What Is The Mathematical Formulation To Derive The Asymptotic Behavior Of The Wave Function Of A Particle In A One-dimensional Infinite Square Well Potential As The Energy Of The Particle Approaches The Continuum Limit, And How Can This Be Used To Explain The Transition From Discrete To Continuous Energy Spectra In Quantum Systems?

To analyze the transition from discrete to continuous energy spectra in quantum systems, consider the infinite square well as the well width ${ L }$ approaches infinity. Initially, the wave functions are sinusoidal and discrete:

$$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$$

with quantized energies:

$$E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}.$$

As ${ L \to \infty }$, the energy levels become densely packed, and the wave functions approximate plane waves:

$$\psi(k, x) \approx \frac{1}{\sqrt{2\pi}} e^{ikx},$$

where ${ k = \frac{n\pi}{L} }$ becomes a continuous variable. This transition illustrates how discrete energy levels merge into a continuous spectrum, characteristic of free particles.

Step-by-Step Explanation and Answer:

1. Infinite Square Well Setup:

   • The potential is zero inside ${ 0 < x < L }$ and infinite outside.
   • The Schrödinger equation inside the well: ${ -\frac{\hbar^2}{2m} \psi''(x) = E \psi(x) }$.
   • Solutions: ${ \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right) }$, with ${ E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} }$.

2. Asymptotic Behavior as ${ L \to \infty }$:

   • Define wave number ${ k = \frac{n\pi}{L} }$, which remains finite as ${ n }$ and ${ L }$ increase.
   • The wave function becomes: ${ \psi_n(x) \approx \sqrt{\frac{2}{L}} \sin(kx) }$.
   • For large ${ L }$, ${ \sqrt{\frac{2}{L}} \sin(kx) }$ approximates ${ \frac{1}{\sqrt{2\pi}} e^{ikx} }$ with delta function normalization.

3. Energy Spectrum Transition:

   • Discrete energies ${ E_n }$ become densely packed as ${ L \to \infty }$, approaching a continuum.
   • The energy levels merge into a continuous spectrum ${ E = \frac{\hbar^2 k^2}{2m} }$.

4. Conclusion:

   • The transition from discrete to continuous energy spectra occurs as the system's size increases, allowing wave functions to behave like free particles with continuous energies.

Final Answer:

The mathematical formulation involves taking the limit of the infinite square well as its width ${ L }$ approaches infinity. In this limit, the discrete energy levels ${ E_n }$ become densely packed, and the wave functions ${ \psi_n(x) }$ approximate plane waves ${ e^{ikx} }$ with continuous wave numbers ${ k }$. This transition explains how a quantum system moves from a discrete to a continuous energy spectrum as it approaches the continuum limit.

Answer:

The transition from discrete to continuous energy spectra occurs as the width ${ L }$ of the infinite square well approaches infinity. Mathematically, the wave functions evolve from standing waves ${ \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right) }$ to plane waves ${ \psi(k, x) \approx \frac{1}{\sqrt{2\pi}} e^{ikx} }$, where ${ k = \frac{n\pi}{L} }$ becomes a continuous variable. Consequently, the discrete energy levels ${ E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} }$ merge into a continuous spectrum ${ E = \frac{\hbar^2 k^2}{2m} }$, illustrating the transition from discrete to continuous energies in quantum systems.

\boxed{ \text{As } L \to \infty, \text{ the wave functions become plane waves and energy levels form a continuum.} }