Spin`Algebra

Spin`Algebra` Reference Guide

Functions

SToMatrix

• SToMatrix[expr]returns matrix representation  of expression expr.
• SToMatrix[expr,BraBasis->bras,KetBasis->kets]returns matrix representation  for a set of bra-vectors bras and ket-vectors kets.

• SToMatrix[expr] uses the full basis of vectors

• The following options can be applied

BraBasis->Automatic uses set of bra-vectors generated from a given set of ket-vectors. If none of these sets is determined uses full set of vectors BraBasis[]. Another possible value is explicitly given set of bra-vectors.

KetBasis->Automatic uses set of ket-vectors generated from a given set of bra-vectors. If none of these sets is determined uses full set of vectors KetBasis[]. Another possible value is explicitly given set of ket-vectors.

• See also: BraBasis, KetBasis, MatrixToS, CenterDot, SMatrix

Further Examples

This loads the packages Spin`

This defines three-spin system

Here is a matrix representation of operator in the eigenbasis of uncoupled spins:

Here are eigenstates of two coupled spins:

This gives a matrix form of operator of exchange interaction, -2·, in the basis of strongly coupled system {a,b}:

Operator of exchange interaction has form -2·. Previous result clearly shows the separation of energy of singlet state (3/2) and three states of the triplet (-1/2).

Operator in the basis of strongly coupled three-spin system:

The part of previous matrix describing the interactions between the doublet (j=1/2) state and quartet (j=3/2) states, both of which are formed from triplet state (j=1) of spins a and b:

See also examples for BraBasis and KetBasis

MatrixToS

• MatrixToS[m] converts the matrix m to corresponding operator form

• MatrixToS[m] needs the dimension of matrix m to be equal to eigenspace of the spin system

• See also: SToMatrix, CenterDot

Further Examples

This loads the packages Spin`

This defines the spin system

The matrix representation of expression -/3 in the basis of eigenstates of multiplets with j=1/2 and j=3/2 formed from the initial spins T and D:

The part of full matrix describing the spin space of the quartet state

Let's redefine spin system:

Previous matrix m can be reformulated now via spin operators of a quartet state:

This is a different presentation of well known operator form:

Thus the diagonal matrix elements of second order operator -/3 of triplet state inherits its form in basis of eigenstates of the quartet, generated from this triplet.   

SpinForm

• SpinForm[expr] shows expr in special form of operator representation
• SpinForm[expr,form] presents the result in the specific form

• For default output of SpinForm the value of global variable $OperatorForm is used. The second argument controls the particular output form of the function. Its possible values are: Ladder, IrreducibleNormalized, Irreducible, Cartesian, Symmetric. For more information refer to description of OperatorForm.

• The output of SpinForm can be in non-canonic form thus to prevent the further evaluation SpinForm puts the result in HoldForm

• See also: CenterDot, $OperatorForm, OperatorForm

Further Examples

This loads the packages Spin`

This defines the spin system

Here is regular output of application of CenterDot:

It returns the operators in IrreducibleNormalized form. But if apply SpinForm to this result it will return in the Cartesian form:

To prevent re-evaluation of the previous expression it is put in the HoldForm:

This clears symbol expr:

Here the operator in Irreducible form:

CenterDot

• CenterDot[a,b,c] gives non-commutative scalar product of operators of angular momentum and its bra- and ket-vectors.

• CenterDot[a, b] can be entered in StandardForm as a · b, a :.: b or a \[CenterDot] b.

• CenterDot[args] treats all args except the functions Bra, Ket and S as commutative. The commutation rules of functions Bra, Ket and S correspond to standard definitions of theory of angular momentum.

• The input form of functions S must have one of the following forms:

1. , , used for the operator of spin i in Cartesian representation; similar notations are used for other projections
2. or are for vector-operator (in Cartesian spin space)
3. and used for rising operator and and used for lowering operator
4. and are equivalent to irreducible non-unit and unit operators T(k,q)
5.  , where p∈{x,X,SpinX,y,Y,SpinY,z,Z,SpinZ,+,SpinPlus,-,SpinMinus} is operator of selective excitation between levels m and m'. The operator has the same matrix elements as Cartesian and ladder operators of effective spin 1/2, formed on states m and m'.

• The input form of bra- and ket-vectors must be of the following form: Bra[i,m] or Ket[i,m], or in more traditional form <i,m⌋ and ⌊i,m>. Here the first argument is spin label, and m is spin projection.

• The input form of operators S determined by the value of option OperatorForm of function SpinSystem

• See also: Bra, Ket, S,  SpinSystem, OperatorForm, SpinForm, CircleTimes

Further Examples

This loads the packages Spin`

By default spin system isn't defined

Nevertheless non-commutative product of operators written in the form of Cartesian or ladder operators can be found:

Here is example of product of two vector operators:

This defines the quartet spin system and sets the output to be transformed via irreducible operators

The result of conversion of second order operator:

Commutator

• Commutator[a,b]=Commutator[a,b,-1]=a·b-b·a.
• Commutator[a,b,0]=b.
• Commutator[a,b,+1]=a·b+b·a
• Commutator[a,b,+n]=

Commutator[a,Commutator[a,b,n-1]]	n>1
Commutator[a,Commutator[a,b,n+1]]	n<-1

.

• Commutator[a, b] can be entered in StandardForm as , {a :.: b}[CTRL]-[-]-. corresponds to Commutator[a, b, 1] and corresponds to Commutator[a, b, n].

• See also: S, CenterDot

Further Examples

This loads the packages Spin`

By default spin system isn't defined

The formal solution of Liouville equation ρ can expanded as . Thus the finite sum of multiple commutator gives approximated solution. Here is an example of developing spin order state by application of a pulse rotating along x direction:

The last term can be computed directly:

CirclePlus

• (+)(+)(+)...represents a matrix which is outer sum of matrices.

• a(+)b=(

	...		0	...	0
...	...	...	...	...	...
	...		0	...	0
0	...	0		...
...	...	...	...	...	...
0	...	0		...

).

• CirclePlus[a,b] can be entered in StandardForm as a(+)b, a :c+: b.

• See also: CircleTimes, SToMatrix, SMatrix

Further Examples

This loads the packages Spin`

This defines spin system:

Here is a matrix form in the basis of strongly coupled system

Here is the matrix of direct sum of matrices of doublet and quartet states

CircleTimes

• (*)(*)(*)...represents a matrix which equals to outer product of matrices.

• a(*)b=(

	...		...		...
...	...	...	...	...	...	...
	...		...		...
...	...	...	...	...	...	...
	...		...		...
...	...	...	...	...	...	...
	...		...		...

).

• CircleTimes[a, b] can be entered in StandardForm as a(*)b, a :c*: b.

• See also: CirclePlus, SToMatrix, SMatrix

Further Examples

This loads the packages Spin`

Here is a direct product of two matrices:

SRotate

• SRotate[a,b]computes the expression a
• SRotate[a,b,c,...]=SRotate[SRotate[a,b],c,...]

• The output form of operators determined by the value of option OperatorForm of function SpinSystem

• See also: SpinSystem, OperatorForm

Further Examples

This loads the packages Spin`

This defines spin system:

The 90° rotation around x-axis:

WignerD

• WignerD[j,β] returns Wigner reduced D-matrix for spin j, rotating on angle β around y-axis. WignerD[j,β] = WignerD[j,{0,β,0}].
• WignerD[j,{α,β,γ}] returns Wigner rotation D-matrix for spin j, rotating on angles {α,β,γ} around axes z, y and z.

• For spin a with the value j Wigner[j,{α,β,γ}] is equivalent to MatrixExp[-I α ] . MatrixExp[-I β ] . MatrixExp[-I γ ] (Rose ME Elementary theory of angular momentum, John Wiley & Sons, Inc., 1963, P. 52).

• References: Edmonds AR 1996, P. 53; Zare RN 1988, P. 85; Rose ME 1963, PP. 48-54

• See also: SRotate, SToMatrix, SMatrix

Further Examples

This loads the packages Spin`

Here is a Wigner reduced matrix for spin 1/2

Here is rotation of matrix SMatrix[1,SpinZ]:

This defines spin system

This another way to find the rotated matrix:

   Last modified: April 10, 2007