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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! ,

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).

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.

14 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! ,

8 Oct 2009: L̂2|a, b〉 = a|a, b〉, L̂z|a, b〉 = b|a, b〉. To find |a, b〉, we could simply proceed by looking for a suitable coordinatebasis to represent L̂2 and L̂z ... However, they do effect the projection:. L̂zL̂|a, b〉 = L̂L̂z|a, b〉 [L̂z, L̂]|a

17 Nov 2009: L = 14µ0. Fµν Fµν. where Fµν = µAν ν Aµ denotes EM field tensor, E = Ȧ iselectric field, and B = A is magnetic field. ... For a classical EM field, the modes of the cavity must satisfyboundary conditions commensurate with perfectly

17 Nov 2009: Fµν Fµν. where Fµν = µAν ν Aµ denotes EM field tensor, E = Ȧ iselectric field, and B = A is magnetic field. ... For a classical EM field, the modes of the cavity must satisfyboundary conditions commensurate with perfectly conducting