Question

5 Uncertainty A: Find the Heisenberg uncertainty relationship between the potential energy V and kinetic energy T for a particle in a harmonic oscillator potential: Note: this is not a particularly interesting uncertainty relationship, but there are only so many interesting examples.

Answer 1

Given data: For Harmonic oscillator potential V(x) = 1/2 m w^2 x^2 Kinetic energy T = p^2/2m {sigma V^2 = 1/2 mw^2 sigma x^2 sigma T^2 = 1/2 m sigma p^2 So we find sigma x^2 = squareroot (x)^2 - (x)^2 sigma p^2 = squareroot < p >^2 - < p >^2 For Harmonic oscillator in Dieral mechanier (matrix) {a = squareroot mw/2h (x + ip/mw) with [a^-1, a^* *] = 1 a^t = mw/2h (x - ip^-1/mw) x = squareroot h/2 mw (a^t + - a) p = + i squareroot h mw/2 (a^t - a) \r\n x^2 = h/2mw (a^t + a)(a^t + a) = h/2mw (a^t^2 + a^2 + a a + a a^t)downarrow Ha^+ a using (*) x^2 = h/2mw (a^t^2 + a^2 + 2a^t a + 1) p^2 = (hmw/2)(a^t - a)(a^t - a) = hmw/2 [-(a^t^2 + a^2) + 2 a'^a + 1] Now < x > = < n/x/n > = squareroot h/2mw < n/(a + a^t)/n > since a'/n > = squareroot n/n - 1 >