# Schrödinger Theory of Electrons in Electromagnetic Fields: New Perspectives

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## Abstract

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## 1. Introduction

- (a)
- as a consequence of the correspondence principle, it is the vector potential $\mathbf{A}\left(\mathbf{r}\right)$ and not the magnetic field $\U0001d4d1\left(\mathbf{r}\right)$ that appears in it. This fact is significant, and is expressly employed to explain, for example, the Bohm–Aharonov [22] effect in which a vector potential can exist in a region of no magnetic field. The magnetic field $\U0001d4d1\left(\mathbf{r}\right)$ appears in the Schrödinger equation only following the choice of gauge;
- (b)
- the characteristics of the potential energy operator $v\left(\mathbf{r}\right)$ are the following:
- (i)
- for the N-electron system, it is assumed that the canonical kinetic and electron-interaction potential energy operators are known. As such, the potential $v\left(\mathbf{r}\right)$ is considered an extrinsic input to the Hamiltonian.
- (ii)
- the potential energy function $v\left(\mathbf{r}\right)$ is assumed known, e.g., it could be Coulombic, harmonic, Yukawa, etc.
- (iii)
- by assumption, the potential $v\left(\mathbf{r}\right)$ is path-independent.

## 2. Stationary State Theory: “Quantal Newtonian” First Law

## 3. New Perspectives

- (i)
- In addition to the external electrostatic $\U0001d4d4\left(\mathbf{r}\right)$ and Lorentz $\U0001d4db\left(\mathbf{r}\right)$ fields, each electron experiences an internal field ${\U0001d4d5}^{\mathrm{int}}\left(\mathbf{r}\right)$. This field via its ${\U0001d4d4}_{\mathrm{ee}}\left(\mathbf{r}\right)$ component is representative not only of Coulomb correlations as one might expect, but also those due to the Pauli exclusion principle due to the antisymmetric nature of the wave function. Additionally, there is a component $\U0001d4e9\left(\mathbf{r}\right)$ representative of the motion of the electrons; a component $\U0001d4d3\left(\mathbf{r}\right)$ representing the density, a fundamental property of the system [29]; and a term $\U0001d4d8\left(\mathbf{r}\right)$ that arises as a consequence of the external magnetic field [27]. Hence, each electron experiences an internal field that encapsulates all the basic properties of the system. As in classical physics, in summing over all the electrons, the contribution of the internal field vanishes, leading thereby to Ehrenfest’s (first law) theorem: $\int \rho \left(\mathbf{r}\right){\U0001d4d5}^{\mathrm{ext}}\left(\mathbf{r}\right)d\mathbf{r}=0$. (In fact, each component of the internal field is shown to separately vanish.).
- (ii)
- The “Quantal Newtonian” first law Equation (4) affords a rigorous physical interpretation of the external electrostatic potential $v\left(\mathbf{r}\right)$: it is the work done to move an electron from some reference point at infinity to its position at $\mathbf{r}$ in the force of a conservative field $\U0001d4d5\left(\mathbf{r}\right)$:$$v\left(\mathbf{r}\right)={\int}_{\infty}^{\mathbf{r}}\U0001d4d5\left({\mathbf{r}}^{\prime}\right)\xb7d{\ell}^{\prime},$$
- (iii)
- What the physical interpretation of the potential $v\left(\mathbf{r}\right)$ further shows is that it can no longer be thought of as an independent entity. It is intrinsically dependent upon all the properties of the system via the various components of the internal field ${\U0001d4d5}^{\mathrm{int}}\left(\mathbf{r}\right)$, and the Lorentz $\U0001d4db\left(\mathbf{r}\right)$ field through the current density $\mathbf{j}\left(\mathbf{r}\right)$. Hence, the potential energy function $v\left(\mathbf{r}\right)$ is comprised of the sum of constituent functions each representative of a property of the system.
- (iv)
- As each component of the internal field ${\U0001d4d5}^{\mathrm{int}}\left(\mathbf{r}\right)$ (and the Lorentz field $\U0001d4db\left(\mathbf{r}\right)$) are obtained from quantal sources that are expectations of Hermitian operators taken with respect to the wave function $\mathsf{\Psi}\left(\mathbf{X}\right)$, we see that the field $\U0001d4d5\left(\mathbf{r}\right)$ is a functional of $\mathsf{\Psi}\left(\mathbf{X}\right)$, i.e., $\U0001d4d5\left(\mathbf{r}\right)=\U0001d4d5\left[\mathsf{\Psi}\right(\mathbf{X}\left)\right]$. Thus, (from Equation (30)), $v\left(\mathbf{r}\right)$ is a functional of $\mathsf{\Psi}\left(\mathbf{X}\right):v\left(\mathbf{r}\right)=v\left[\mathsf{\Psi}\right(\mathbf{X}\left)\right]$. The functional $v\left[\mathsf{\Psi}\right(\mathbf{X}\left)\right]$ is exactly known [via Equation (30)].
- (v)
- On substituting the functional $v\left[\mathsf{\Psi}\right(\mathbf{X}\left)\right]$ into Equation (1), the Schrödinger equation may then be written as$$\left[\frac{1}{2}\sum _{i}{({\widehat{\mathbf{p}}}_{i}+\mathbf{A}\left({\mathbf{r}}_{i}\right))}^{2}+\frac{1}{2}\underset{i,j}{{\sum}^{\prime}}\frac{1}{|{\mathbf{r}}_{i}-{\mathbf{r}}_{j}|}+\sum _{i}v\left[\mathsf{\Psi}\right]\left({\mathbf{r}}_{i}\right)\right]\mathsf{\Psi}\left(\mathbf{X}\right)=E\mathsf{\Psi}\left(\mathbf{X}\right),$$$$\left[\frac{1}{2}\sum _{i}{({\widehat{\mathbf{p}}}_{i}+\mathbf{A}\left({\mathbf{r}}_{i}\right))}^{2}+\frac{1}{2}\underset{i,j}{{\sum}^{\prime}}\frac{1}{|{\mathbf{r}}_{i}-{\mathbf{r}}_{j}|}+\sum _{i}{\int}_{\infty}^{{\mathbf{r}}_{i}}\U0001d4d5\left[\mathsf{\Psi}\right]\left(\mathbf{r}\right)\xb7d\ell \right]\mathsf{\Psi}\left(\mathbf{X}\right)=E\mathsf{\Psi}\left(\mathbf{X}\right).$$

- (vi)
- Observe that in writing the Schrödinger equation as in Equations (31) and (32), the magnetic field $\U0001d4d1\left(\mathbf{r}\right)$ now appears in the Hamiltonian explicitly via the Lorentz field $\U0001d4db\left(\mathbf{r}\right)$ (see Equation (30)). It is the intrinsic self-consistent nature of the equation that demands the presence of $\U0001d4d1\left(\mathbf{r}\right)$ in the Hamiltonian. In other words, since the Hamiltonian $\widehat{H}\left[\mathsf{\Psi}\right]$ is being determined self-consistently, all the information of the physical system—electrons and fields—must be incorporated in it. (Of course, equivalently, the field $\U0001d4d1\left(\mathbf{r}\right)$ could be expressed in terms of the vector potential $\mathbf{A}\left(\mathbf{r}\right)$. This then shows that when written in self-consistent form, there exists another component of the Hamiltonian involving the vector potential.)
- (vii)
- The presence of a solely electrostatic external field $\U0001d4d4\left(\mathbf{r}\right)=-\mathbf{\nabla}v\left(\mathbf{r}\right)$ is a special case of the stationary state theory discussed above. This case then constitutes the description of matter as defined in the Introduction.

## 4. Examples: Quantum Dots in a Ground and Excited State

## 5. Time-Dependent Theory: “Quantal Newtonian” Second Law

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References and Notes

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**Figure 2.**The electron-interaction ${\U0001d4d4}_{\mathrm{ee}}\left(\mathbf{r}\right)$, kinetic $\U0001d4e9\left(\mathbf{r}\right)$, and differential density $\U0001d4d3\left(\mathbf{r}\right)$ components of the internal field ${\U0001d4d5}^{\mathrm{int}}\left(\mathbf{r}\right)$ for a quantum dot in a magnetic field in its ground state. The sums $\U0001d4d3\left(\mathbf{r}\right)+\U0001d4e9\left(\mathbf{r}\right)$, and $-{\U0001d4d4}_{\mathrm{ee}}\left(\mathbf{r}\right)+\U0001d4e9\left(\mathbf{r}\right)+\U0001d4d3\left(\mathbf{r}\right)=-{k}_{\mathrm{eff}}r$ with ${k}_{\mathrm{eff}}=1$ are also plotted.

**Figure 3.**Same as in Figure 2 but for a quantum dot in a magnetic field in its first excited singlet state with ${k}_{\mathrm{eff}}=0.471716$.

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Sahni, V.; Pan, X.-Y. Schrödinger Theory of Electrons in Electromagnetic Fields: New Perspectives. *Computation* **2017**, *5*, 15.
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Sahni, Viraht, and Xiao-Yin Pan. 2017. "Schrödinger Theory of Electrons in Electromagnetic Fields: New Perspectives" *Computation* 5, no. 1: 15.
https://doi.org/10.3390/computation5010015