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Re: Particle in box > quantized E, particle in ? > quantized S
Posted:
Jun 14, 1996 8:53 AM


Jack Sarfatti <sarfatti@ix.netcom.com> wrote in part:
>A space of functions (mod orthogonalization) that >vanish at the two ends x=0 and x=L like the standing waves sin(pi n >x/L) n = 0, 1, 2 ... This Hilbert space has a denumerable infinity of >dimensions. These real functions form a basis or frame of reference >which allow complexvalued superpositions.
>You were wrong to claim "You got it. The complex numbers C  >and thus every space C^n, or more generally every Hilbert space  has >both a real inner product and a symplectic structure." OK, if that's >true what is the symplectic structure of the Hilbert space of wave >functions of the particle on the line segment above?
This is too easy! As John said (cut), v^w = Im<v,w>. Let f_n = sin(pi n x/L) sqrt(2/L). Then <f_n,f_m> = delta(m,n), where delta is the Kronecker delta. As a real vector space, {f_n, i f_n, n = 0, 1, 2 ...} forms a basis, so f_n ^ f_m = 0, f_n ^ i f_m = delta(m,n), and i f_n ^ i f_n = 0. So in fact {f_n, i f_n} is a symplectonormal basis (or whatever one calls such a thing). If v = (sum) v_n f_n + (sum) V_n i f_n and w = (sum) w_n f_n + (wum) W_n i f_n, then v ^ w = (sum) v_n W_n  (sum) V_n w_n. This was a straightforward calculation, incidentally.
>What is the >symplectic structure of the Fock space with basis a*^n0>?
Try it for yourself! Just remember, v^w = Im<v,w>. That's all you need to know.
 Toby toby@ugcs.caltech.edu



