Spin`Algebra

Spin`Algebra` Reference Guide

Elements

S

• S[args] represent operator of angular momentum

• S[a,b,...] can be entered in StandardForm as , S [CTRL]-[-] a,b,c...

• S has attributes HoldAll and Locked

• See also: Bra, Ket, CenterDot, SToMatrix, MatrixToS

Further Examples

This loads the packages Spin`

This sets undefined spin system

The commutator of two operators in Cartesian representation:

Bra

• Bra[a,m] represents bra-vector of a state with projection m of spin a.

• Bra[a, m] can be entered in StandardForm as <a,m⌋, its template can be entered as :bra:.

• Bra has attributes HoldAll and Locked

• See also: Ket, S, CenterDot, SToMatrix, MatrixToS

Further Examples

This loads the packages Spin`

This sets undefined spin system

Here is an action of rising operator on the bra-vector with projection 1:

Here is a definition of projective operator and its action on the state a⌊a,j>·⌊b,j>:

Ket

• Ket[a,m] represents bra-vector of state with projection m of spin a.

• Ket[a, m] can be entered in StandardForm as ⌊a,m>, its template can be entered as :ket:.

• Ket has attributes HoldAll and Locked

• See also: Bra, S, CenterDot, SToMatrix, MatrixToS

Further Examples

This loads the packages Spin`

This generates a list ket-vectors for spin 2 labeled as a

SpinX

• SpinX is an abstract variable of package Spin` used to label x-projection in Cartesian coordinates.

• The output of SpinX in StandardForm is printed as X

• The symbols x and X entered as arguments of operator S during the computation within the functions like CenterDot, SpinForm etc... are converted to SpinX.

• SpinX has attribute Locked

• See also: SpinY, SpinZ, SpinPlus, SpinMinus, Bra, Ket, CenterDot, SToMatrix, MatrixToS

Further Examples

This loads the packages Spin`

This sets a spin system for one spin labeled as 1+2:

Function SpinForm by default uses the representation $OperatorForm, which is set in previous evaluation as Cartesian:

All entered functions after the interpretation have identical form:

Operator form Cartesian doesn't belong to the basic (noncomputable) form of representation. To prevent the infinite loop function SpinForm puts the results in hold form:

SpinY

• SpinY is an abstract variable of package Spin` used to label y-projection in Cartesian coordinates.

• The output of SpinY in StandardForm is printed as Y

• The symbols y and Y entered as arguments of operator S during the computation within functions like CenterDot, SpinForm etc... are converted to SpinY.

• SpinY has attributes Protected and Locked

• See also: SpinX, SpinZ, SpinPlus, SpinMinus, Bra, Ket, CenterDot, SToMatrix, MatrixToS

Further Examples

For further examples refer to SpinX

SpinZ

• SpinZ is an abstract variable of package Spin` used to label z-projection in Cartesian coordinates.

• The output of SpinZ in StandardForm is printed as Z

• The symbols z and Z entered as arguments of operator S during the computation within functions like CenterDot, SpinForm etc... are converted to SpinZ.

• SpinZ has attributes Protected and Locked

• See also: SpinX, SpinY, SpinPlus, SpinMinus, Bra, Ket, CenterDot, SToMatrix, MatrixToS

Further Examples

For further examples refer to SpinX

SpinPlus

• SpinPlus is an abstract variable of package Spin`, used to label rising operator

• The output of SpinPlus in StandardForm is printed as +

• The symbol + entered as argument of operator S during the computation within functions like CenterDot, SpinForm etc... is converted to SpinPlus.

• SpinPlus has attributes Protected and Locked

• See also: SpinX, SpinY, SpinZ, SpinMinus, Bra, Ket, CenterDot, SToMatrix, MatrixToS

Further Examples

This loads the packages Spin`

This sets undefined spin system

The product of the lowering and rising operators:

SpinMinus

• SpinMinus is an abstract variable of package Spin`Algebra`, used to label rising operator

• The output of SpinPlus in StandardForm is printed as -

• The symbol - entered as argument of operator S during the computation within functions like CenterDot,  SpinForm etc... is converted to SpinMinus.

• SpinMinus has attributes Protected and Locked

• See also: SpinX, SpinY, SpinZ, SpinPlus, Bra, Ket, CenterDot, SToMatrix, MatrixToS

Further Examples

For further examples refer to SpinPlus

SMatrix

• SMatrix[args] generates matrix representation of spin operator
• SMatrix[j,x] generates matrix representation of spin operator

• SMatrix doesn't need SpinSystem to be defined

• The first argument of SMatrix is a value of spin state and the second must be either SpinX, SpinY and SpinZ for Cartesian representation, and SpinZ, SpinPlus and SpinMinus for Ladder representation, or numbers in a special structure for some other representations.

• See also: Bra, Ket, CenterDot, SToMatrix, MatrixToS

Further Examples

This loads the packages Spin`

If spin system with one spin of the same value is defined:

Then the same matrix can be get by function SToMatrix:

Thus, SMatrix is more simple and faster but SToMatrix is more general and flexible, it can be applied to expression of any complexity.

BraMatrix

• BraMatrix[j,m] generates matrix representation of bra-vector with spin value j and projection m

• BraMatrix doesn't need SpinSystem to be defined

• See also: Bra, Ket, KetMatrix, SMatrix, CircleTimes

Further Examples

This loads the packages Spin`

This is BraMatrix of a triplet state

Here is scalar product of BraMatrix and KetMatrix:

The order of positions is crucial. This gives a product of KetMatrix acting on BraMatrix:

KetMatrix

• KetMatrix[j,m] generates matrix representation of bra-vector with spin value j and projection m

• KetMatrix doesn't need SpinSystem to be defined

• See also: Bra, Ket, BraMatrix, SMatrix, CircleTimes

Further Examples

This loads the packages Spin`

This is KetMatrix of a triplet state

Here is SMatrix of selective transition, it equivalent to direct product of BraMatrix and KetMatrix

   Last modified: April 10, 2007