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The Heisenberg Uncertainty Relations — R. Shankar
Physics — Quantum Mechanics · harmonic-oscillator-path-integral
Problem Show that for the harmonic oscillator with
L = 1 2 m x ˙ 2 − 1 2 m ω 2 x 2 \mathscr{L} = \tfrac{1}{2}m\dot{x}^2 - \tfrac{1}{2}m\omega^2 x^2 L = 2 1 m x ˙ 2 − 2 1 m ω 2 x 2 U ( x , t ; x ′ ) = A ( t ) exp { i m ω 2 ℏ sin ω t [ ( x 2 + x ′ 2 ) cos ω t − 2 x x ′ ] } U(x, t; x') = A(t) \exp\left\{\frac{im\omega}{2\hbar \sin\omega t}\left[(x^2 + x'^2)\cos\omega t - 2xx'\right]\right\} U ( x , t ; x ′ ) = A ( t ) exp { 2ℏ sin ω t imω [ ( x 2 + x ′2 ) cos ω t − 2 x x ′ ] } A ( t ) A(t) A ( t ) Browse the problem library