Introduction to Quantum Mechanics · Problem 3.42

Formalism — David J. Griffiths

Physics — Quantum Mechanics · coherent_states

Problem

Coherent states of the harmonic oscillator. Among the stationary states of the harmonic oscillator (Equation 2.68) only n=0n = 0 hits the uncertainty limit (σxσp=ℏ/2\sigma_x \sigma_p = \hbar/2); in general, σxσp=(2n+1)ℏ/2\sigma_x \sigma_p = (2n+1)\hbar/2, as you found in Problem 2.12. But certain linear combinations (known as coherent states) also minimize the uncertainty product. They are (as it turns out) *eigenfunctions of the lowering operator*: a−∣α⟩=α∣α⟩a_- |\alpha\rangle = \alpha |\alpha\rangle (the eigenvalue α\alpha can be any complex number). (a) Calculate ⟨x⟩\langle x \rangle, ⟨x2⟩\langle x^2 \rangle, ⟨p⟩\langle p \rangle, ⟨p2⟩\langle p^2 \rangle in the state ∣α⟩|\alpha\rangle. *Hint:* Use the technique in Example 2.5, and remember that a+a_+ is the hermitian conjugate of a−a_-. Do *not* assume α\alpha is real. (b) Find σx\sigma_x and σp\sigma_p; show that σxσp=ℏ/2\sigma_x \sigma_p = \hbar/2. (c) Like any other wave function, a coherent state can be expanded in terms of energy eigenstates: ∣α⟩=∑n=0∞cn∣n⟩.|\alpha\rangle = \sum_{n=0}^{\infty} c_n |n\rangle. Show that the expansion coefficients are cn=αnn!c0.c_n = \frac{\alpha^n}{\sqrt{n!}} c_0. (d) Determine c0c_0 by normalizing ∣α⟩|\alpha\rangle. *Answer:* exp⁡(−∣α∣2/2)\exp\left(-|\alpha|^2/2\right). (e) Now put in the time dependence: ∣n⟩→e−iEnt/ℏ∣n⟩,|n\rangle \to e^{-iE_n t/\hbar}|n\rangle, and show that ∣α(t)⟩|\alpha(t)\rangle remains an eigenstate of a−a_-, but the eigenvalue evolves in time: α(t)=e−iωtα.\alpha(t) = e^{-i\omega t} \alpha. So a coherent state *stays* coherent, and continues to minimize the uncertainty product. (f) Based on your answers to (a), (b), and (e), find ⟨x⟩\langle x \rangle and σx\sigma_x as functions of time. It helps if you write the complex number α\alpha as α=Cmω2ℏeiϕ\alpha = C\sqrt{\frac{m\omega}{2\hbar}} e^{i\phi} for real numbers CC and ϕ\phi. *Comment:* In a sense, coherent states behave quasi-classically. (g) Is the ground state (∣n=0⟩|n = 0\rangle) itself a coherent state? If so, what is the eigenvalue?
Intuitive explanation

Coherent states are special quantum superpositions that saturate the uncertainty principle—they behave like classical oscillators with minimal quantum fuzziness. The key insight is that these states are eigenfunctions of the lowering operator a−a_-, which lets you use ladder-operator algebra to compute expectation values directly without expanding in the energy basis. This eigenvalue property persists under time evolution, so coherent states remain "coherent" (uncertainty-minimizing) forever, making them the closest quantum analog to a classical harmonic oscillator.

Intuitive explanation

Coherent states are special quantum superpositions that saturate the uncertainty principle—they behave like classical oscillators with minimal quantum fuzziness. The key insight is that these states are eigenfunctions of the lowering operator a−a_-, which lets you use ladder-operator algebra to compute expectation values directly without expanding in the energy basis. This eigenvalue property persists under time evolution, so coherent states remain "coherent" (uncertainty-minimizing) forever, making them the closest quantum analog to a classical harmonic oscillator.

Related theory

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