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Angular momentum (Read 5284 times)
Gerrit-Jan Linker
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Angular momentum
10.02.08 at 21:47:24
 
Angular momentum
 
Angular momentum L = r x p =
 
| i j k |
| x y z  |
| px py pz |
= (xpz - zpy)i - (xpz - zpx)j + (xpy - ypx)k = Lxi + Lyj + Lzk
 
where:
Position of the particle r = xi +yj +zk
i, j and k are the orthogonal unit vectors
Linear momentum p = pxi + pyj + pzk
px = (1/i)δ/δx, py = (1/i)δ/δy, pz = (1/i)δ/δz
 
Angular momentum can be expressed in terms if its components:
L = Lxi + Lyj + Lzk
 
Lx= (xpz - zpy)
Ly= - (xpz - zpx) = (zpx - xpz)
Lz= (xpy - ypx)
 
The magnitude of the angular momentum L is related to its components by the normal expression for constructing the magnitude of a vector:  
L2 = L2x + L2y + L2z
 
The components of L do not commute. This means that they cannot be determined simultaneously (Heisenberg's indeterminate principle applies):
[Lx,Ly]=iLz
[Lx,Lz]=iLx
[Lz,Lx]=iLy
 
The components of L do commute with the operator corresponding to the the square of the angular momentum:
[Lx,L2]=0
[Ly,L2]=0
[Lz,L2]=0
 
Notation:
For orbital angular momenta the notation is: l and ml (integral quantum numbers)
For internal angular momentum the notation is: s ms (possibly half integral quantum numbers)
For general disucssions the notation is: j and mj
 
Shift operators
L+ = Lx + iLy : the raising operator ; step up
L- = Lx - iLy : the lowering operator ; step down
 
The shift operators are the adjoint of eachother:
L+ = L-
L- = L+
 
See also:
Angular momentum
http://en.wikipedia.org/wiki/Angular_momentum
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« Last Edit: 14.02.08 at 08:06:26 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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Re: Angular momentum
Reply #1 - 11.02.08 at 21:06:08
 
Derivation of Lz(L+|Ylm>) = (m+1)(L+|Ylm>),  
the functions L+|Ylm> are eigenfunctions of Lz with eigenvalues (m+1)
 
We will use the following:
The definition: L+ = L+ + i Ly
The commutators:
[Ly,Lz] = LyLz - LzLy = iLx , it follows that LzLy = LyLz - iLx
[Lz,Lx] = LzLx - LxLz = iLy , it follows that LzLx = iLy + LxLz
The eigenvalue equation: Lz|Ylm> = m|Ylm>
 
 
Now, the derivation:
LzL+ = Lz(Lx + iLy)  
      = LzLx + iLzLy (Use the expressions for LzLx and LzLy derived from the commutators)
      = iLy + LxLz + i(LyLz - iLx)
      = iLy + LxLz + iLyLz +Lx
      = (Lx + iLy) + (Lx +iLy)Lz
      = L+ + L+Lz = L+ (1 + Lz) = L+ (Lz + 1)
 
We therefore conclude that LzL+ = L+ (Lz + 1). We will use this in the original equation:
 
Lz(L+|Ylm>) =
LzL+ |Ylm> =
L+ (Lz + 1) |Ylm> =
L+Lz|Ylm> + L+|Ylm>
 
We will use the eigenvalue equation: Lz|Ylm> = m|Ylm>
 
= L+m|Ylm> + L+|Ylm>
= (m+1)L+|Ylm>
= (m+1) (L+|Ylm>)
 
Hence: Lz(L+|Ylm>) = (m+1)(L+|Ylm>)
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« Last Edit: 11.02.08 at 21:27:50 by Gerrit-Jan Linker »  

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Re: Angular momentum
Reply #2 - 14.02.08 at 08:16:16
 
Eigenvalues of L2 and Lz
 
The eigenvalue equations are:
L2|Ylm> = βl|Ylm>
Lz|Ylm> = m|Ylm>
 
The magnitude of angular momentum is confined to the values √l(l+1) with l=0,1/2,1,...
 
The components on an arbitrary z-axis are limited to 2l+1 values ml with ml=l,l-1,...,-l
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« Last Edit: 14.02.08 at 08:25:21 by Gerrit-Jan Linker »  

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Re: Angular momentum
Reply #3 - 16.02.08 at 12:59:05
 
Angular momentum of electrons
 
Electrons can have 2 possible spin moments: s = 1/2 and ms = + and -
 
Associated with ms = + we have an eigenfunction |α> and associated with ms = - we have an eigenfunction |> of the sz operator.
 
We can write the following eigenvalue equations for electrons:
 
sz|α> = +|α>  
sz|> = -|>  
 
s+|α> = 0
s+|> = |α>  
 
s-|α> = |>  
s-|> = 0  
 
s2|α> = +|α>  
s2|> = +|>  
 
Matrix representations:
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« Last Edit: 16.02.08 at 14:42:34 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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Spin and multiplicity
Reply #4 - 26.02.08 at 12:44:08
 
Spin and multiplicity
 
Electrons with their individual spins of sj=l/2 can combine in various ways to lead to a state of given total spin.  
 
The total spin S=Esj. Since spin is a vector, there are various ways of combining individual spins, but the net result is that a molecule can have spin Sof 0, 1/2, 1, .... These states have a multiplicity of 2S+1 = 1, 2, 3, ...,that is, there is only one way (a singlet) of orienting a spin of 0, two ways (a doublet) of orienting a spin of 1/2, three ways (a triplet) of orienting a spin of 1, and so on.  
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« Last Edit: 26.02.08 at 12:44:23 by Gerrit-Jan Linker »  

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