Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems Quantum Mechanics: Concepts and Applications · Problem 5.14
Three-Dimensional Problems — Nouredine Zettili
Physics — Quantum Mechanics · expectation-values-lm-states
Problem (a) Show that the following expectation values between ∣ l m ⟩ |lm\rangle ∣ l m ⟩ ⟨ L ^ x ⟩ = ⟨ L ^ y ⟩ = 0 \langle \hat{L}_x \rangle = \langle \hat{L}_y \rangle = 0 ⟨ L ^ x ⟩ = ⟨ L ^ y ⟩ = 0 ⟨ L ^ x 2 ⟩ = ⟨ L ^ y 2 ⟩ = 1 2 [ l ( l + 1 ) ℏ 2 − m 2 ℏ 2 ] \langle \hat{L}_x^2 \rangle = \langle \hat{L}_y^2 \rangle = \frac{1}{2}\left[l(l+1)\hbar^2 - m^2\hbar^2\right] ⟨ L ^ x 2 ⟩ = ⟨ L ^ y 2 ⟩ = 2 1 [ l ( l + 1 ) ℏ 2 − m 2 ℏ 2 ] Δ L x Δ L y ≥ ℏ 2 m / 2 \Delta L_x \Delta L_y \geq \hbar^2 m/2 Δ L x Δ L y ≥ ℏ 2 m /2 Δ L x = ⟨ L x 2 ⟩ − ⟨ L x ⟩ 2 \Delta L_x = \sqrt{\langle L_x^2 \rangle - \langle L_x \rangle^2} Δ L x = ⟨ L x 2 ⟩ − ⟨ L x ⟩ 2 Browse the problem library