaAs, the dielectric constant $ϵ=12.4$, and the band effective mass $m^{🟉}=0.067m$, where m is the free electron mass. But the most significant difference between artificial and natural atoms is that the binding potential of the electrons in the former is harmonic and not Coulombic. This has been confirmed both theoretically [10] as well as by an experiment [5,6,7]. As a consequence, the parameters of the ‘artificial atom’ differ from those of a natural atom, e.g., the size is about an order of magnitude greater. As the proposed experiment involves a quantum dot in the Wigner regime, we note that there exists work on the creation of magnetically induced Wigner crystals [11,12,13,14]. Hence, such a Wigner-regime two-electron quantum dot can be physically created. When this is achieved, we suggest employing the single-electron-capacitance spectroscopy method of Ashoori et al. [5] for the experimental search for the ground triplet-state spectral line.

(B) The equation describing the quantum dot in a magnetic field is the Schrödinger–Pauli [15] equation, for which the Hamiltonian explicitly accounts for the electron spin moment and its interaction with the external magnetic field. Therefore, in addition to the paramagnetic and diamagnetic components, there is a magnetization (spin) component to the current density. There has been considerable theoretical work [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] on the two-electron quantum dot in a magnetic field for both ground and excited states, principally in the high-electron-density low-electron-correlation regime. The present work shows that there exists an exact analytical solution for a ground triplet state in the Wigner low-electron-density regime [8,9] characterized by an electron interaction energy much greater than the kinetic energy. Further, this expression is valid for an arbitrary value of the magnetic field, weak or strong. Thus, the ground triplet state is theoretically predicted by an exact solution to the Schrödinger–Pauli equation. We note that in prior work [31], we derived the analytical solution for the ground singlet state in the Wigner regime together with many of its properties, including the energy. This provides additional information for experimentalists. Hence, from a theoretical perspective, for the two-electron Wigner-regime quantum dot in a magnetic field, both a ground triplet and singlet state exist simultaneously.

To put this work in context, there are two other relevant points of interest:

(a) The spin moments of any bound two-electron system can be polarized to create a triplet state by applying a high magnetic field. There has been earlier theoretical work [6,7,32,33,34] on such spin polarization in the case of the two-electron quantum dot. In such work, the solutions of the interacting Schrödinger–Pauli equation are treated, for example, as finite configuration-interaction functions of single-particle orbitals or via perturbation theory. The energy is studied as a function of the magnetic field. It is observed that there is a flip of the ground singlet state to a triplet state at a specific value of the magnetic field: spin polarization occurs due to a high magnetic field, known as the Zeeman effect. These calculations are numerical, and not exact in a rigorous mathematical sense. We refer the reader to the review articles [6,7] and other similar work for a summary of this.

The present work, however, is distinct in the following ways: (i) The ground triplet state obtained in our work is an exact, closed-form analytical and correlated wave function solution of the Schrödinger–Pauli equation; (ii) we show that the solution exists solely in the high-electron-correlation Wigner regime; (iii) being an exact solution, it satisfies all requisite symmetry properties and electron coalescence constraints of a triplet-state wave function exactly; (iv) as noted above, in [31], we derived an analytical correlated wave function for the ground singlet state in the Wigner regime. It can be shown analytically that it is possible for both the ground triplet and singlet states to exist in the same magnetic field. In fact it can be shown that the singlet state can exist for a magnetic field that is greater than that of the triplet state. This proves that the expression derived for the ground triplet state does not represent a high-magnetic-field spin-polarized state.

(b) Another two-electron system is the three-dimensional Hooke’s atom [35,36,37], in which the electrons are also bound by a harmonic potential. In this case, there is no magnetic field present. For this system, we also derived an exact closed-form analytical correlated wave function solution for the ground triplet state of the corresponding Schrödinger equation. This solution also exists in the high-electron correlation Wigner regime. Thus, there exists a ground triplet state for such an atom even in the absence of a magnetic field. We further note that it is also possible to derive [38,39], for Hooke’s atom, an exact analytical correlated wave function for the ground singlet state in the Wigner regime of this atom. The existence of such a ground triplet state of a 3D ‘artificial atom’ contrasts with that of the natural He atom.

In Section 2, we present the general analytical form of a Wigner regime ground state wave function solution to the two-dimensional Schrödinger–Pauli equation for a two-electron quantum dot in a magnetic field in which the two electrons have the same spin moment. For a particular choice of solution, in Section 3, we explain that this wave function satisfies all established and recently derived [40] requisite symmetry properties and coalescence constraints of a ground triplet state. These properties are the following: the PP of antisymmetry; the Wave Function Identity; odd parity; odd parity about each point of electron–electron coalescence; the node electron–electron coalescence constraint [41,42,43,44]; and the zero-node structure. To prove the simultaneous existence of a Wigner-regime ground singlet state at the same value of the magnetic field, we provide in Appendix A an example of such an exact analytical solution. In our concluding remarks of Section 4, we discuss the significance to quantum mechanics of the probable experimental observation of the triplet-state spectral line.

2. Ground-State Triplet Wave Function of the Two-Electron Quantum Dot in a Magnetic Field: General Form

In this section, we present the expression derived by us of the general form of the exact solution to the ground state of a two-electron quantum dot in a magnetic field in a triplet state. (For the procedure for obtaining a general solution to the corresponding Schrödinger–Pauli equation, we refer the reader to [16,17,18,19].) Consider a two-electron semiconductor (of dielectric constant $ϵ$) quantum dot in a magnetic field ($\mathcal{B}\left(\mathbf{r}\right)=\mathbf{\nabla }\times \mathbf{A}\left(\mathbf{r}\right)$, with $\mathbf{A}\left(\mathbf{r}\right)$ representing the vector potential). The electrons, of charge $-e$ and spin angular momentum vector $\mathbf{s}$, are bound by a harmonic field $\mathcal{E}\left(\mathbf{r}\right)$ such that $(-e)\mathcal{E}\left(\mathbf{r}\right)=-\mathbf{\nabla }v\left(\mathbf{r}\right)$, where the scalar potential $v\left(\mathbf{r}\right)={\displaystyle \frac{1}{2}}{k}_0{r}^2={\displaystyle \frac{1}{2}}{m}^{🟉}{w}_0^2{r}^2$, with ($k_0,w_0$) representing the harmonic binding force constant and frequency, and $m^{🟉}$ the band effective mass. The corresponding Schrödinger–Pauli equation is

(1)$\begin{array}{ccc}\hfill [{\displaystyle \frac{1}{2m^{🟉}}}\sum _{k=1}^2({\widehat{\mathbf{p}}}_k& +& {\displaystyle \frac{e}{c}}\mathbf{A}\left({\mathbf{r}}_k\right){)}^2+{\displaystyle \frac{g^{🟉}{\mu }_B^{🟉}}{2\hslash }}\sum _{k=1}^2\mathcal{B}\left({\mathbf{r}}_k\right)·{\mathbf{s}}_k+{\displaystyle \frac{{\left(e^{🟉}\right)}^2}{|{\mathbf{r}}_1-{\mathbf{r}}_2|}}\hfill \\ & +& {\displaystyle \frac{1}{2}}{m}^{🟉}{\omega }_0^2\sum _{k=1}^2{r}_k^2]\mathsf{\Psi }({\mathbf{x}}_1,{\mathbf{x}}_2)=E\mathsf{\Psi }({\mathbf{x}}_1,{\mathbf{x}}_2),\hfill \end{array}$

where $\widehat{\mathbf{p}}=-i\hslash \mathbf{\nabla }$ represents the canonical momentum operator, and the effective (starred) properties are as follows: $g^{🟉}$ is the gyromagnetic ratio, ${\mu }_B^{🟉}=e^{🟉}\hslash /2m^{🟉}c$ is the Bohr magneton, and ${\left(e^{🟉}\right)}^2=e^2/ϵ$ is the screened charge. There exist exact closed-form analytical solutions to this differential equation for an infinite set of values of an effective force constant. Because the solutions are exact, they inherently account for the electron correlations due to both the Pauli principle and Coulomb repulsion. The solutions to the equation are of the form

(2)$\mathsf{\Psi }({\mathbf{x}}_1,{\mathbf{x}}_2)=\psi ({\mathbf{r}}_1,{\mathbf{r}}_2)\chi ({\zeta }_1,{\zeta }_2),$

with $\psi ({\mathbf{r}}_1,{\mathbf{r}}_2)$ representing the correlated spatial and $\chi ({\zeta }_1,{\zeta }_2)$ the spin components, and $\mathbf{x}=\mathbf{r}\zeta ,\mathbf{r}$ and $\zeta$ being the spatial and spin coordinates. The reason for the existence of such correlated analytical solutions is that the Schrödinger–Pauli differential equation is separable into a center of mass and a relative coordinate component. The center-of-mass differential equation is the harmonic oscillator equation. The differential equation for the relative coordinate, which explicitly accounts for the Coulomb electron–electron interaction, can be solved in closed analytical form.

We consider the two electrons to have the same spin moment. In the symmetric gauge $\mathbf{A}\left(\mathbf{r}\right)={\displaystyle \frac{1}{2}}\mathcal{B}\left(\mathbf{r}\right)\times \mathbf{r}$ with $\mathcal{B}\left(\mathbf{r}\right)=\mathcal{B}{\mathbf{i}}_z$, there exist an infinite number of closed-form analytical solutions of the Schrödinger–Pauli equation with zero and a finite number of nodes [16,17]. The number of nodes is associated with the solution to the relative coordinate differential equation. The angular momentum quantum number, $m=0,\pm 1,\pm 2$, … Each solution corresponds to a number p of terms in a polynomial. For each value of $|m|$ and $p\ge 4$ terms, there are at least two solutions with different numbers of nodes, different values of an effective force constant $k_{\mathrm{eff}}$, and different energies. The higher the value of p, the more solutions are obtained. For example, for $|m|=1$ and $p=10$ terms, there are five solutions with a variety numbers of nodes, $k_{\mathrm{eff}}$ values, and energies. In the present work, we are concerned solely with the zero-node solutions to the relative coordinate differential equation, which, for each value of m, lead to the lowest value of $k_{\mathrm{eff}}$ and to the lowest value of the energy corresponding to a ground state. Thus, for these exact correlated solutions, a state is defined by the number of nodes n, the angular momentum quantum number m, and the value p of terms in a polynomial involving the relative coordinate.

In effective atomic units ${\left(e^{🟉}\right)}^2=\hslash =m^{🟉}=c=1$, the general form of the spatial component function $\psi ({\mathbf{r}}_1,{\mathbf{r}}_2)$ of the zero-node solutions are of the analytical form

(3)${\psi }_{n=0,m,p}({\mathbf{r}}_1,{\mathbf{r}}_2)=N{e}^{im\alpha }{e}^{-{\displaystyle \frac{1}{2}}\sqrt{k_{\mathrm{eff},\mathrm{lowest}}}(r_1^2+r_2^2)}{\mathsf{\Phi }}_{m,p}(|{\mathbf{r}}_2-{\mathbf{r}}_1|),$

(4)${\mathsf{\Phi }}_{m,p}(|{\mathbf{r}}_2-{\mathbf{r}}_1|)=(|{\mathbf{r}}_2-{\mathbf{r}}_1{|)}^{|m|}[1+c_1(|{\mathbf{r}}_2-{\mathbf{r}}_1|)+c_2(|{\mathbf{r}}_2-{\mathbf{r}}_1{|)}^2+\dots +c_{p-1}(|{\mathbf{r}}_2-{\mathbf{r}}_1{|)}^{p-1}],$

where the subscript n corresponds to the number of nodes of the solution to the relative coordinate differential equation, which, in this case, is zero; N is the normalization constant; $m=\pm 1,\pm 2,\dots$ is the angular momentum quantum number; $\alpha$ is the angle of the relative coordinate vector $\mathbf{r}={\mathbf{r}}_2-{\mathbf{r}}_1$; $|\mathbf{r}|=r$; $k_{\mathrm{eff},\mathrm{lowest}}={\omega }_0^2+{\omega }_L^2$ is the lowest value of the effective force constant, with ${\omega }_L=\mathcal{B}/2$ representing the Larmor frequency; ${\mathsf{\Phi }}_{m,p}(|{\mathbf{r}}_2-{\mathbf{r}}_1|)$ is a finite polynomial, with p representing the number of terms; and $c_1,c_2,\dots ,c_{p-1}$ represents the coefficients. Such solutions exist for all $p\ge 2$. For the component of the wave function which arises from the center-of-mass differential equation, we employ the solution corresponding to the ground state of the harmonic oscillator. Finally, the corresponding spin functions $\chi ({\zeta }_1,{\zeta }_2)$ are always such that ${\zeta }_1={\zeta }_2$, i.e., the spin moments of the electrons are the same. The corresponding solutions to Equation (1) for zero nodes are designated as ${\mathsf{\Psi }}_{n=0,m,p}({\mathbf{x}}_1,{\mathbf{x}}_2)={\psi }_{n=0,m,p}({\mathbf{r}}_1,{\mathbf{r}}_2)\chi ({\zeta }_1={\zeta }_2)$. The Pauli principle antisymmetry of the solutions ${\mathsf{\Psi }}_{n=0,m,p}({\mathbf{x}}_1,{\mathbf{x}}_2)$ is due to the phase factor $e^{im\alpha }$, and this is explained below.

3. Particular Wigner-Regime Ground-State Triplet Wave Function

In this section, we consider a particular ground-state triplet wave function of the form of Equations (2)–(4), and note its well-behavedness and exact satisfaction of all requisite ground triplet-state properties. Such an analysis confirms that the expression derived does in fact correspond to a ground triplet-state wave function. The specific solution considered is characterized by the finite polynomial ${\mathsf{\Phi }}_{m,p}\left(\right(|{\mathbf{r}}_2-{\mathbf{r}}_1|)$, where $p=4$; the angular momentum quantum number $m=1$, which corresponds to electrons of the same spin moment; $\sqrt{k_{\mathrm{eff},\mathrm{lowest}}}=2.7564\times {10}^{-2}$, for which the normalization constant $N=8.0318\times {10}^{-4}$; the coefficients $c_1={\displaystyle \frac{1}{3}}$; $c_2=3.1330\times {10}^{-2}$; $c_3=8.6359\times {10}^{-4}$; and the phase factor is $e^{i\alpha }$.

(i) Figure 1 is a plot of the Real and Imaginary parts of ${\psi }_{0,1,4}({\mathbf{r}}_1,{\mathbf{r}}_2)$ as a function of the electron coordinates ${\mathbf{r}}_1(r_1,{\theta }_1)$ and ${\mathbf{r}}_2(r_2,{\theta }_2)$. It is evident that the function ${\psi }_{0,1,4}({\mathbf{r}}_1,{\mathbf{r}}_2)$ is single-valued, smooth, and bounded (the function ${\psi }_{0,1,4}({\mathbf{r}}_1,{\mathbf{r}}_2)$ is also plotted for the negative values $(-{\mathbf{r}}_1,-{\mathbf{r}}_2)$ to demonstrate its parity (see (iv) below)).

(ii) The function ${\mathsf{\Psi }}_{0,1,4}({\mathbf{x}}_1,{\mathbf{x}}_2)$ satisfies the Pauli principle (PP): $\mathsf{\Psi }[e_1\left({\mathbf{r}}_1{\zeta }_1\right),e_2\left({\mathbf{r}}_2{\zeta }_2\right)]=-\mathsf{\Psi }[e_1\left({\mathbf{r}}_2{\zeta }_2\right),e_2\left({\mathbf{r}}_1{\zeta }_1\right)]$. Since ${\zeta }_1={\zeta }_2$, the spin component $\chi ({\zeta }_1,{\zeta }_2)$ is symmetric in an interchange of the coordinates ${\zeta }_1,{\zeta }_2$. The spatial component ${\psi }_{0,1,4}({\mathbf{r}}_1,{\mathbf{r}}_2)$ is antisymmetric in an interchange of ${\mathbf{r}}_1,{\mathbf{r}}_2$, i.e., $\psi ({\mathbf{r}}_1,{\mathbf{r}}_2)=-\psi ({\mathbf{r}}_2,{\mathbf{r}}_1)$. This is due to the phase factor $e^{i\alpha }$. The magnitude of the relative vector $\mathbf{r}$ does not change, so the polynomial ${\mathsf{\Phi }}_{1,4}\left(r\right)$ remains unchanged, but its angle $\alpha$ (angle of vector $\mathbf{r}$ that points from the tip of ${\mathbf{r}}_1$ to the tip of ${\mathbf{r}}_2)$ changes to $\alpha +\pi$ when the electrons interchange their positions, thus changing the sign of the phase factor.

(iii) The function ${\mathsf{\Psi }}_{0,1,4}({\mathbf{x}}_1,{\mathbf{x}}_2)$ satisfies the Wave Function Identity: $\mathsf{\Psi }[e_1\left({\mathbf{r}}_1{\zeta }_1\right),e_2$$\left({\mathbf{r}}_2{\zeta }_2\right)] = \mathsf{\Psi }[e_1(-{\mathbf{r}}_2{\zeta }_1),e_2(-{\mathbf{r}}_1{\zeta }_2)]$. This symmetry operation involves an interchange of the spatial coordinates of the electrons whilst keeping their spin moments unchanged, followed by an inversion. When ${\mathbf{r}}_1$ is changed to $-{\mathbf{r}}_2$, and ${\mathbf{r}}_2$ is changed to $-{\mathbf{r}}_1$, the center-of-mass coordinate $\mathbf{R}=({\mathbf{r}}_1+{\mathbf{r}}_2)/2$ becomes $-\mathbf{R}$, but its magnitude is unchanged, as is the magnitude of the relative coordinate $\mathbf{r}$. It is $|\mathbf{R}|$ that contributes to the function $\mathsf{\Psi }({\mathbf{x}}_1,{\mathbf{x}}_2)$ (note that ${\displaystyle \frac{1}{2}}(r_1^2+r_2^2)$ in the exponent in Equation (3) is equivalent to $R^2+{\displaystyle \frac{1}{4}}{r}^2$). (In [16,40], the discovery of a new symmetry property of two-electron systems referred to as the Wave Function Identity is described. This property then leads to the understanding that the parity of all singlet states is even, and that of all triplet states is odd. This identity also shows that the parity of singlet states about each point of electron–electron coalescence is even, and that of triplet states is odd).

(iv) The parity of the function ${\psi }_{0,1,4}({\mathbf{r}}_1,{\mathbf{r}}_2)$ is odd for electrons of parallel spin, as must be the case, i.e., $\mathsf{\Psi }({\mathbf{r}}_1{\zeta }_1,{\mathbf{r}}_2{\zeta }_2)=-\mathsf{\Psi }(-{\mathbf{r}}_1{\zeta }_1,-{\mathbf{r}}_2{\zeta }_2)$ (this follows from the Wave Function Identity). This may also be seen from the phase factor $e^{i\alpha }$. When ${\mathbf{r}}_1,{\mathbf{r}}_2$ are inverted to $-{\mathbf{r}}_1,-{\mathbf{r}}_2$, the magnitude of the relative vector $\mathbf{r}$ remains unchanged so that the polynomial ${\mathsf{\Phi }}_{1,4}\left(\mathbf{r}\right)$ is unchanged, but the relative vector angle $\alpha$ changes to $\alpha +\pi$, thus changing the sign of the function ${\psi }_{0,1,4}({\mathbf{r}}_1,{\mathbf{r}}_2)$ (See Figure 1).

(v) The function ${\psi }_{0,1,4}({\mathbf{r}}_1,{\mathbf{r}}_2)$ satisfies the node electron–electron coalescence condition. The integral form of the electron–electron and electron–nucleus coalescence constraints in dimensions $D\ge 2$ is derived in [16,41,42,43,44]. For $D=2$, the electron–electron coalescence constraint is

(5)$\psi ({\mathbf{r}}_1,{\mathbf{r}}_2)=\psi ({\mathbf{r}}_2,{\mathbf{r}}_2)(1 + |{\mathbf{r}}_2-{\mathbf{r}}_1|)+({\mathbf{r}}_2-{\mathbf{r}}_1)·\mathbf{C}\left({\mathbf{r}}_2\right),$

where $\mathbf{C}\left({\mathbf{r}}_2\right)$ is an unknown vector. As the spins of the electrons are the same, the probability of two electrons being at the same physical position is zero, and thus, in Equation (5), ${\psi }_{0,1,4}({\mathbf{r}}_2,{\mathbf{r}}_2)=0$, and the spatial component function ${\psi }_{0,1,4}({\mathbf{r}}_1,{\mathbf{r}}_2)$ vanishes.

(vi) The parity of ${\psi }_{0,1,4}({\mathbf{r}}_1,{\mathbf{r}}_2)$ about each point of electron–electron coalescence is odd because this is a triplet state.

(vii) The function ${\mathsf{\Psi }}_{0,1,4}({\mathbf{x}}_1,{\mathbf{x}}_2)$ is square integrable, as all properties of the system are determinable.

In Figure 2a, we plot the density $\rho \left(\mathbf{r}\right)=<{\mathsf{\Psi }}_{0,1,4}({\mathbf{x}}_1,{\mathbf{x}}_2)|\widehat{\rho }\left(\mathbf{r}\right)|{\mathsf{\Psi }}_{0,1,4}({\mathbf{x}}_1,{\mathbf{x}}_2)>$, where $\widehat{\rho }\left(\mathbf{r}\right)={\displaystyle {\sum }_{k=1}^2}\delta ({\mathbf{r}}_k-\mathbf{r})$. Figure 2b is a plot of the radial probability density $r\rho \left(r\right)$. Figure 3a is a plot of the physical current density $\mathbf{j}\left(\mathbf{r}\right)=<{\mathsf{\Psi }}_{0,1,4}({\mathbf{x}}_1,{\mathbf{x}}_2)|\widehat{\mathbf{j}}\left(\mathbf{r}\right)|{\mathsf{\Psi }}_{0,1,4}({\mathbf{x}}_1,{\mathbf{x}}_2)>$, where the operator $\widehat{\mathbf{j}}\left(\mathbf{r}\right)$ is a sum of its paramagnetic ${\widehat{\mathbf{j}}}_p\left(\mathbf{r}\right)$, diamagnetic ${\widehat{\mathbf{j}}}_d\left(\mathbf{r}\right)$, and magnetization ${\widehat{\mathbf{j}}}_m\left(\mathbf{r}\right)$ components, where ${\widehat{\mathbf{j}}}_p\left(\mathbf{r}\right)={\displaystyle \frac{1}{2}}{\displaystyle {\sum }_{k=1}^2}[{\widehat{\mathbf{p}}}_k\delta ({\mathbf{r}}_k-\mathbf{r})+\delta ({\mathbf{r}}_k-\mathbf{r}){\widehat{\mathbf{p}}}_k]$, ${\widehat{\mathbf{j}}}_d\left(\mathbf{r}\right)=\widehat{\rho }\left(\mathbf{r}\right)\mathbf{A}\left(\mathbf{r}\right)$, ${\widehat{\mathbf{j}}}_m\left(\mathbf{r}\right)=-\mathbf{\nabla }\times \widehat{\mathbf{m}}\left(\mathbf{r}\right)$, where the magnetization density operator $\widehat{\mathbf{m}}\left(\mathbf{r}\right)={\displaystyle {\sum }_{k=1}^2}{\mathbf{s}}_k\delta ({\mathbf{r}}_k-\mathbf{r})$. The flow line contours of the current density $\mathbf{j}\left(\mathbf{r}\right)$ are plotted in Figure 3b. The white regions in the contour plots indicate the low- and high-current-density zones.

The canonical kinetic energy $T=<{\mathsf{\Psi }}_{0,1,4}({\mathbf{x}}_1,{\mathbf{x}}_2)|{\displaystyle \frac{1}{2}}{\displaystyle {\sum }_{k=1}^2}{\widehat{p}}_k^2|{\mathsf{\Psi }}_{0,1,4}({\mathbf{x}}_1,{\mathbf{x}}_2)>=$ $0.031086{(\mathrm{a}.\mathrm{u}.)}^{🟉}$; the electron interaction energy $E_{ee}=<{\mathsf{\Psi }}_{0,1,4}({\mathbf{x}}_1,{\mathbf{x}}_2)|1/|{\mathbf{r}}_1-{\mathbf{r}}_2||{\mathsf{\Psi }}_{0,1,4}({\mathbf{x}}_1,{\mathbf{x}}_2)$$>=0.068809{(\mathrm{a}.\mathrm{u}.)}^{🟉}$; the electromagnetic energy $E_{em}=<{\mathsf{\Psi }}_{0,1,4}({\mathbf{x}}_1,{\mathbf{x}}_2)|{\displaystyle \frac{1}{2}}{k}_{\mathrm{eff},\mathrm{lowest}}{r}^2|{\mathsf{\Psi }}_{0,1,4}$$({\mathbf{x}}_1,{\mathbf{x}}_2)>=0.065491{(\mathrm{a}.\mathrm{u}.)}^{🟉}$; the total energy $E_{\mathrm{tot}}=0.16539{(\mathrm{a}.\mathrm{u}.)}^{🟉}$.

(viii) Observe that both the radial probability density in Figure 2b and the physical current density in Figure 3a, representing the ground triplet state of interest, exhibit a single shell, as they must.

(ix) The function ${\psi }_{0,1,4}({\mathbf{r}}_1,{\mathbf{r}}_2)$ corresponds to the Wigner low-electron-density high-electron-correlation regime. This is evident from the structure of the density $\rho \left(\mathbf{r}\right)$ in Figure 2a, typical of this regime: there is a local minimum at the origin, and the maximum occurs well away from it. That the function ${\psi }_{0,1,4}({\mathbf{r}}_1,{\mathbf{r}}_2)$ corresponds to this regime is further confirmed by the ratio $E_{\mathrm{ee}}/T=221\%$, a key characteristic of the Wigner system.

(x) Here are a few general remarks with regard to the nodal structure of the exact correlated solutions of the Schrödinger–Pauli equation for the two-electron quantum dot in a magnetic field. The nodal structure of the wave function solutions is governed by the solutions to the relative coordinate differential equation. The state of the system is defined by these nodes. As expected, the ground triplet state has zero nodes.

The phase factor $e^{i\alpha }$ provides a finite value ($\ne$0) for the total wave function at any position, because when the Real part of the wave function vanishes, the Imaginary part is finite, and vice versa. For example, the Real part of the wave function $\psi ({\mathbf{r}}_1,{\mathbf{r}}_2)$ is zero when the projections of the vectors ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ on the x-axis are the same. The total wave function is then Imaginary (finite). On the other hand, the Imaginary part of the wave function is zero when the projection of the vectors ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ on the y-axis are the same. The total wave function is then Real (finite). Thus, there are no nodes in the triplet ground-state wave function due to the phase factor.

The only positions at which the total wave function vanishes (other than at infinity) are when the radial and angular coordinates of the individual electrons are the same, i.e., when both the projections on the x-axis are the same and the projections on the y-axis are the same. This corresponds to the case of electron–electron coalescence discussed in (v) above. The ground triplet-state wave function satisfies the node electron–electron coalescence constraint. Thus, there are an infinite number of such coalescence nodes. These nodes differ from the nodes representing the state of the system (this indicates whether the state is ground or excited) which exist in the solutions of the relative coordinate differential equation.

4. Concluding Remarks

The purpose of this paper is two-fold: (a) The first aim of this study is to reveal the theoretical existence of the ground triplet state of a 2D two-electron quantum dot in the Wigner regime in the presence of a magnetic field. We accomplish this by deriving the exact closed-form analytical correlated wave function solution to the corresponding Schrödinger–Pauli equation (we note that a corresponding ground singlet state also exists in the Wigner regime for the same value of the magnetic field [31] (see Appendix A). (b) The second aim of this study is to propose an experimental investigation of such a system to confirm the existence of this state. Current technology allows for the creation of Wigner-regime quantum dots.

The experimental search for the ground triplet-state spectral line must be near that of the ground singlet line of the quantum dot [31]. We expect the difference between the ground singlet and triplet states (for the same magnetic field) to be due principally to correlations arising from the Pauli principle. Although the wave function of the ground triplet state obeys the Pauli principle of antisymmetry, it is not a Slater determinant of single-particle orbitals, but a correlated function. Hence, it is not possible to separate the contribution of the Pauli correlations from those due to Coulomb repulsion of the electrons. At present, what is known is the combined quantum-mechanical Pauli–Coulomb (exchange–correlation) contribution to the total energy. In future work, we propose determining each contribution separately via quantal density functional theory [45] mapping from the interacting ground triplet state to a noninteracting fermion, also in a ground triplet state, possessing the same density and physical current density. Such a calculation would also provide the correlation contribution to the kinetic energy.

If an experimental spectral line for a ground triplet state is observed for the Wigner-regime quantum dot, it would imply that the existence of such states is a function of the binding potential. This idea is buttressed by the fact that there also exists an exact analytical solution to the Schrödinger equation for the three-dimensional Hooke’s atom for a ground triplet state in the Wigner regime. Hence, if the triplet state exists experimentally, one could conclude that there is a fundamental difference between the Coulombic and harmonic binding potentials. For the Coulombic potential, no ground triplet state can exist for the two-electron bound system, whereas for the harmonic potential, such a state can exist in both the presence and absence of a magnetic field. Or, put another way, such a ground triplet state does not exist for the natural two-electron atom, viz. He, but does exist for the human-made two-electron two-dimensional ‘artificial atom’. As noted previously, the absence of the ground triplet state for the He atom implies that there is no solution to the corresponding Schrödinger equation. However, as there does exist such a solution to the Schrödinger–Pauli equation for the quantum dot in a magnetic field, we think it highly likely that the triplet-state spectral line will be discovered experimentally.

Author Contributions

Conceptualization, M.S. and V.S.; Formal analysis, M.S. and V.S.; Writing—original draft, M.S. and V.S.; Writing—review & editing, M.S. and V.S. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

All data generated and analyzed in this study are available from the corresponding author on reasonable request.

Conflicts of Interest

The authors declare no competing interests.

Footnotes

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Figures

Figure 1 Plot of ${\psi }_{0,1,4}({\mathbf{r}}_1,{\mathbf{r}}_2)$ as function of $r_1$ and $r_2$ for ${\theta }_1={30}^{\circ },{\theta }_2={25}^{\circ }$: (a) Real and (b) Imaginary parts.

Figure 2 (a) Electron density $\rho \left(r\right)$; (b) radial probability density $r\rho \left(r\right)$.

Figure 3 (a) Physical current density $\mathbf{j}\left(\mathbf{r}\right)$; (b) flow line contours of $\mathbf{j}\left(\mathbf{r}\right)$. The arrows in the contour lines represent the direction of the physical current density.

Appendix A. Particular Wigner-Regime Ground-State Singlet Wave Function

The Wigner-regime ground singlet state wave function solution [31] to the Schrödinger–Pauli equation Equation (1) is of the same form as that of Equations (3) and (4) in this text. The corresponding parameters are as follows: the number of nodes $n=0$; $m=0$, which corresponds to electrons with an opposite spin moment; the effective force constant $k_{\mathrm{eff},\mathrm{lowest}}=5.6064\times {10}^{-4}$; the number of terms in the polynomial ${\mathsf{\Phi }}_{m=0,p}(|{\mathbf{r}}_2-{\mathbf{r}}_1|)$ is $p=5$; the normalization constant $N=9.2666\times {10}^{-5}$; the coefficients of the polynomial are $c_1=1$; $c_2=2.2632\times {10}^{-2}$; $c_3=1.7254\times {10}^{-3}$; and $c_4=4.0855\times {10}^{-4}$.

The wave function satisfies the PP; the Wave Function Identity; has zero nodes; has even parity; has even parity about each point of electron–electron coalescence; satisfies the cusp electron–electron coalescence constraint; exhibits a density of the Wigner-regime form; and has a radial probability density that exhibits a single shell.

The energy values in effective atomic units (a.u.)^🟉 are as follows: the total energy $E=1.4207\times {10}^{-1}$; the canonical kinetic energy $T=2.1771\times {10}^{-2}$; the electron interaction energy $E_{ee}=6.5683\times {10}^{-2}$; the electromagnetic energy $E_{em}=5.4613\times {10}^{-2}$; and the ratio $E_{ee}/T=302\%$, which further indicates that the solution is in the Wigner high-electron-correlation regime.

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