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Angular Momentum Operators

rename
Updated 2009-03-31 00:11

Angular Momentum Operators
in Quantum Mechanics

These are useful in the field of Magnetic Resonance, which includes Nuclear Magnetic Resonance (NMR), Electron Paramagnetic Resonance (EPR), and ENDOR. They are also useful in the fields of Spintronics and Quantum Computing.

 

In the following, the expressions in the left column are equivalent to the expressions in the right column. Quantum mechanical operators are generally Hermitian matrices.

General Matrix Operators

Left ColumnRight Column
Ajkelement of matrix A
in j-th row and k-th column of A
Ajk< j | A | k >
I=Identity Matrixhas Imn=1 if m=n and 0 if m≠n
I=Identity Matrixa diagonal matrix with all nonzero elements equal to 1
I=Identity Matrixa matrix with diagonal elements equal to 1
and all other elements equal to 0
I=Identity MatrixΣn | n >< n | when summed over a complete basis
A BΣk Ajk Bkl = Σk < j | A | k > < k | B | l >
B AΣk Bjk Akl = Σk < j | B | k > < k | A | l >
A B CΣk Σl Ajk Bkl Clm =
Σk Σl < j | A | k > < k | B | l > < l | C | m >
Tr(A)=Trace of AΣm Amm = sum of A's diagonal elements
Tr(A B)Tr(B A) = Σm Σn Bmn Anm
Tr(A B C)Tr(B C A) = Tr(C A B)
Tr(C B A)Tr(B A C) = Tr(A C B)
A-1inverse of A
Ainverse of A-1
A A-1I
A-1 AI
A I A
I AA
memorize

 

Left ColumnRight Column
[A,B]- = commutator of A and BA B - B A
[A,A]-0
[A,B]+ = anti-commutator of A and BA B + B A
[A,A]+2 A2
ATtranspose of A
A=ATa symmetric matrix A
A+complex conjugate
transpose of A
A=A+a Hermitian matrix A
eiAθΣn=0 (iAθ)n/(n!)
ecos(θ) + i sin(θ) = Σn=0 (iθ)n/(n!)
cos(θ)Re[e] = Σeven n=0 (iθ)n/(n!)
           = Σn=0 (-1)n θ2n/[(2n)!]
sin(θ)Im[e] = Σodd n=1 in-1 θn/(n!)
           = Σn=0 (-1)n θ2n+1/[(2n+1)!]
memorize

Angular Momentum Operators

In the following, L = Lx x + Ly y + Lz z is an angular momentum vector where
Lx, Ly, Lz are angular momentum operators and
x, y, z are unit vectors along the cartesian coordinate x, y, z axes.


L can represent:
an electron spin (often denoted S with S2=s(s+1)=3/4, s=1/2, Sz=mS=-1/2 or +1/2),
a nuclear spin (often denoted I),
an angular momentum (often denoted L), or
any combination of the above (see Adding Angular Momenta for more details).


In the following, l and m are quantum numbers.
l is an integer or half-integer like 0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5, etc.
m is a quantum number that ranges from -l to +l in steps of 1.
For example, l=2 gives m=-2, -1, 0, 1, 2
while l=3/2 gives m=-3/2, -1/2, 1/2, 3/2.
Thus, for a particular l, there are 2l+1 different values of m.

Left ColumnRight Column
Lz | l m >m | l m >
L+ | l m >sqrt[l(l+1) - m(m+1)] | l m+1 >
L- | l m >sqrt[l(l+1) - m(m-1)] | l m-1 >
L+ | l m >sqrt[l2 + l - m2 - m] | l m+1 >
L- | l m >sqrt[l2 + l - m2 + m] | l m-1 >
L+ | l m >sqrt[(l+m+1)(l-m)] | l m+1 >
L- | l m >sqrt[(l-m+1)(l+m)] | l m-1 >
L+Lx + i Ly
L-Lx - i Ly
Lx(1/2)(L+ + L-)
Ly(-i/2)(L+ - L-)
L2Lx2 + Ly2 + Lz2
Lx2 + Ly2L2 - Lz2
L2 | l m >l(l+1) | l m >
(Lx2 + Ly2) | l m >[l(l+1) - m2] | l m >
Tr(Lx)0
Tr(Ly)0
Tr(Lz)0
Tr(L+)0
Tr(L-)0
Tr(L2)l(l+1)(2l+1)
Tr(Lx2)=Tr(Ly2)=Tr(Lz2)l(l+1)(2l+1)/3
L+ L-L2 - Lz2 + Lz
L- L+L2 - Lz2 - Lz
memorize

Commutation Relations

Left ColumnRight Column
[Lx,Ly]-i Lz
[Ly,Lz]-i Lx
[Lz,Lx]-i Ly
[Ly,Lx]--i Lz
[Lz,Ly]--i Lx
[Lx,Lz]--i Ly
[Lx,Lx]-0
[Ly,Ly]-0
[Lz,Lz]-0
[Lx,L2]-0
[Ly,L2]-0
[Lz,L2]-0
[L+,L2]-0
[L-,L2]-0
[Lx,L+]--Lz
[Lx,L-]-Lz
[Ly,L+]--i Lz
[Ly,L-]--i Lz
[Lz,L+]-L+
[Lz,L-]- -L-
[L+,L-]-2 Lz
[L-,L+]--2 Lz
memorize

Rotations About Different Axes

Rotations About x
Left ColumnRight Column
e-iLxθ Lx eiLxθLx
e-iLxθ Ly eiLxθLy cos(θ) + Lz sin(θ)
e-iLxθ Lz eiLxθLz cos(θ) - Ly sin(θ)
memorize

 

Rotations About y
Left ColumnRight Column
e-iLyθ Lx eiLyθLx cos(θ) - Lz sin(θ)
e-iLyθ Ly eiLyθLy
e-iLyθ Lz eiLyθLz cos(θ) + Lx sin(θ)
memorize

 

Rotations About z
Left ColumnRight Column
e-iLzθ Lx eiLzθLx cos(θ) + Ly sin(θ)
e-iLzθ Ly eiLzθLy cos(θ) - Lx sin(θ)
e-iLzθ Lz eiLzθLz
e-iLzθ L+ eiLzθL+ [cos(θ) - i sin(θ)] = L+ e-iθ
e-iLzθ L- eiLzθL- [cos(θ) + i sin(θ)] = L- e
memorize

Special Cases

In the (2l+1)x(2l+1) matrices listed below,
each row and column represents a different m value.

The column gives the initial value of m,
and the row gives the final value of m after the operator has acted on it.

Special Cases with l=1/2
(electron spins and 1H nuclei obey these)

 

In the 2x2 matrices listed below for l=1/2,

the columns from left to right have m=1/2 and m=-1/2 respectively, while

the rows from top to bottom have m=1/2 and m=-1/2 respectively.

 

Left ColumnRight Column
eiL+θ for l=1/21 + i L+ θ
eiL-θ for l=1/21 + i L- θ
eiLxθ for l=1/2cos(θ/2) + i 2 Lx sin(θ/2)
eiLyθ for l=1/2cos(θ/2) + i 2 Ly sin(θ/2)
eiLzθ for l=1/2cos(θ/2) + i 2 Lz sin(θ/2)
L+ for l=1/2
    0       1  
0 0
L- for l=1/2
    0       0  
1 0
Lx for l=1/2
    0       1/2
1/2 0
Ly for l=1/2
    0      -i/2
i/2 0
Lz for l=1/2
    1/2     0  
0 -1/2
Lx2=Ly2=Lz2 for l=1/2
    1/4     0  
0 1/4
L2 for l=1/2
    3/4     0  
0 3/4
memorize

Special Cases with l=1, 3/2, or 2

 

In the (2l+1)x(2l+1) matrices listed below for l=1, 3/2, or 2,

the columns from left to right have m=+l, +l-1, +l-2, ..., -l+2, -l+1, and -l respectively, while

the rows from top to bottom have m=+l, +l-1, +l-2, ..., -l+2, -l+1, and -l respectively.

 

Left ColumnRight Column
L+ for l=1
    0     sqrt(2)   0   
0 0 sqrt(2)
0 0 0
L- for l=1
    0       0       0   
sqrt(2) 0 0
0 sqrt(2) 0
Lx for l=1
    0     sqrt(2)/2 0    
sqrt(2)/2 0 sqrt(2)/2
0 sqrt(2)/2 0
Ly for l=1
    0   -isqrt(2)/2 0     
isqrt(2)/2 0 -isqrt(2)/2
0 isqrt(2)/2 0
Lz for l=1
    1       0       0
0 0 0
0 0 -1
Lx2 for l=1
    1/2     0       1/2
0 1 0
1/2 0 1/2
Ly2 for l=1
    1/2     0      -1/2
0 1 0
-1/2 0 1/2
Lz2 for l=1
    1       0       0
0 0 0
0 0 1
L2 for l=1
    2       0       0
0 2 0
0 0 2
memorize

 

Left ColumnRight Column
L+ for l=3/2
    0     sqrt(3)   0       0
0 0 sqrt(4) 0
0 0 0 sqrt(3)
0 0 0 0
L- for l=3/2
    0       0       0       0
sqrt(3) 0 0 0
0 sqrt(4) 0 0
0 0 sqrt(3) 0
Lx for l=3/2
    0     sqrt(3)/2 0       0
sqrt(3)/2 0 sqrt(4)/2 0
0 sqrt(4)/2 0 sqrt(3)/2
0 0 sqrt(3)/2 0
Ly for l=3/2
    0   -isqrt(3)/2 0       0
isqrt(3)/2 0 -isqrt(4)/2 0
0 isqrt(4)/2 0 -isqrt(3)/2
0 0 isqrt(3)/2 0
Lz for l=3/2
    3/2     0       0       0
0 1/2 0 0
0 0 -1/2 0
0 0 0 -3/2
Lx2 for l=3/2
    3/4     0     sqrt(3)/2 0
0 7/4 0 sqrt(3)/2
sqrt(3)/2 0 7/4 0
0 sqrt(3)/2 0 3/4
Ly2 for l=3/2
    3/4     0    -sqrt(3)/2 0
0 7/4 0 -sqrt(3)/2
-sqrt(3)/2 0 7/4 0
0 -sqrt(3)/2 0 3/4
Lz2 for l=3/2
    9/4     0       0       0
0 1/4 0 0
0 0 1/4 0
0 0 0 9/4
L2 for l=3/2
   15/4     0       0       0
0 15/4 0 0
0 0 15/4 0
0 0 0 15/4
memorize

 

Left ColumnRight Column
L+ for l=2
    0     sqrt(4)   0       0       0
0 0 sqrt(6) 0 0
0 0 0 sqrt(6) 0
0 0 0 0 sqrt(4)
0 0 0 0 0
L- for l=2
    0       0       0       0       0
sqrt(4) 0 0 0 0
0 sqrt(6) 0 0 0
0 0 sqrt(6) 0 0
0 0 0 sqrt(4) 0
Lx for l=2
    0     sqrt(4)/2 0       0       0
sqrt(4)/2 0 sqrt(6)/2 0 0
0 sqrt(6)/2 0 sqrt(6)/2 0
0 0 sqrt(6)/2 0 sqrt(4)/2
0 0 0 sqrt(4)/2 0
Ly for l=2
    0   -isqrt(4)/2 0       0       0
isqrt(4)/2 0 -isqrt(6)/2 0 0
0 isqrt(6)/2 0 -isqrt(6)/2 0
0 0 isqrt(6)/2 0 -isqrt(4)/2
0 0 0 isqrt(4)/2 0
Lz for l=2
    2       0       0       0       0
0 1 0 0 0
0 0 0 0 0
0 0 0 -1 0
0 0 0 0 -2
Lx2 for l=2
    1       0     sqrt(3/2) 0       0
0 5/2 0 3/2 0
sqrt(3/2) 0 3 0 sqrt(3/2)
0 3/2 0 5/2 0
0 0 sqrt(3/2) 0 1
Ly2 for l=2
    1       0    -sqrt(3/2) 0       0
0 5/2 0 -3/2 0
-sqrt(3/2) 0 3 0 -sqrt(3/2)
0 -3/2 0 5/2 0
0 0 -sqrt(3/2) 0 1
Lz2 for l=2
    4       0       0       0       0
0 1 0 0 0
0 0 0 0 0
0 0 0 1 0
0 0 0 0 4
L2 for l=2
    6       0       0       0       0
0 6 0 0 0
0 0 6 0 0
0 0 0 6 0
0 0 0 0 6
memorize

References:

C. P. Slichter's "Principles of Magnetic Resonance", 3rd Edition, 1996
http://books.google.com/books?id=zgnrRkaIhFoC
J. A. Weil and J. R. Bolton's "Electron Paramagnetic Resonance", 2nd Edition, 2006
http://books.google.com/books?id=qjLpMw9ZgPIC