Table of Contents
- Introduction
- Classical vs Quantum Oscillator
- Potential and Schrödinger Equation
- Dimensionless Variables and Rescaling
- Series Solution and Hermite Polynomials
- Energy Quantization
- Wavefunctions and Normalization
- Probability Densities and Node Structure
- Ladder Operators: Creation and Annihilation
- Algebraic Solution Using Operators
- Properties of Number States
- Uncertainty and Coherent States
- Time Evolution and Phase Space
- Quantum vs Classical Motion
- Applications Across Physics
- Conclusion
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.
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
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.
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.
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.
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: \( \frac{1}{2} \hbar \omega \)
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
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
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:
\[
\hat{H} = \hbar \omega \left( \hat{a}^\dagger \hat{a} + \frac{1}{2} \right)
\]
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:
\[
|n\rangle = \frac{(\hat{a}^\dagger)^n}{\sqrt{n!}} |0\rangle
\]
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
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:
\[
|\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle
\]
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.
14. Quantum vs Classical Motion
| Aspect | Classical Oscillator | Quantum Harmonic Oscillator |
|---|---|---|
| Energy | Continuous | Discrete levels |
| Trajectory | Deterministic path | Probabilistic distribution |
| Zero-point | Absent | \( E_0 = \frac{1}{2}\hbar\omega \) |
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
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.
