Quantum Harmonic Oscillator: A Foundation of Quantum Systems

Table of Contents

1. Introduction
2. Classical vs Quantum Oscillator
3. Potential and Schrödinger Equation
4. Dimensionless Variables and Rescaling
5. Series Solution and Hermite Polynomials
6. Energy Quantization
7. Wavefunctions and Normalization
8. Probability Densities and Node Structure
9. Ladder Operators: Creation and Annihilation
10. Algebraic Solution Using Operators
11. Properties of Number States
12. Uncertainty and Coherent States
13. Time Evolution and Phase Space
14. Quantum vs Classical Motion
15. Applications Across Physics
16. Conclusion

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1. Introduction

The quantum harmonic oscillator (QHO) is one of the most important models in quantum mechanics. Its simplicity, solvability, and wide applicability make it a cornerstone in atomic physics, quantum optics, field theory, and beyond.

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2. Classical vs Quantum Oscillator

Classical:

• Mass \( m \) in potential \( V(x) = \frac{1}{2} m \omega^2 x^2 \)
• Oscillates sinusoidally with constant amplitude and period

Quantum:

• Quantized energy levels
• Wavefunctions spread out with increasing quantum number
• Position and momentum described probabilistically

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3. Potential and Schrödinger Equation

The time-independent Schrödinger equation is:

\[
-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi(x) = E \psi(x)
\]

This differential equation can be solved exactly via power series or operator methods.

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4. Dimensionless Variables and Rescaling

Define:

\[
\xi = \sqrt{\frac{m\omega}{\hbar}} x, \quad \epsilon = \frac{2E}{\hbar \omega}
\]

Equation becomes:

\[
\frac{d^2 \psi}{d\xi^2} + (\epsilon – \xi^2)\psi = 0
\]

Solution leads to Hermite polynomials and exponential envelopes.

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5. Series Solution and Hermite Polynomials

General solution:

\[
\psi_n(\xi) = N_n H_n(\xi) e^{-\xi^2/2}
\]

• \( H_n(\xi) \): Hermite polynomial of degree \( n \)
• \( N_n \): normalization constant

Quantization arises from requirement of normalizable wavefunctions.

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6. Energy Quantization

Allowed energy levels:

\[
E_n = \hbar \omega \left( n + \frac{1}{2} \right), \quad n = 0, 1, 2, \dots
\]

• Equally spaced levels
• Zero-point energy: \( \frac{1}{2} \hbar \omega \)

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7. Wavefunctions and Normalization

\[
\psi_n(x) = \left( \frac{m\omega}{\pi \hbar} \right)^{1/4} \frac{1}{\sqrt{2^n n!}} H_n(\xi) e^{-\xi^2/2}
\]

• Even/odd parity depending on \( n \)
• Higher \( n \): more nodes and wider spread

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8. Probability Densities and Node Structure

• \( |\psi_n(x)|^2 \): probability of finding the particle at \( x \)
• \( n \) nodes for \( \psi_n(x) \)
• For large \( n \), classical turning points and envelope match classical predictions

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9. Ladder Operators: Creation and Annihilation

Define operators:

\[
\hat{a} = \frac{1}{\sqrt{2\hbar m \omega}} (m\omega \hat{x} + i\hat{p}), \quad \hat{a}^\dagger = \frac{1}{\sqrt{2\hbar m \omega}} (m\omega \hat{x} – i\hat{p})
\]

Satisfy:

\[
[\hat{a}, \hat{a}^\dagger] = 1
\]

Hamiltonian becomes:

\[
\hat{H} = \hbar \omega \left( \hat{a}^\dagger \hat{a} + \frac{1}{2} \right)
\]

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10. Algebraic Solution Using Operators

Use:

\[
\hat{a}^\dagger |n\rangle = \sqrt{n+1} |n+1\rangle, \quad \hat{a} |n\rangle = \sqrt{n} |n-1\rangle
\]

Construct higher states from ground state:

\[
|n\rangle = \frac{(\hat{a}^\dagger)^n}{\sqrt{n!}} |0\rangle
\]

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11. Properties of Number States

• \( \hat{N} = \hat{a}^\dagger \hat{a} \) gives particle number
• \( \langle x \rangle = \langle p \rangle = 0 \) in all \( |n\rangle \)
• Variances increase with \( n \), maintaining uncertainty relation

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12. Uncertainty and Coherent States

Ground state:

\[
\Delta x \Delta p = \frac{\hbar}{2}
\]

Coherent states minimize uncertainty like ground state but exhibit classical motion:

\[
|\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle
\]

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13. Time Evolution and Phase Space

Each state evolves with phase:

\[
|\psi_n(t)\rangle = e^{-i E_n t/\hbar} |\psi_n\rangle
\]

Coherent states evolve like classical oscillators in phase space — a key link between quantum and classical worlds.

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14. Quantum vs Classical Motion

Aspect	Classical Oscillator	Quantum Harmonic Oscillator
Energy	Continuous	Discrete levels
Trajectory	Deterministic path	Probabilistic distribution
Zero-point	Absent	\( E_0 = \frac{1}{2}\hbar\omega \)

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15. Applications Across Physics

• Vibrational modes in molecules
• Phonons in solid-state systems
• Quantum optics and laser theory
• Field quantization in quantum electrodynamics
• Trapped ions and cavity QED

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16. Conclusion

The quantum harmonic oscillator is a model of exceptional importance. Its solvability and elegant algebraic structure provide a basis for many advanced theories. From foundational quantum mechanics to cutting-edge technologies, mastering this system is essential for any quantum physicist or engineer.

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