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9 Oct 2009: Show that the operators,. a =. 2z̄. z. 4. a† =. 2. z. z̄. 4. b =. 2z. z̄. 4. b† =. 2. z̄. z. 4. fulfil the commutation relations [a a†] = [b b†] = 1 ... d) Show that the angular momentum operator takes the form,. L̂z = (b†b a†a).

9 Nov 2014: form a group operation. This means that:. 1. a b = b a. ... 4. All numbers obey the associativeness law:(a b) c = a (b c).

9 Jun 2004: σxx(ω,q) acoustic attenuation & velocity shift. Acoustic attenuation in B-field [Pippard, 1957]. ... x/ρ 0. B/Bω. Dynamic modulationStatic modulation. ql = 600, E = 0.01, ωτ = 0, 20.

7 Jan 2019: d. dt(γmv) = e(E v B). ——————. (b) [unseen calculation] Suppose that B = 0, that E is constant and that att = 0 the particle has velocity v0. ... d2v. dt2=. e. mγ. dv. dtB =. (e. mγ. )2(v B) B =. (e. mγ. )2(vB2 B(v B)). In terms of the

14 Oct 2009: z̄, z̄] [z, z]) = 1 ,. and similarly [b, b†] = 1. ... z. 4. ) (z. z̄. 4. )= (a†a b†b). (e) In the coordinate representation, the condition a|0, 0〉 = 0 translates to theequation.

7 Jan 2019: rr$"B<1&u="B1"%=r.6r'[ l8?<7=([="F Q1=&1¡&u<="')?'="%:r'C1|1"%&h). ... 2:')?'="%: $"%' % ri"%$h)0A:mr|=<1&¡&,-8 "%' % ri"B<7@ <1@(h[=<1/. (

21 Jan 2018: d. dt(γmv) = e(E v B). ——————. (b) [unseen calculation] Suppose that B = 0, that E is constant and that att = 0 the particle has velocity v0. ... d2v. dt2=. e. mγ. dv. dtB =. (e. mγ. )2(v B) B =. (e. mγ. )2(vB2 B(v B)). In terms of the

21 Oct 2009: harmonic oscillator). Angular momentum: raising and lowering operators. L̂2|a, b〉 = a|a, b〉, L̂z |a, b〉 = b|a, b〉. ... L̂|a, b〉. if L̂z |a, b〉 = b|a, b〉, L̂|a, b〉 is either zero, or an eigenstate of L̂zwith eigenvalue b! ,

20 Jan 2016: m. (a, ,b ). z. M. (r, ,z). B B. (a) Show that, up to irrelevant constants and ignoring gravity, the Lagrangian forthe system is: [6]. ... 2. (b) Find the corresponding equations of motion for the particle and the bead.

7 Jan 2019: a uniform static magneticfield perpendicular to the plane, B = Bẑ (E > 0, B > 0). ... 6]. (b) Compute the behaviour of the zero-field magnetic susceptibilityχ = (φ/B)|B=0 close to the transition (both above and below).