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Topic: Particle in box --> quantized E, particle in ? --> quantized S
Replies: 1   Last Post: Jun 14, 1996 8:53 AM

 Toby Bartels Posts: 206 Registered: 12/6/04
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 complex-valued 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*^n|0>?

Try it for yourself!
Just remember, v^w = Im<v,w>. That's all you need to know.

-- Toby
toby@ugcs.caltech.edu