1D Coulomb Problem Energy Spectrum

The 1D Schrödinger Equation and its Challenges

Let's restate the 1D Schrödinger equation for the longitudinal part of the wavefunction, $f_{\nu }(z)$:

$\left(-\frac{{\hslash }^2}{2\mu }\frac{d^2}{d{z}^2}+V_{NM}^{\text{eff}}(z)\right)f_{\nu }(z)=E_{NM\nu }^{1D}{f}_{\nu }(z)$

The effective potential, $V_{NM}^{\text{eff}}(z)$, which results from averaging the 3D Coulomb interaction over the 2D Landau orbital, has a specific shape:

• For large distances ($|z|\gg l_B$, where $l_B=\sqrt{\hslash /eB}$ is the magnetic length), it asymptotically approaches the familiar 1D Coulomb potential: $V_{NM}^{\text{eff}}(z)\sim \frac{-e^2}{4\pi ϵ|z|}$.
• For small distances ($|z|\lesssim l_B$), the potential is "softened" and deviates from the $1/|z|$ form. The finite radius of the cyclotron orbit prevents the electron and hole from occupying the exact same $z=0$ plane, thus removing the divergence and leading to a potential that varies logarithmically near the origin.

This complex shape prevents a straightforward solution. We will use the elegant method developed by Hasegawa & Howard, which involves finding separate asymptotic solutions for the "exterior" (large $|z|$) and "interior" (small $|z|$) regions and then matching them.

For a comprehensive and rigorous treatment of this method, the following paper is an excellent resource. While it's framed for hydrogenic ions, the mathematical physics is directly analogous to our exciton problem (an electron-hole pair is essentially a hydrogen atom with different masses).

Step 1: The Exterior Solution (Large $|z|$)

In the exterior region, where $|z|\gg l_B$, the potential is effectively the pure 1D Coulomb potential. The Schrödinger equation becomes:
$\left(-\frac{{\hslash }^2}{2\mu }\frac{d^2}{d{z}^2}-\frac{e^2}{4\pi ϵ|z|}\right)f(z)=E^{1D}f(z)$

The solutions to this well-known equation for bound states ($E^{1D}<0$) are given in terms of Whittaker functions.

The key insight is that the bound-state energies of this 1D Coulomb problem can be written as:
$E_{NM\nu }^{1D}=-\frac{R{y}^{\ast }}{{\tilde{\nu }}^2}$
where $R{y}^{\ast }=\frac{\mu e^4}{2(4\pi ϵ{)}^2{\hslash }^2}$ is the exciton effective Rydberg. The value $\tilde{\nu }$ is not necessarily an integer; it is the "effective principal quantum number" that we need to determine.

Step 2: The Interior Solution (Small $|z|$)

In the interior region ($|z|\ll l_B$), the softened potential dominates. An approximate solution can be found using perturbation theory. The paper carries out this calculation for even-parity states.

The crucial outcome of this step is an expression for the logarithmic derivative $\eta (z)=f^{\prime }(z)/f(z)$ of the wavefunction in both the interior and exterior regions.

Step 3: Matching and Deriving the Energy Spectrum

Now we perform the key step: we demand that the wavefunction and its derivative are continuous. This is equivalent to equating the logarithmic derivatives of the interior and exterior solutions at a matching point $z_0$:
${\eta }_{\text{int}}(z_0)={\eta }_{\text{ext}}(z_0)$

This procedure leads to a transcendental equation that determines the allowed values of $\tilde{\nu }$, and thus the energy eigenvalues.

Let's summarize the results for the energy spectrum of the 1D motion, which is indexed by a new quantum number $\nu =0,1,2,\dots$ corresponding to the longitudinal states.

Tightly Bound States (Even Parity, $\nu =0$)

The lowest energy state ($\nu =0$) for each Landau level is the "tightly bound" ground state. Its effective quantum number $\tilde{\nu }$ is much less than 1, and its binding energy is very large. Equation (206) from the paper shows that, to a good approximation, $\tilde{\nu }$ is given by:
$\frac{1}{\tilde{\nu }}\approx \ln \left(\frac{{\gamma }^{\ast }}{2}\right)+2\ln \tilde{\nu }-\text{const.}$
Here, ${\gamma }^{\ast }=(a_B^{\ast }/l_B{)}^2\propto B$ is the square of our original dimensionless field parameter. This equation must be solved numerically, but its structure tells us that $1/\tilde{\nu }$ grows roughly as $\ln {\gamma }^{\ast }$.

This implies the binding energy of the ground state increases approximately as the square of the logarithm of the magnetic field:
$|E_{N,M,0}^{1D}|=\frac{R{y}^{\ast }}{{\tilde{\nu }}^2}\propto (\ln B{)}^2$
This is a hallmark of the high-field regime: the strong confinement dramatically increases the binding energy of the ground state.

Loosely Bound States (Even Parity, $\nu =2,4,\dots$)

For the excited states, $\tilde{\nu }$ approaches an integer value. These states are more hydrogen-like and less tightly bound. The energy levels can be expressed using a quantum defect ${\delta }_{\nu }$, similar to Rydberg series in atoms:
$E_{N,M,\nu }^{1D}\approx -\frac{R{y}^{\ast }}{(\nu /2+{\delta }_{\nu }{)}^2}\text{for }\nu =2,4,6,\dots$
The quantum defect ${\delta }_{\nu }$ depends logarithmically on the magnetic field, shifting the series relative to a pure hydrogenic model.

Odd Parity States ($\nu =1,3,\dots$)

For odd-parity states, the wavefunction $f(z)$ has a node at $z=0$. This means the particle is unlikely to be found near the origin, where the effective potential differs most from the pure $1/|z|$ form. Consequently, to a first approximation, these states behave like the odd states of a true 1D hydrogen atom, with energies:
$E_{N,M,\nu }^{1D}\approx -\frac{R{y}^{\ast }}{((\nu +1)/2{)}^2}\text{for }\nu =1,3,5,\dots$

The Full Magneto-Exciton Energy Spectrum

We have now assembled all the pieces. The total energy for an exciton in the high-field limit is given by:

$E_{N,M,\nu }(B)=E_g+\left(N+\frac{1}{2}\right)\hslash {\omega }_c+E_{NM\nu }^{1D}(B)$

This equation describes a rich and complex spectrum:

• The spectrum is organized into Landau fan diagrams, a series of primary energy levels indexed by the Landau quantum number $N=0,1,2,\dots$, whose energy increases linearly with $B$.
• Attached to each Landau level is an entire hydrogen-like series of 1D bound states, indexed by the longitudinal quantum number $\nu =0,1,2,\dots$.
• The ground state of each series ($\nu =0$) is a deeply bound state whose binding energy increases as $(\ln B{)}^2$.
• The excited states of each series ($\nu >0$) are more loosely bound and follow a Rydberg-like formula with a field-dependent quantum defect.

This is precisely the structure that leads to the complex fan diagrams observed in magneto-spectroscopy experiments.

An Alternative Solution Method for Cu₂O

The Hasegawa-Howard method is powerful and general. For specific materials, simpler approximations can be used. Your interest in Cu₂O makes this next resource particularly relevant. The authors approximate the complex effective potential with a simpler analytical function, which allows a direct solution.

This alternative reinforces the central concept: once the problem is reduced to a 1D Schrödinger equation, various mathematical techniques can be employed to find the eigenvalues, depending on the desired accuracy and generality.

Conclusion

In this lesson, we have successfully solved the 1D Coulomb problem that governs the longitudinal motion of an exciton in a strong magnetic field.

Key Takeaways:

• The 1D Schrödinger equation for the longitudinal motion can be solved approximately by matching asymptotic solutions for small and large $z$.
• The solution reveals a full hydrogen-like series of bound states for each Landau level, indexed by a longitudinal quantum number $\nu$.
• The ground state of each series ($\nu =0$) is a tightly bound state with a binding energy that increases as $(\ln B{)}^2$.
• The excited states ($\nu >0$) are loosely bound and follow a Rydberg-like energy formula.
• The total energy spectrum $E_{N,M,\nu }(B)$ consists of a fan of Landau levels, each serving as the ionization limit for its own series of 1D Coulomb states.

Preview of the Next Lesson

We have now developed a detailed picture of the exciton energy levels in both the low-field (diamagnetic shift, Zeeman splitting) and high-field (Landau levels) regimes. But what happens in between? In the next lesson, we will venture into the challenging intermediate-field regime ($\gamma \approx 1$), where neither the Coulomb nor the magnetic interaction dominates. We will see how the breakdown of separability leads to non-integrable dynamics and the fascinating phenomenon of quantum chaos.