The hydrogen atom is a central system in quantum physics, often used as a model to understand the electronic structure of atoms. Solving the Schrödinger equation for this atom relies on the spherical symmetry of the problem and the Coulomb potential between the proton (nucleus) and the electron.

1. Schrödinger Equation in the Coulomb Potential

The Schrödinger equation for a particle of mass $m$ in a central potential $V(r) = -\frac{e^2}{4\pi \epsilon_0 r}$ is given by:

$$-\frac{\hbar^2}{2m} \nabla^2 \psi + V(r)\psi = E\psi$$

In spherical coordinates, due to the radial symmetry, the wavefunction $\psi(r, \theta, \phi)$ can be separated as:

$$\psi(r, \theta, \phi) = R(r) Y_l^m(\theta, \phi)$$

where:

• $R(r)$ is the radial part of the wavefunction, depending only on the distance $r$,
• $Y_l^m(\theta, \phi)$ are the spherical harmonics dependent on angles $\theta$ and $\phi$,
• $l$ is the orbital quantum number, and $m$ its magnetic sublevel.

The radial part satisfies an independent differential equation:

$$\frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{dR}{dr} \right) + \left[ \frac{2m}{\hbar^2} \left( E – V(r) \right) – \frac{l(l+1)}{r^2} \right] R(r) = 0$$

2. Solving the Radial Equation

To solve this equation, we introduce the dimensionless variable $\rho = \frac{r}{a_0}$, where $a_0$ is the Bohr radius:

$$a_0 = \frac{4\pi \epsilon_0 \hbar^2}{me^2}$$

The solution for $R(r)$ is a combination of exponential functions and associated Laguerre polynomials:

$$R_{n,l}(r) = N_{n,l} \, \rho^l e^{-\rho / n} L_{n-l-1}^{2l+1}(\rho)$$

where:

• $n$ is the principal quantum number,
• $l$ is the orbital quantum number,
• $L_{n-l-1}^{2l+1}(\rho)$ are associated Laguerre polynomials,
• $N_{n,l}$ is a normalization constant.

For the ground state ($n = 1, l = 0$), the solution simplifies to:

$$R_{1,0}(r) = \frac{2}{\sqrt{a_0^3}} e^{-r / a_0}$$

3. Radial Density and Probability

The radial probability density, which describes the likelihood of finding the electron at a distance $r$, is given by:

$$P(r) = |R(r)|^2 r^2$$

For $n = 1, l = 0$, this probability density becomes:

$$P(r) = \frac{4}{a_0^3} e^{-2r / a_0} r^2$$

This shows an exponential decay modulated by a geometric factor $r^2$. This combination reflects the duality between the electron’s radial localization and spherical symmetry.

From the Hydrogen Atom to General Waves: A Universal Decomposition

The solution for the hydrogen atom is built on a combination of exponentials ($e^{-r}$) and polynomial terms. This structure is typical in wave or field modeling. A key idea in mathematical physics is that all waves or fields can be decomposed into sums of complex exponentials, similar to Fourier series.

4. Wave Decomposition into Exponentials

The decomposition of a function or wave $f(r)$ can be generalized as sums or integrals of the form:

$$f(r) = \int A(k) e^{-kr} \, dk$$

where:

• $A(k)$ is an amplitude dependent on $k$,
• $e^{-kr}$ represents an elementary component.

This idea is analogous to Fourier series, where periodic functions are expressed as sums of $e^{i\omega t}$, but here we handle non-periodic or localized functions.

In BeeTheory, this principle is generalized to describe any wave or field using terms of the form $A e^{-kr}$, encompassing not only quantum solutions like those of the hydrogen atom but also models for gravity or fundamental interactions.

BeeTheory and Summations of $e^{-R}$

In BeeTheory, the central idea is to extend this decomposition to all wave-like interactions. We know that:

1. Electromagnetic waves (solutions of Maxwell’s equations) decompose into spherical harmonics and exponentials.
2. Quantum solutions for atoms already use exponential bases like $e^{-r/a}$.
3. Gravitational interactions and potentials like Yukawa’s (in particle physics) are modeled with exponential decays.

5. The Universal Link: Any Wave as a Superposition

BeeTheory proposes that any wave-like interaction (be it electromagnetic, gravitational, or otherwise) can be modeled as a sum of terms $A e^{-R}$, where $R$ generalizes distance or a coordinate:

$$\Phi(R) = \sum_{i} A_i e^{-k_i R}$$

This approach:

• Unifies classical solutions (Maxwell, Schrödinger) and modern ones (screened potentials like Yukawa),
• Provides a simplified vision of fundamental interactions,
• Offers a framework to simulate or describe complex phenomena.

6. Extending to All Waves

• Gravity: In quantum frameworks, the gravitational potential can be viewed as a sum of $e^{-R}$ terms (a gravitational screening model).
• Quantum Physics: Quantum states, such as those in the hydrogen atom, already demonstrate this exponential basis.
• Cosmology: Fluctuations in the cosmic microwave background or gravitational waves can be expressed using exponential terms.

By unifying interaction models through sums of $e^{-R}$, BeeTheory offers a general framework for modeling all forms of waves, whether in a quantum, classical, or cosmological context.

If you’d like to dive deeper into this theory or explore its applications, BeeTheory is designed to provide accessible and powerful modeling tools to unify physical phenomena under a common wave-based framework.