Re: angular momentum raising lowering ladder operators
- From: "jambaugh" <ego@xxxxxxxxxxxxxxx>
- Date: Tue, 26 Dec 2006 20:37:16 +0000 (UTC)
corpsicle wrote:
hello,
are there any angular momentum operators for, say, a hydrogen atom,
that raise and lower the angular quantum number l (ell) up and down,
as opposed to the Lx + iLy type operators that move the magnetic
quantum number m, "sideways" as it were but leave l unchanged?
so what i guess i'm asking is there some kind of differential
equation that generates the l+1'th spherical harmonic from the l'th,
or, or, something!
If I understand your question correctly you are asking for the
representation of the orbital angular momentum operators in terms of x
and d/dx (or P).
Note that iLx = yd/dz - zd/dy and iLy = zd/dx - xd/dz,
(I may be off by a sign, also differential operators are of course
partial derivatives).
Hence the ladder operator:
L+ = z d/dx - x d/dz - iyd/dz + izd/dy
= z(d/dx + i d/dy) - (x + iy)d/dz
= z d/d(x-iy) - (x+iy)d/dz
[Note: dx/diy = -id/dy]
This is for raising and lowering angular momentum in the x-y plane
centered about the origin.
To act about a point (a,b,c) replace x with (x-a) y with (y-b) and z
with (z-c).
Double check it by acting on spherical harmonics written in rectangular
coordinates.
[Multiplied by a suitable power of r so as to give simplest form.]
e.g. something like:
Y(x,y,z) = z^u(x+iy)^v(x-iy)^w
Lz = i(yd/dx - xd/dy)
Lz[Y] = (v-w)Y
etc...
Regards,
James
.
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- From: corpsicle
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