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# Chapter 12: Harmonic Oscillators and Other Spatially-varying Wells

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*) = *m*ω^{2}*x*^{2}/2, is perhaps the most ubiquitous potential energy function in physics. Several 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}

# Table of Contents

## 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.

## Alternate Visualizations

- Section 12.3: Classical and Quantum-Mechanical Probabilities.
- Section 12.4: Wave Packet Dynamics.
- Problem 12.3: Two-state superpositions in the harmonic oscillator.

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

*V*(*x*) = *V*(*x*_{0}) + (*x* − *x*_{0}) *dV*(x)/*dx*|_{x = x0 }+ ((*x* − *x*_{0})^{2}/2!) *d*^{2}*V*(*x*)/*dx*^{2}|_{x = x0}+…

If the original potential energy is symmetric about *x* = 0, we can expand about *x*_{0} = 0 to yield

V(*x*) = V(*x*_{0}) + (*x*) *dV*(*x*)/*dx*|_{x = 0 }+ (*x*^{2}/2!) *d*^{2}*V*(*x*)/*dx*^{2}|_{x = 0 }+ …

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.