you have a pair of ladders, a+ and a- then:

- a+- = x +- iy
- thus a+- a-+ = (xx + yy) +- i[x,y]
- thus 1/2 [a+,a-] = i[x,y] = (a+a-) - 1/2{a+,a-}
- thus 1/2 {a+,a-} = (xx + yy) = (a+a-) - 1/2[a+,a-]
- [H,[x,y]] = 0
- [H,xx + yy] = 0
- [H,x] not = 0
- [H,y] not = 0
- thus [H,a+-] not = 0
- thus [H,a+-a-+] = 0
- [[a+,a-],a+-] = ka+- where [k,H] = 0

The result : if Y is an eigenfunction of {a+,a-} then so is (a+ Y) and so is (a- Y), each with different eigenvalues, shifted by the eigenvalue of [a+,a-].

This theory is invariant under [a+,a-] <-> {a+,a-}

The result of this theory (quantitative) is:

HY = EY {a+,a-} Y = n Y [a+,a-] Y = m Y [[a+,a-],a+-] = l+- a+- n,l,m commute with a+ and a- and H a+-a-= = +-(1/2[a+,a-] +- 1/2{a+,a-} ) = +- 1/2 ( m +- n ) {a+,a-}(a+ Y) = (n - 2m - l+)(a+ Y) {a+,a-}(a- Y) = (n + 2m + l-)(a- Y)

Two common cases are:

[x,y] = 1

{a+,a-} = N

this is a Harmonic Oscillator (N+1/2=H), or creation operator

causes a linear step of operator N

[x,[x,y]] = y

this is a spin or angular momentum. This system is also characterized
by [x,y]=z and xy = z/2 (where z is defined as [x,y])

This is also a representation of the spherical symmetry group.

Finally : the hard problem:

Given N, find an x and y which create raising and lowering operators
for N.

i.e. N = 2(xx + yy) , and we have the various constraints
above on x and y.

If anyone knows the answer to this problem, let me know.

Charles Bloom / cb at my domain Send Me Email

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