Question

a) Discuss why the de Broglie wavelength λ corresponding to a momentum p (p wavenumber given by k # 2n/A) leads to a representation of p by the operator p as (h/) (d/dx) hk, where k is the b) Using theoperao orm of p given in part a, show that,pih c) The total energy of a simple harmonic oscillator of mass M and spring constant K can be written as H- p2/M + ke . If the mass is displaced from the equilibrium point, it also oscillates at frequency ω = (K/M)1/2 We also defined operators a, and a given by a+= (-ip +mux)/(2hmu)1/2 and a.s(ip-max)/(2hma)1/2. Using the commutator for and p, show that [â.,a]- -1 Using these commutators we also showed that H be written as H (aa+-%) h ii) (a,a + %) hu. Show that H may also i Given this information, show that if is an eigenfunction of H with corresponding eigenvalue E (that is, Hys Ey), then y of H with eigenvalue E +hw a-V is proportional to another eigenfunction

Answer 1

Let psi (x) = A e^(kx - wt) partial differential psi/partial differential x = I k psi: p = hstrok k doublerightarrow k = p/hstrok partial differential psi/partial differential x = i p psi/hstrok doublerightarrow p psi = hstrok/i partial differential/partial differential x psi therefore p^cap = hstrok/i partial differential/partial differential x [x^cap, p^cap] = [x^cap, hstrok/i partial differential/partial differential x] psi = x hstrok/i partial differential psi/partial differential x - hstrok/i partial differential/partial differential x (x, psi) = x hstrok/i middot partial differential psi/partial differential x - hstrok/i x partial differential psi/partial differential x - hstrok /i psi [c^cap, p^cap] = -hstrok/i = i hstrok a^cap_+ = -i p^cap + mw x^cap/(2 hstrok mw)^1/2: a^cap = (I p^cap + mw x^cap)/(2 hstrok mw)^1/2 [a^cap_+, a^cap] = 1/2 hstrok mw [(-ip^cap + mwx^cap), (I p^cap + mw x^cap)] = 1/2 hstrok {[p^cap, p^cap] + imw [x^cap, p^cap] - imw [p^cap, x^cap] + (mw)^2
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