Module 8: Summary, Cheat Sheet & Further Reading
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β Module 7 | β Course Index
1 π Congratulations β Youβve Completed the QHO Course!
You have journeyed from a classical spring to the quantum foundations of LIGO. Every equation was derived from first principles. You now understand why this system is called the most important in all of physics.
βThe harmonic oscillator is the drosophila of quantum mechanics.β β C. Cohen-Tannoudji
2 Complete Reference: All Key Results
2.1 The Fundamentals
2.1.1 Classical SHO
\[\ddot{x} + \omega^2 x = 0, \quad x(t) = A\cos(\omega t + \phi), \quad E = \tfrac{1}{2}m\omega^2 A^2\]
2.1.2 The QHO Hamiltonian (three forms)
\[\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\hat{x}^2 = \hbar\omega\!\left(\hat{N} + \tfrac{1}{2}\right) = \hbar\omega\!\left(\hat{a}^\dagger\hat{a} + \tfrac{1}{2}\right)\]
2.1.3 Energy Spectrum
\[E_n = \hbar\omega\!\left(n + \tfrac{1}{2}\right), \quad n = 0,1,2,\ldots \quad \text{(equally spaced, zero-point energy } E_0 = \tfrac{1}{2}\hbar\omega\text{)}\]
2.2 Ladder Operators
2.2.1 Definitions and Inversions
\[\hat{a} = \sqrt{\frac{m\omega}{2\hbar}}\!\left(\hat{x} + \frac{i\hat{p}}{m\omega}\right), \quad \hat{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}}\!\left(\hat{x} - \frac{i\hat{p}}{m\omega}\right)\] \[\hat{x} = \frac{x_0}{\sqrt{2}}(\hat{a} + \hat{a}^\dagger), \quad \hat{p} = \frac{i\hbar}{\sqrt{2}x_0}(\hat{a}^\dagger - \hat{a}), \quad x_0 = \sqrt{\frac{\hbar}{m\omega}}\]
2.2.2 Commutators
\[[\hat{a},\hat{a}^\dagger] = 1, \quad [\hat{N},\hat{a}] = -\hat{a}, \quad [\hat{N},\hat{a}^\dagger] = \hat{a}^\dagger\] \[[\hat{H},\hat{a}] = -\hbar\omega\hat{a}, \quad [\hat{H},\hat{a}^\dagger] = +\hbar\omega\hat{a}^\dagger\]
2.2.3 Action on Fock States
\[\hat{a}|n\rangle = \sqrt{n}\,|n-1\rangle, \quad \hat{a}^\dagger|n\rangle = \sqrt{n+1}\,|n+1\rangle, \quad |n\rangle = \frac{(\hat{a}^\dagger)^n}{\sqrt{n!}}|0\rangle\]
2.3 Wavefunctions
2.3.1 General Formula
\[\psi_n(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\frac{1}{\sqrt{2^n n!}} H_n\!\left(\frac{x}{x_0}\right)e^{-x^2/2x_0^2}\] - Parity: \(\psi_n(-x) = (-1)^n\psi_n(x)\) - Nodes: \(\psi_n\) has exactly \(n\) zeros - Ground state: \(\psi_0(x) = (\pi x_0^2)^{-1/4}\,e^{-x^2/2x_0^2}\) (Gaussian)
2.3.2 Expectation Values in \(|n\rangle\)
\[\langle\hat{x}\rangle = 0, \quad \langle\hat{p}\rangle = 0, \quad \langle\hat{x}^2\rangle = \left(n+\tfrac{1}{2}\right)x_0^2, \quad \langle\hat{p}^2\rangle = \left(n+\tfrac{1}{2}\right)\frac{\hbar^2}{x_0^2}\] \[\Delta x\,\Delta p = \left(n + \tfrac{1}{2}\right)\hbar \geq \frac{\hbar}{2}, \quad \langle T\rangle = \langle V\rangle = \frac{E_n}{2}\]
2.4 Time Evolution
2.4.1 Time-Evolved States
\[|n,t\rangle = e^{-i(n+1/2)\omega t}|n\rangle\] \[|\Psi(t)\rangle = \sum_n c_n e^{-i(n+1/2)\omega t}|n\rangle, \quad \text{Period } T = \frac{2\pi}{\omega}\]
2.4.2 Heisenberg Picture / Ehrenfest
\[\hat{a}(t) = e^{-i\omega t}\hat{a}(0), \quad \hat{x}(t) = x_0\cos\omega t\,\hat{x}(0) + \frac{\sin\omega t}{m\omega}\hat{p}(0)\] \[\frac{d^2}{dt^2}\langle\hat{x}\rangle = -\omega^2\langle\hat{x}\rangle \quad \text{(classical EOM for any quantum state)}\]
2.5 Coherent States
2.5.1 Definition and Expansion
\[\hat{a}|\alpha\rangle = \alpha|\alpha\rangle, \quad |\alpha\rangle = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle = \hat{D}(\alpha)|0\rangle\] \[\hat{D}(\alpha) = e^{\alpha\hat{a}^\dagger - \alpha^*\hat{a}} = e^{-|\alpha|^2/2}e^{\alpha\hat{a}^\dagger}e^{-\alpha^*\hat{a}}\]
2.5.2 Properties
\[\langle\hat{x}\rangle = \sqrt{2}x_0\,\text{Re}(\alpha), \quad \langle\hat{p}\rangle = \frac{\sqrt{2}\hbar}{x_0}\,\text{Im}(\alpha)\] \[\Delta x = \frac{x_0}{\sqrt{2}}, \quad \Delta p = \frac{\hbar}{\sqrt{2}x_0}, \quad \Delta x\,\Delta p = \frac{\hbar}{2}\] \[P(n) = e^{-|\alpha|^2}\frac{|\alpha|^{2n}}{n!} \quad \text{(Poisson)}, \quad \bar{n} = |\alpha|^2, \quad \Delta N = |\alpha|\] \[|\alpha(t)\rangle = e^{-i\omega t/2}|\alpha e^{-i\omega t}\rangle, \quad |\langle\beta|\alpha\rangle|^2 = e^{-|\alpha-\beta|^2}\] \[\frac{1}{\pi}\int|\alpha\rangle\langle\alpha|\,d^2\alpha = \hat{I} \quad \text{(over-completeness)}\]
2.6 Squeezed States
2.6.1 Squeeze Operator and Action
\[\hat{S}(\xi) = \exp\!\left(\tfrac{1}{2}\xi^*\hat{a}^2 - \tfrac{1}{2}\xi\hat{a}^{\dagger 2}\right), \quad \xi = re^{i\theta}\] \[\hat{S}^\dagger(r)\hat{a}\hat{S}(r) = \hat{a}\cosh r - \hat{a}^\dagger\sinh r \quad \text{(Bogoliubov)}\]
2.6.2 Squeezed Vacuum and Coherent Squeezed States
\[|0,r\rangle = \hat{S}(r)|0\rangle, \quad |\alpha,r\rangle = \hat{D}(\alpha)\hat{S}(r)|0\rangle\] \[\Delta x = \frac{x_0}{\sqrt{2}}e^{-r}, \quad \Delta p = \frac{\hbar}{\sqrt{2}x_0}e^{+r}, \quad \Delta x\,\Delta p = \frac{\hbar}{2}\] \[\langle\hat{N}\rangle_{\text{sq.vac}} = \sinh^2 r, \quad Q < 0 \text{ (sub-Poissonian, non-classical)}\]
3 Download the Cheat Sheet
3.1 π QHO Cheat Sheet (PDF)
A print-ready, two-page LaTeX-compiled cheat sheet with every formula from this course β ready for your exam, your desk, or your soul.
β¬οΈ Download PDF Cheat Sheet
LaTeX source also available: qho_cheatsheet.tex
4 Further Reading
4.0.1 π Griffiths β Introduction to Quantum Mechanics (3rd ed.)
Chapter 2, Section 2.3: The harmonic oscillator β best first introduction. Clear, pedagogical, lots of examples.
Best for: Beginners and undergraduates. The analytic (Hermite polynomial) method is done very clearly.
4.0.2 π Shankar β Principles of Quantum Mechanics (2nd ed.)
Chapter 7: The harmonic oscillator β the definitive algebraic treatment. Exercise 7.4.9 introduces coherent states.
Best for: Anyone who wants to understand the why behind the algebra. The notation is clean and the derivations are rigorous.
4.0.3 π Cohen-Tannoudji, Diu & LaloΓ« β Quantum Mechanics (Vol. 1 & 2)
Chapter V: The harmonic oscillator β extremely systematic. Complement GV covers coherent states.
Best for: The most complete treatment. Read this when you want to make sure you havenβt missed anything.
4.0.4 π Walls & Milburn β Quantum Optics (2nd ed.)
Chapter 2-3: Squeezed states, coherent states, phase space methods.
Best for: Graduate students going into quantum optics or quantum information. The definitive reference for squeezed states.
4.0.5 π Gerry & Knight β Introductory Quantum Optics
Chapter 3: Coherent states and squeezed states with quantum optical applications.
Best for: A gentler introduction to quantum optics than Walls & Milburn.
5 What Comes Next?
Having mastered the QHO, you are now ready for:
- Quantum Field Theory: The QHO is the building block of every quantum field β scalar, electromagnetic, fermionic.
- Quantum Optics: Glauber-Sudarshan P-function, optical phase space, quantum state tomography.
- Circuit QED: Superconducting qubits are quantized LC circuits (QHOs), the platform of modern quantum computers.
- Many-Body Physics: Phonons, magnons, plasmons β all are collections of QHOs.
- Perturbation Theory: Anharmonic corrections to the QHO (\(\lambda x^4\) potential) are the first test of perturbation theory.