Tru Physics

Chapter 19: The Harmonic Oscillator

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19.1 Introduction to the Harmonic Oscillator

The harmonic oscillator is a fundamental model in physics that describes the behavior of a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. In quantum mechanics, the harmonic oscillator is a key model used to understand a variety of physical systems, such as the motion of atoms in a molecule or the behavior of a mass attached to a spring.

19.2 The Quantum Harmonic Oscillator Potential

The potential energy function for a harmonic oscillator is given by:

V(x) = \dfrac{1}{2}kx^2

where k is the spring constant and x is the displacement from equilibrium. In quantum mechanics, we use the time-independent Schrödinger equation to analyze the behavior of a particle in a harmonic oscillator potential:

-\dfrac{\hbar^2}{2m}\dfrac{d^2\psi}{dx^2} + \dfrac{1}{2}kx^2\psi(x) = E\psi(x)

19.3 Energy Levels and Wavefunctions

Solving the time-independent Schrödinger equation for the quantum harmonic oscillator yields a set of quantized energy levels and corresponding wavefunctions. The energy levels are given by:

E_n = \hbar\omega_0\left(n+\dfrac{1}{2}\right)

where n is a non-negative integer, and \omega_0 = \sqrt{\frac{k}{m}} is the angular frequency of the oscillator. The corresponding wavefunctions are the product of a Gaussian function and Hermite polynomials:

\psi_n(x) = \dfrac{1}{\sqrt{2^nn!\sqrt{\pi}a}}e^{-\dfrac{x^2}{2a^2}}H_n\left(\dfrac{x}{a}\right)

where a = \sqrt{\frac{\hbar}{m\omega_0}}, and H_n(x) are the Hermite polynomials.

19.4 Zero-Point Energy

An important feature of the quantum harmonic oscillator is the existence of zero-point energy. Even in the ground state (the lowest energy state, with n=0), the harmonic oscillator has a non-zero energy:

E_0 = \dfrac{1}{2}\hbar\omega_0

This result demonstrates that a quantum harmonic oscillator can never truly be at rest, as there is always some residual energy.

19.5 Applications of the Quantum Harmonic Oscillator

The quantum harmonic oscillator is a widely applicable model in various branches of physics, including molecular physics, solid-state physics, and quantum field theory. For instance, it can describe the vibrational motion of atoms in molecules, phonons in solids, and even the behavior of particles in a quantized field.

Chapter Summary

The harmonic oscillator is an essential model in quantum mechanics that describes the behavior of systems subjected to a restoring force proportional to their displacement from equilibrium. Solving the time-independent Schrödinger equation for the quantum harmonic oscillator yields quantized energy levels and corresponding wavefunctions. Key features of the quantum harmonic oscillator include the existence of zero-point energy and the wide range of applications it has across various branches of physics.

Continue to Chapter 20: The Measurement Problem

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