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From: Darwin123 on 29 Mar 2010 16:16
On Mar 29, 1:56 pm, Darwin123 <drosen0...(a)yahoo.com> wrote:
> On Mar 27, 7:55 pm, "Jay R. Yablon" <jyab...(a)nycap.rr.com> wrote:
>
> > "Jay R. Yablon" <jyab...(a)nycap.rr.com> wrote in messagenews:8170i5Fep2U1(a)mid.individual.net...
> >  H_0=P^2+V(Q) = (P+i sqrt(V(Q))(P-i sqrt(V(Q))   (3)
>
> > Don't hold me to this; I am simply exploring.
>
>     As long as you don't hold me to my criticisms, as I am only trying
> to help you out. So far, I have two problems with this.

I told you not to hold me too it. I made a slight boo-boo. I forgot
the factor of 1/2 in front of the k. So if,
w^2=k/u
where k is the spring constant and "u" is the mass, then,
H=(1/2u)P^2+(k/2)Q^2
Let us use a harmonic oscillator with a frequency 2w. Thus k goes to
4k. So,
this case:
H'=(1/2u)P^2+2kQ^2
One can map the eigenvalues of H onto the eigenevalues of H'.
One can make a transformation on P and Q to find a P' and Q' such
that:
H'=(1/2u)P'^2+(k/2)Q'^2
The factor of 1/2 is unimportant for the point I was trying to
make. However, I felt obliged to correct myself.
The point. One can redefine creation operators "trans(a)" to
"trans(a')", annihilation operators "a", position operators P to P'
and momentum operators Q to Q' to satisfy the new Hamiltonian. One
problem is mapped to the other.
Physicists often solve a complex problem by "mapping" it onto a
simpler problem. I think the harmonic oscillator problem is important
only because it is one of the simplest problems. Many complicated
problems are mapped onto this problem.
Another way to solve a complex problem is to subtract the results
of a simple problem from the complex solution. The difference may be a
simple solution. This sort of thing is called "perturbation theory."
Again, the importance of the harmonic oscillator problem may be based
on the fact that the solution of the harmonic oscillator is relatively
simple.

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