[Paper Review] A chart for the energy levels of the square quantum well

Marco Chiani|arXiv (Cornell University)|Sep 30, 2016

Advanced Physical and Chemical Molecular Interactions8 references3 citations

TL;DR

This paper introduces a universal, normalized energy-level chart for a particle in a one-dimensional square quantum well, derived by scaling energy and potential using $8m w^2 / h^2$. The chart graphically displays all allowed (potential, energy) pairs for both even and odd parity states across arbitrary system parameters, enabling direct visualization of quantization, level counting, and the infinite-well limit without recalculating for each case.

ABSTRACT

A chart for the quantum mechanics of a particle of mass $m$ in a one-dimensional potential well of width $w$ and depth $V_0$ is derived. The chart is obtained by normalizing energy and potential through multiplication by ${8 m}{w^2} / h^2$, and gives directly the allowed couples (potential, energy), providing insights on the relation between the parameters and the number of allowed energy levels.

Motivation & Objective

• To develop a universal graphical tool for determining allowed energy levels in a finite square quantum well.
• To eliminate the need for case-specific plots by normalizing energy and potential using $8m w^2 / h^2$.
• To provide direct insight into the dependence of quantized energy levels on $V_0$, $m$, and $w$ through a single, reusable chart.
• To clarify the relationship between finite and infinite potential well limits using the normalized framework.

Proposed method

• Normalization of energy $E$ and potential depth $V_0$ via $\tilde{E} = E \cdot 8m w^2 / h^2$ and $\tilde{V}_0 = V_0 \cdot 8m w^2 / h^2$ to make the problem dimensionless and universal.
• Derivation of closed-form inverse relations $\tilde{V}_0(\tilde{E})$ for even and odd parity states using trigonometric identities from the transcendental Schrödinger equation solutions.
• Plotting $\sqrt{\tilde{V}_0} = \sqrt{\tilde{E}} \cdot |\sec(\pi \sqrt{\tilde{E}} / 2)|$ (even) and $\sqrt{\tilde{V}_0} = \sqrt{\tilde{E}} \cdot |\csc(\pi \sqrt{\tilde{E}} / 2)|$ (odd) with only positive derivative segments to ensure physical validity.
• Use of the resulting chart to read off allowed $\tilde{E}$ values for any given $\tilde{V}_0$, and vice versa, for arbitrary $m$, $w$, and $V_0$.
• Validation of the chart’s accuracy by comparing graphical estimates with numerical solutions for a specific electron-in-well example.

Experimental results

Research questions

• RQ1 How can the energy levels of a finite square quantum well be universally represented without recalculating for each $V_0$, $m$, and $w$?
• RQ2 What is the functional relationship between normalized potential depth $\tilde{V}_0$ and normalized energy $\tilde{E}$ for bound states?
• RQ3 How does the number of bound states depend on the system parameters, and can it be directly read from a single universal chart?
• RQ4 What is the limiting behavior of the energy levels as $V_0 \to \infty$, and how does the chart reflect this?

Key findings

• The number of bound states for a given $\tilde{V}_0$ is $N = \lfloor \sqrt{\tilde{V}_0} \rfloor + 1$, directly readable from the chart.
• The chart correctly predicts four bound states for $\tilde{V}_0 = 13$, with normalized energies $\tilde{E} \approx \{0.72, 2.89, 6.25, 10.89\}$, matching numerical solutions $\{0.72, 2.85, 6.30, 10.73\}$ within 1%.
• The infinite well limit is recovered as $\tilde{V}_0 \to \infty$, with energy levels at $\tilde{E} = k^2$, corresponding to $E = k^2 h^2 / (8m w^2)$.
• A solution with $E = V_0$ exists only when $\sqrt{\tilde{V}_0}$ is an integer, specifically $\tilde{V}_0 = (2k)^2$ for even parity and $\tilde{V}_0 = (2k+1)^2$ for odd parity.
• The chart reveals that energy quantization depends only on the product $V_0 m w^2$, not on individual parameters.