Simple quantum systems like the particle in a box and harmonic oscillator are key to understanding quantum mechanics. They show how energy levels become quantized when particles are confined, leading to discrete energy states and unique wavefunctions.

These systems illustrate fundamental quantum principles like zero-point energy and the uncertainty principle. By studying them, we gain insights into more complex quantum systems and their behavior at the atomic and subatomic levels.

Particle in an Infinite Potential Well

Solving the Schrödinger Equation

• The one-dimensional infinite potential well consists of a particle confined between two impenetrable walls at $x = 0$ and $x = L$
• The time-independent Schrödinger equation for a particle in a one-dimensional infinite potential well is given by $-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E\psi$, where:
  • $\hbar$ is the reduced Planck's constant
  • $m$ is the mass of the particle
  • $E$ is the energy
  • $\psi$ is the wavefunction
• The potential energy $V(x)$ is:
  • Zero inside the well ($0 < x < L$)
  • Infinite outside the well ($x \leq 0$ and $x \geq L$)
• The boundary conditions for the wavefunction are $\psi(0) = \psi(L) = 0$, as the wavefunction must vanish at the walls of the infinite potential well

Energy Levels and Wavefunctions

• The allowed energy levels for a particle in a one-dimensional infinite potential well are quantized and given by $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$, where:
  • $n$ is a positive integer ($n = 1, 2, 3, ...$)
  • The energy levels depend on the mass of the particle ($m$) and the size of the well ($L$)
• The corresponding wavefunctions for each energy level are given by $\psi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n\pi x}{L})$, where:
  • $n$ is a positive integer ($n = 1, 2, 3, ...$)
  • The wavefunctions are sinusoidal functions with nodes at the walls of the well
• The probability density for finding the particle at a specific position $x$ within the well is given by $|\psi_n(x)|^2$, which exhibits:
  • Nodes: points of zero probability
  • Antinodes: points of maximum probability

Quantum Harmonic Oscillator

Potential Energy and Schrödinger Equation

• The quantum harmonic oscillator describes a particle subject to a quadratic potential energy, such as a mass attached to a spring obeying Hooke's law
• The potential energy for a quantum harmonic oscillator is given by $V(x) = \frac{1}{2}kx^2$, where:
  • $k$ is the spring constant
  • $x$ is the displacement from the equilibrium position
• The time-independent Schrödinger equation for a quantum harmonic oscillator is given by $-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + \frac{1}{2}kx^2\psi = E\psi$, where:
  • $\hbar$ is the reduced Planck's constant
  • $m$ is the mass of the particle
  • $E$ is the energy
  • $\psi$ is the wavefunction

Energy Levels and Wavefunctions

• The allowed energy levels for a quantum harmonic oscillator are quantized and given by $E_n = (n + \frac{1}{2})\hbar\omega$, where:
  • $n$ is a non-negative integer ($n = 0, 1, 2, ...$)
  • $\omega$ is the angular frequency of the oscillator, given by $\omega = \sqrt{\frac{k}{m}}$
• The ground state energy ($n = 0$) of a quantum harmonic oscillator is non-zero and given by $E_0 = \frac{1}{2}\hbar\omega$, known as the zero-point energy
• The wavefunctions for a quantum harmonic oscillator are given by $\psi_n(x) = \frac{1}{\sqrt{2^n n!}} (\frac{m\omega}{\pi\hbar})^{\frac{1}{4}} \exp(-\frac{m\omega x^2}{2\hbar}) H_n(\sqrt{\frac{m\omega}{\hbar}} x)$, where:
  • $H_n(x)$ are the Hermite polynomials
  • The wavefunctions are Gaussian-like functions modulated by Hermite polynomials
• The probability density for finding the particle at a specific position $x$ is given by $|\psi_n(x)|^2$, which exhibits a Gaussian-like distribution centered around the equilibrium position

Particle in a Box vs Harmonic Oscillator

Similarities

• Both the particle in a box and the harmonic oscillator are fundamental model systems in quantum mechanics
• Both systems exhibit quantized energy levels and wavefunctions
• The ground state energy is non-zero for both systems

Differences

• The particle in a box is confined within a finite region ($0 < x < L$) by infinite potential walls, while the harmonic oscillator is subject to a quadratic potential energy $V(x) = \frac{1}{2}kx^2$ that extends to infinity
• The energy levels of a particle in a box are proportional to $n^2$, where $n$ is a positive integer, while the energy levels of a harmonic oscillator are proportional to $(n + \frac{1}{2})$, where $n$ is a non-negative integer
• The ground state energy of a particle in a box depends on the size of the box ($L$), while the ground state energy of a harmonic oscillator depends on the angular frequency ($\omega$) of the oscillator
• The wavefunctions of a particle in a box are sinusoidal functions with nodes at the walls of the box, while the wavefunctions of a harmonic oscillator are Gaussian-like functions modulated by Hermite polynomials
• The probability density for a particle in a box exhibits nodes and antinodes within the box, while the probability density for a harmonic oscillator is centered around the equilibrium position and decays exponentially away from it
• The energy level spacing for a particle in a box decreases with increasing box size ($L$), while the energy level spacing for a harmonic oscillator remains constant and is determined by the angular frequency ($\omega$)

Quantization and Zero-Point Energy

Quantization

• Quantization is a fundamental concept in quantum mechanics, stating that certain physical quantities, such as energy and angular momentum, can only take on discrete values rather than a continuous spectrum
• In the case of a particle in a box, the allowed energy levels are quantized and given by $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$, where $n$ is a positive integer, demonstrating the concept of energy quantization
• Similarly, for a harmonic oscillator, the allowed energy levels are quantized and given by $E_n = (n + \frac{1}{2})\hbar\omega$, where $n$ is a non-negative integer, showcasing energy quantization

Zero-Point Energy

• Zero-point energy is the lowest possible energy state of a quantum system, which is non-zero even when the system is at its ground state ($n = 0$)
• In the case of a harmonic oscillator, the zero-point energy is given by $E_0 = \frac{1}{2}\hbar\omega$, which is a direct consequence of the Heisenberg uncertainty principle and the quantization of energy levels
• The presence of zero-point energy has important implications in various physical systems, such as:
  • The stability of molecules
  • The behavior of phonons in solids
  • The Casimir effect (attractive force between two uncharged, conducting plates due to quantum fluctuations)
• The concept of quantization and zero-point energy can be applied to other simple quantum systems to understand their energy levels and quantum behavior, such as:
  • A particle in a finite potential well
  • A particle in a two-dimensional box
  • A rigid rotor (a model for molecular rotations)