Chapter 12: Harmonic Oscillators and Other Spatially-varying Wells

Introduction

In this chapter we will consider eigenstates of potential energy functions that are spatially varying, V(x) ≠ constant. We begin with the most recognizable of these problems, that of the simple harmonic oscillator, V(x) = mω²x²/2, is perhaps the most ubiquitous potential energy function in physics. Many systems in nature exactly exhibit the harmonic oscillator's potential energy, but many more systems approximately exhibit the form of the harmonic oscillator's potential energy.¹

Sections

Section 12.1: The Classical Harmonic Oscillator.
Section 12.2: The Quantum-mechanical Harmonic Oscillator.
Section 12.3: Classical and Quantum-Mechanical Probabilities.
Section 12.4: Wave Packet Dynamics.
Section 12.5: Ramped Infinite and Finite Wells.
Section 12.6: Exploring Other Spatially-varying Wells.

Problems

Problem 12.1: Compare classical and quantum harmonic oscillator probability distributions.
Problem 12.2: A particle is in a 1-d dimensionless harmonic oscillator potential.
Problem 12.3: Two-state superpositions in the harmonic oscillator.
Problem 12.4: A particle is confined to a box with an added unknown potential energy function.
Problem 12.5: Describe the effect of the added potential energy function.
Problem 12.6: Determining the properties of half wells.
Problem 12.7: Determining the properties of half wells.

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¹A generic potential energy function, V(x), can be expanded in a Taylor series to yield

V(x) = V(x₀) + (x − x₀) dV(x)/dx|_{x =
x0}+ ((x − x₀)²/2!) d²V(x)/dx²|_{x = x0}+… (12.1)

If the original potential energy is symmetric about x = 0, we can expand about x₀ = 0 to yield

V(x) = V(x₀) + (x) dV(x)/dx|_{x = 0}+ (x²/2!) d²V(x)/dx²|_{x
= 0}+ … (12.2)

The leading non-constant term is in the form of a harmonic oscillator, and thus this potential can be approximately treated as a harmonic oscillator.