Quantum eigenvalues and eigenfunctions of an electron confined between conducting planes

This expository paper derives the electrostatic potential arising from image charges for an electron confined between grounded conducting planes, coupling a hydrogen-like system with a particle-in-a-box model, and solves the resulting Schrödinger equation to analyze the transition between large and small separation limits and the associated tunneling level splitting.

The Gist

The Big Picture: An Electron in a Glass Box

Imagine you have a tiny, mischievous electron. Usually, this electron is free to roam, but in this story, we trap it inside a very specific room: a gap between two giant, perfectly conducting metal walls (like the plates of a capacitor).

In the real world, if you put a charged object near a metal wall, the metal reacts. It rearranges its own electrons to cancel out the charge, creating an "image" of the object on the other side of the wall (just like a mirror).

Now, imagine our electron is stuck between two mirrors facing each other.

1. The electron sees a reflection in the left wall.
2. That reflection sees a reflection in the right wall.
3. That reflection sees a reflection in the left wall... and so on, forever.

The electron is essentially trapped in an infinite hallway of mirrors. This paper asks: How does this infinite hallway of "ghost" electrons change the behavior of the real electron?

The Problem: The Math is a Nightmare

To figure out how the electron moves, we need to know the "force field" (potential energy) it feels.

• The Old Way: Physicists used to try to solve this by adding up the pull of every single "ghost" electron one by one. But because there are infinite ghosts, the math is incredibly slow and messy. It's like trying to count every grain of sand on a beach by picking them up one at a time.
• The New Way (The Paper's Contribution): The author, Don MacMillen, found a clever shortcut. He realized that all those infinite reflections can be summarized by a single, neat mathematical formula involving a special function called the Digamma function.
  • Analogy: Instead of counting every grain of sand, he found a formula that tells you exactly how much sand is on the beach in one second.

The Solution: The "Double-Well" Trap

When the author plugged this new formula into the rules of quantum mechanics (Schrödinger's equation), he discovered something fascinating about the shape of the trap:

1. The Shape: The potential energy looks like a "W".

   • There are two deep valleys (wells) right next to the walls.
   • There is a high hill (barrier) in the middle.
   • Why? The electron is attracted to the walls (because of the image charges), but it's repelled by the other "ghost" electrons in the middle.

2. The Two Extreme Scenarios:
   The paper explores what happens when you change the distance ($L$) between the walls.

   • Scenario A: The Walls are Far Apart (Large $L$)

     • What happens: The electron doesn't know the other wall exists. It just sees the wall right next to it.
     • The Result: The electron acts like it's stuck to a single wall, forming a "bound state" (like a planet orbiting a star, but stuck to a wall).
     • The Twist: Because there are two walls, the electron has two identical places to hide. This creates two energy levels that are almost exactly the same (degenerate).

   • Scenario B: The Walls are Very Close (Small $L$)

     • What happens: The electron is squeezed so tight that the two "W" valleys merge into one big, flat valley.
     • The Result: The electron behaves like a classic "Particle in a Box." It bounces back and forth between the walls like a ping-pong ball. The energy levels follow a simple, predictable pattern.

The Magic Trick: Tunneling and Splitting

The most exciting part of the paper is what happens in the middle, as the walls move from far apart to close together.

• Tunneling: In quantum mechanics, particles can sometimes "ghost" through barriers they shouldn't be able to cross.
• The Split: When the walls are far apart, the electron is stuck in the left valley OR the right valley. But as the walls get closer, the barrier in the middle gets lower. The electron starts to "tunnel" through the middle, spending time in both valleys at once.
• The Result: This tunneling causes the two identical energy levels to split apart. One level goes slightly up, and the other goes slightly down.
  • Analogy: Imagine two identical tuning forks. If you tap them, they hum at the exact same pitch. But if you connect them with a rubber band (the tunneling), they start to vibrate slightly differently—one slightly higher, one slightly lower. The paper calculates exactly how much they split based on the distance between the walls.

How They Solved It: The "Spectral" Method

To get these answers, the author didn't just use a calculator; he used a super-smart computer technique called a Spectral Method.

• Analogy: Imagine trying to draw a smooth curve on a piece of paper.
  • Old Method: You plot a dot every inch, then connect them. It's okay, but jagged.
  • Spectral Method: You plot dots only where the curve is tricky (near the walls) and fewer dots where it's smooth. Then, you use a special mathematical "lens" (Chebyshev polynomials) to reconstruct the perfect curve from those few dots.
• The author wrote a short computer program (less than 40 lines of code!) to do this. It was so efficient that it solved the problem in about 2 seconds on a standard laptop.

Why Should You Care?

This isn't just a theoretical puzzle.

1. Real Materials: This setup mimics what happens in advanced materials like bilayer graphene (a super-thin, super-strong carbon material) or when scientists use microscopes to look at single atoms on surfaces.
2. Future Tech: Understanding how electrons behave in these "squeezed" spaces is crucial for building better nanotechnology and quantum computers.
3. Education: The author shows that with modern tools, these complex problems can be solved by undergraduates, not just PhDs with supercomputers.

Summary

The paper takes a classic physics problem (an electron between two mirrors), finds a shortcut to describe the infinite reflections, and uses a smart computer trick to show how the electron's energy changes as the walls move. It reveals that as the walls get closer, the electron's "tunneling" ability causes its energy levels to split, bridging the gap between two famous quantum models: the "bound electron" and the "particle in a box."

Technical Summary

1. Problem Statement

The paper addresses the quantum mechanical problem of an electron confined between two infinite, parallel, grounded conducting planes separated by a distance $L$.

• Physical Context: Unlike the standard "particle in a box" (PIB) where the potential is infinite outside the box and zero inside, this system involves a real electron interacting with its own induced image charges on the grounded planes.
• The Potential: The electron experiences an electrostatic potential generated by an infinite series of image charges. This potential creates a symmetric double-well structure.
• Boundary Conditions: The wavefunction must vanish at the planes ($x=0$ and $x=L$).
• Goal: To derive a closed-form expression for this potential and solve the Schrödinger equation to find the energy eigenvalues and eigenfunctions across different regimes of $L$ (small vs. large separation).

2. Methodology

A. Derivation of the Electrostatic Potential

The author starts with the classic series solution for the potential of a point charge between parallel plates derived by Kellogg (1929).

• Image Charge Series: The potential $U$ is initially expressed as an infinite sum of image charges with alternating signs.
• Modifications: To find the potential energy acting on the electron due to its own images (excluding self-energy), the author:
  1. Removes the self-interaction term.
  2. Introduces a factor of $1/2$ to account for the energy of assembly (interaction of a charge with its own image).
• Closed-Form Solution: By grouping terms and utilizing the properties of the Digamma function ($\psi$) and the Euler-Mascheroni constant ($\gamma$), the infinite series is converted into a compact analytical expression:
  $$V(x) = \frac{1}{4L} \left[ \psi\left(\frac{x}{L}\right) + \psi\left(1 - \frac{x}{L}\right) + 2\gamma \right]$$
  where $a = x/L$.
• Convergence Analysis: The paper notes that while the original image-charge series converges, it does so extremely slowly (requiring thousands of terms for high precision). The closed-form Digamma expression provides immediate, high-precision evaluation.

B. Numerical Solution of the Schrödinger Equation

To solve the time-independent Schrödinger equation $H\psi = E\psi$ with the derived potential, the author employs a Spectral Method (specifically a Chebyshev collocation method).

• Hamiltonian: In scaled units, the Hamiltonian is:
  $$H(x, L) = -\frac{1}{2}\frac{d^2}{dx^2} + V(x)$$
• Discretization:
  • The domain is mapped to Chebyshev grid points, which cluster near the boundaries ($x=0, L$). This clustering is crucial because the potential behaves like a Coulomb singularity near the walls.
  • A second-order differentiation matrix ($D^{(2)}$) is constructed to represent the kinetic energy operator.
  • The potential energy is evaluated directly at the grid points and added to the diagonal of the kinetic energy matrix.
• Boundary Conditions: Homogeneous Dirichlet boundary conditions ($\psi=0$ at walls) are enforced by truncating the differentiation matrix to exclude the boundary points.
• Implementation: The solution is implemented in Julia using the LinearAlgebra and SpecialFunctions libraries, resulting in a concise codebase (<40 lines).

3. Key Contributions

1. Compact Potential Expression: The paper provides a clear, short derivation of the closed-form potential using Digamma functions, contrasting it with the slow convergence of the traditional image-charge summation.
2. Spectral Technique Application: It demonstrates the efficiency of spectral collocation methods (Chebyshev points) for solving 1D Schrödinger equations with singular potentials and strict boundary conditions, avoiding the dense matrix integration required by Rayleigh-Ritz methods.
3. Unified Regime Analysis: The work bridges two limiting cases:
   • Small $L$ (High Energy): The system behaves like a Particle in a Box (PIB).
   • Large $L$ (Low Energy): The system behaves like a bound image charge state near a single plane (degenerate double well).
4. Tunneling Splitting Analysis: The paper analytically and numerically characterizes the energy level splitting ($\Delta E$) between the ground and first excited states as a function of $L$, attributing it to quantum tunneling through the central barrier.

4. Results

A. Energy Levels and Limits

• Large $L$ Regime ($L \to \infty$):
  • The energy levels converge to those of a bound image charge: $E_n \approx -\frac{1}{32n^2}$.
  • Due to the symmetry of the double well, these levels appear as nearly degenerate pairs (splitting is exponentially small).
  • Numerical results for $L=10,000$ match the theoretical limit to high precision (see Table II in the paper).
• Small $L$ Regime ($L \to 0$):
  • The energy levels approach the ideal PIB values: $E_N \approx \frac{1}{2}(\frac{\pi N}{L})^2$.
  • The "quantum defect" (the deviation from the ideal PIB integer $N$) is calculated. It starts at $\sim 10\%$ for the ground state but rapidly decreases to $\sim 0.2\%$ by $N=10$.
  • A log-log plot of Energy vs. Quantum Number $N$ shows a slope of 2, confirming the quadratic dependence characteristic of the PIB.

B. Wavefunction Evolution

• Small $L$: The ground state wavefunction is concentrated in the center of the box (PIB-like).
• Large $L$: The wavefunction splits into two localized peaks near the conducting planes, representing the symmetric and anti-symmetric combinations of an electron bound to a single image charge.
• Transition: As $L$ increases, the wavefunction evolves from a single central peak to a double-peak structure.

C. Tunneling Splitting

• The energy splitting $\Delta E$ between the first two states decreases exponentially as $L$ increases.
• The author derives an analytical approximation for the splitting based on the wavefunction value and derivative at the barrier peak:
  $$\Delta E(L) \approx \frac{L}{16} e^{-L/4} \left(1 - \frac{L}{8}\right)$$
• Numerical results align closely with this analytical prediction, confirming the tunneling mechanism.

D. Numerical Stability

• The paper highlights that for a spectral method with $M$ grid points, only the first $M/2$ eigenvalues are physically accurate. Higher eigenvalues suffer from "spectral pollution" and do not represent true solutions to the differential equation.

5. Significance and Future Directions

• Educational Value: The paper demonstrates that modern computational tools (Julia, spectral methods) make complex quantum problems accessible to advanced undergraduates.
• Physical Relevance: The model is relevant to:
  • Image Potential States: Observed in surface physics and scanning quantum dot microscopy.
  • 2D Materials: Applications in bilayer graphene and induced surface states.
  • Radiative Corrections: The setup serves as a baseline for calculating QED radiative corrections (Lamb shift analogs) in confined geometries.
• Future Work: The author suggests extending the study to include radiative corrections, analyzing higher excited state splittings, and calculating the induced surface charge density as a function of $L$ and quantum number.

In summary, MacMillen provides a rigorous yet accessible treatment of a classic electrostatic problem transformed into a quantum mechanical double-well system, utilizing modern numerical techniques to elucidate the transition between particle-in-a-box and image-charge bound states.