Spin

Quantum mechanical angular momemtum.

class sympy.physics.quantum.spin.Rotation

Wigner D operator in terms of Euler angles.

Defines the rotation operator in terms of the Euler angles defined by the z-y-z convention for a passive transformation. That is the coordinate axes are rotated first about the z-axis, giving the new x’-y’-z’ axes. Then this new coordinate system is rotated about the new y’-axis, giving new x’‘-y’‘-z’’ axes. Then this new coordinate system is rotated about the z’‘-axis. Conventions follow those laid out in [R292].

Parameters :

alpha : Number, Symbol

First Euler Angle

beta : Number, Symbol

Second Euler angle

gamma : Number, Symbol

Third Euler angle

See also

WignerD

Symbolic Wigner-D function

D

Wigner-D function

d

Wigner small-d function

References

[R292]

(1, 2) Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.

Examples

A simple example rotation operator:

>>> from sympy import pi
>>> from sympy.physics.quantum.spin import Rotation
>>> Rotation(pi, 0, pi/2)
R(pi,0,pi/2)

With symbolic Euler angles and calculating the inverse rotation operator:

>>> from sympy import symbols
>>> a, b, c = symbols('a b c')
>>> Rotation(a, b, c)
R(a,b,c)
>>> Rotation(a, b, c).inverse()
R(-c,-b,-a)

classmethod D(j, m, mp, alpha, beta, gamma)

Wigner D-function.

Returns an instance of the WignerD class corresponding to the Wigner-D function specified by the parameters.

Parameters :

j : Number

Total angular momentum

m : Number

Eigenvalue of angular momentum along axis after rotation

mp : Number

Eigenvalue of angular momentum along rotated axis

alpha : Number, Symbol

First Euler angle of rotation

beta : Number, Symbol

Second Euler angle of rotation

gamma : Number, Symbol

Third Euler angle of rotation

See also

WignerD

Symbolic Wigner-D function

Examples

Return the Wigner-D matrix element for a defined rotation, both numerical and symbolic:

>>> from sympy.physics.quantum.spin import Rotation
>>> from sympy import pi, symbols
>>> alpha, beta, gamma = symbols('alpha beta gamma')
>>> Rotation.D(1, 1, 0,pi, pi/2,-pi)
WignerD(1, 1, 0, pi, pi/2, -pi)

classmethod d(j, m, mp, beta)

Wigner small-d function.

Returns an instance of the WignerD class corresponding to the Wigner-D function specified by the parameters with the alpha and gamma angles given as 0.

Parameters :

j : Number

Total angular momentum

m : Number

Eigenvalue of angular momentum along axis after rotation

mp : Number

Eigenvalue of angular momentum along rotated axis

beta : Number, Symbol

Second Euler angle of rotation

See also

WignerD

Symbolic Wigner-D function

Examples

Return the Wigner-D matrix element for a defined rotation, both numerical and symbolic:

>>> from sympy.physics.quantum.spin import Rotation
>>> from sympy import pi, symbols
>>> beta = symbols('beta')
>>> Rotation.d(1, 1, 0, pi/2)
WignerD(1, 1, 0, 0, pi/2, 0)

class sympy.physics.quantum.spin.WignerD

Wigner-D function

The Wigner D-function gives the matrix elements of the rotation operator in the jm-representation. For the Euler angles \(\alpha\), \(\beta\), \(\gamma\), the D-function is defined such that:

\[\begin{split}<j,m| \mathcal{R}(\alpha, \beta, \gamma ) |j',m'> = \delta_{jj'} D(j, m, m', \alpha, \beta, \gamma)\end{split}\]

Where the rotation operator is as defined by the Rotation class [R293].

The Wigner D-function defined in this way gives:

\[D(j, m, m', \alpha, \beta, \gamma) = e^{-i m \alpha} d(j, m, m', \beta) e^{-i m' \gamma}\]

Where d is the Wigner small-d function, which is given by Rotation.d.

The Wigner small-d function gives the component of the Wigner D-function that is determined by the second Euler angle. That is the Wigner D-function is:

\[D(j, m, m', \alpha, \beta, \gamma) = e^{-i m \alpha} d(j, m, m', \beta) e^{-i m' \gamma}\]

Where d is the small-d function. The Wigner D-function is given by Rotation.D.

Note that to evaluate the D-function, the j, m and mp parameters must be integer or half integer numbers.

Parameters :

j : Number

Total angular momentum

m : Number

Eigenvalue of angular momentum along axis after rotation

mp : Number

Eigenvalue of angular momentum along rotated axis

alpha : Number, Symbol

First Euler angle of rotation

beta : Number, Symbol

Second Euler angle of rotation

gamma : Number, Symbol

Third Euler angle of rotation

See also

Rotation

Rotation operator

References

[R293]

(1, 2) Varshalovich, D A, Quantum Theory of Angular Momentum. 1988.

Examples

Evaluate the Wigner-D matrix elements of a simple rotation:

>>> from sympy.physics.quantum.spin import Rotation
>>> from sympy import pi
>>> rot = Rotation.D(1, 1, 0, pi, pi/2, 0)
>>> rot
WignerD(1, 1, 0, pi, pi/2, 0)
>>> rot.doit()
sqrt(2)/2

Evaluate the Wigner-d matrix elements of a simple rotation

>>> rot = Rotation.d(1, 1, 0, pi/2)
>>> rot
WignerD(1, 1, 0, 0, pi/2, 0)
>>> rot.doit()
-sqrt(2)/2

class sympy.physics.quantum.spin.JxKet

Eigenket of Jx.

See JzKet for the usage of spin eigenstates.

See also

JzKet

Usage of spin states

class sympy.physics.quantum.spin.JxBra

Eigenbra of Jx.

See JzKet for the usage of spin eigenstates.

See also

JzKet

Usage of spin states

class sympy.physics.quantum.spin.JyKet

Eigenket of Jy.

See JzKet for the usage of spin eigenstates.

See also

JzKet

Usage of spin states

class sympy.physics.quantum.spin.JyBra

Eigenbra of Jy.

See JzKet for the usage of spin eigenstates.

See also

JzKet

Usage of spin states

class sympy.physics.quantum.spin.JzKet

Eigenket of Jz.

Spin state which is an eigenstate of the Jz operator. Uncoupled states, that is states representing the interaction of multiple separate spin states, are defined as a tensor product of states.

Parameters :

j : Number, Symbol

Total spin angular momentum

m : Number, Symbol

Eigenvalue of the Jz spin operator

See also

JzKetCoupled

Coupled eigenstates

TensorProduct

Used to specify uncoupled states

uncouple

Uncouples states given coupling parameters

couple

Couples uncoupled states

Examples

Normal States:

Defining simple spin states, both numerical and symbolic:

>>> from sympy.physics.quantum.spin import JzKet, JxKet
>>> from sympy import symbols
>>> JzKet(1, 0)
|1,0>
>>> j, m = symbols('j m')
>>> JzKet(j, m)
|j,m>

Rewriting the JzKet in terms of eigenkets of the Jx operator: Note: that the resulting eigenstates are JxKet’s

>>> JzKet(1,1).rewrite("Jx")
|1,-1>/2 - sqrt(2)*|1,0>/2 + |1,1>/2

Get the vector representation of a state in terms of the basis elements of the Jx operator:

>>> from sympy.physics.quantum.represent import represent
>>> from sympy.physics.quantum.spin import Jx, Jz
>>> represent(JzKet(1,-1), basis=Jx)
Matrix([
[      1/2],
[sqrt(2)/2],
[      1/2]])

Apply innerproducts between states:

>>> from sympy.physics.quantum.innerproduct import InnerProduct
>>> from sympy.physics.quantum.spin import JxBra
>>> i = InnerProduct(JxBra(1,1), JzKet(1,1))
>>> i
<1,1|1,1>
>>> i.doit()
1/2

Uncoupled States:

Define an uncoupled state as a TensorProduct between two Jz eigenkets:

>>> from sympy.physics.quantum.tensorproduct import TensorProduct
>>> j1,m1,j2,m2 = symbols('j1 m1 j2 m2')
>>> TensorProduct(JzKet(1,0), JzKet(1,1))
|1,0>x|1,1>
>>> TensorProduct(JzKet(j1,m1), JzKet(j2,m2))
|j1,m1>x|j2,m2>

A TensorProduct can be rewritten, in which case the eigenstates that make up the tensor product is rewritten to the new basis:

>>> TensorProduct(JzKet(1,1),JxKet(1,1)).rewrite('Jz')
|1,1>x|1,-1>/2 + sqrt(2)*|1,1>x|1,0>/2 + |1,1>x|1,1>/2

The represent method for TensorProduct’s gives the vector representation of the state. Note that the state in the product basis is the equivalent of the tensor product of the vector representation of the component eigenstates:

>>> represent(TensorProduct(JzKet(1,0),JzKet(1,1)))
Matrix([
[0],
[0],
[0],
[1],
[0],
[0],
[0],
[0],
[0]])
>>> represent(TensorProduct(JzKet(1,1),JxKet(1,1)), basis=Jz)
Matrix([
[      1/2],
[sqrt(2)/2],
[      1/2],
[        0],
[        0],
[        0],
[        0],
[        0],
[        0]])

class sympy.physics.quantum.spin.JzBra

Eigenbra of Jz.

See the JzKet for the usage of spin eigenstates.

See also

JzKet

Usage of spin states

class sympy.physics.quantum.spin.JxKetCoupled

Coupled eigenket of Jx.

See JzKetCoupled for the usage of coupled spin eigenstates.

See also

JzKetCoupled

Usage of coupled spin states

class sympy.physics.quantum.spin.JxBraCoupled

Coupled eigenbra of Jx.

See JzKetCoupled for the usage of coupled spin eigenstates.

See also

JzKetCoupled

Usage of coupled spin states

class sympy.physics.quantum.spin.JyKetCoupled

Coupled eigenket of Jy.

See JzKetCoupled for the usage of coupled spin eigenstates.

See also

JzKetCoupled

Usage of coupled spin states

class sympy.physics.quantum.spin.JyBraCoupled

Coupled eigenbra of Jy.

See JzKetCoupled for the usage of coupled spin eigenstates.

See also

JzKetCoupled

Usage of coupled spin states

class sympy.physics.quantum.spin.JzKetCoupled

Coupled eigenket of Jz

Spin state that is an eigenket of Jz which represents the coupling of separate spin spaces.

The arguments for creating instances of JzKetCoupled are j, m, jn and an optional jcoupling argument. The j and m options are the total angular momentum quantum numbers, as used for normal states (e.g. JzKet).

The other required parameter in jn, which is a tuple defining the \(j_n\) angular momentum quantum numbers of the product spaces. So for example, if a state represented the coupling of the product basis state \(|j_1,m_1\rangle\times|j_2,m_2\rangle\), the jn for this state would be (j1,j2).

The final option is jcoupling, which is used to define how the spaces specified by jn are coupled, which includes both the order these spaces are coupled together and the quantum numbers that arise from these couplings. The jcoupling parameter itself is a list of lists, such that each of the sublists defines a single coupling between the spin spaces. If there are N coupled angular momentum spaces, that is jn has N elements, then there must be N-1 sublists. Each of these sublists making up the jcoupling parameter have length 3. The first two elements are the indicies of the product spaces that are considered to be coupled together. For example, if we want to couple \(j_1\) and \(j_4\), the indicies would be 1 and 4. If a state has already been coupled, it is referenced by the smallest index that is coupled, so if \(j_2\) and \(j_4\) has already been coupled to some \(j_{24}\), then this value can be coupled by referencing it with index 2. The final element of the sublist is the quantum number of the coupled state. So putting everything together, into a valid sublist for jcoupling, if \(j_1\) and \(j_2\) are coupled to an angular momentum space with quantum number \(j_{12}\) with the value j12, the sublist would be (1,2,j12), N-1 of these sublists are used in the list for jcoupling.

Note the jcoupling parameter is optional, if it is not specified, the default coupling is taken. This default value is to coupled the spaces in order and take the quantum number of the coupling to be the maximum value. For example, if the spin spaces are \(j_1\), \(j_2\), \(j_3\), \(j_4\), then the default coupling couples \(j_1\) and \(j_2\) to \(j_{12}=j_1+j_2\), then, \(j_{12}\) and \(j_3\) are coupled to \(j_{123}=j_{12}+j_3\), and finally \(j_{123}\) and \(j_4\) to \(j=j_{123}+j_4\). The jcoupling value that would correspond to this is:

((1,2,j1+j2),(1,3,j1+j2+j3))

Parameters :

args : tuple

The arguments that must be passed are j, m, jn, and jcoupling. The j value is the total angular momentum. The m value is the eigenvalue of the Jz spin operator. The jn list are the j values of argular momentum spaces coupled together. The jcoupling parameter is an optional parameter defining how the spaces are coupled together. See the above description for how these coupling parameters are defined.

See also

JzKet

Normal spin eigenstates

uncouple

Uncoupling of coupling spin states

couple

Coupling of uncoupled spin states

Examples

Defining simple spin states, both numerical and symbolic:

>>> from sympy.physics.quantum.spin import JzKetCoupled
>>> from sympy import symbols
>>> JzKetCoupled(1, 0, (1, 1))
|1,0,j1=1,j2=1>
>>> j, m, j1, j2 = symbols('j m j1 j2')
>>> JzKetCoupled(j, m, (j1, j2))
|j,m,j1=j1,j2=j2>

Defining coupled spin states for more than 2 coupled spaces with various coupling parameters:

>>> JzKetCoupled(2, 1, (1, 1, 1))
|2,1,j1=1,j2=1,j3=1,j(1,2)=2>
>>> JzKetCoupled(2, 1, (1, 1, 1), ((1,2,2),(1,3,2)) )
|2,1,j1=1,j2=1,j3=1,j(1,2)=2>
>>> JzKetCoupled(2, 1, (1, 1, 1), ((2,3,1),(1,2,2)) )
|2,1,j1=1,j2=1,j3=1,j(2,3)=1>

Rewriting the JzKetCoupled in terms of eigenkets of the Jx operator: Note: that the resulting eigenstates are JxKetCoupled

>>> JzKetCoupled(1,1,(1,1)).rewrite("Jx")
|1,-1,j1=1,j2=1>/2 - sqrt(2)*|1,0,j1=1,j2=1>/2 + |1,1,j1=1,j2=1>/2

The rewrite method can be used to convert a coupled state to an uncoupled state. This is done by passing coupled=False to the rewrite function:

>>> JzKetCoupled(1, 0, (1, 1)).rewrite('Jz', coupled=False)
-sqrt(2)*|1,-1>x|1,1>/2 + sqrt(2)*|1,1>x|1,-1>/2

Get the vector representation of a state in terms of the basis elements of the Jx operator:

>>> from sympy.physics.quantum.represent import represent
>>> from sympy.physics.quantum.spin import Jx
>>> from sympy import S
>>> represent(JzKetCoupled(1,-1,(S(1)/2,S(1)/2)), basis=Jx)
Matrix([
[        0],
[      1/2],
[sqrt(2)/2],
[      1/2]])

class sympy.physics.quantum.spin.JzBraCoupled

Coupled eigenbra of Jz.

See the JzKetCoupled for the usage of coupled spin eigenstates.

See also

JzKetCoupled

Usage of coupled spin states

sympy.physics.quantum.spin.couple(expr, jcoupling_list=None)

Couple a tensor product of spin states

This function can be used to couple an uncoupled tensor product of spin states. All of the eigenstates to be coupled must be of the same class. It will return a linear combination of eigenstates that are subclasses of CoupledSpinState determined by Clebsch-Gordan angular momentum coupling coefficients.

Parameters :

expr : Expr

An expression involving TensorProducts of spin states to be coupled. Each state must be a subclass of SpinState and they all must be the same class.

jcoupling_list : list or tuple

Elements of this list are sub-lists of length 2 specifying the order of the coupling of the spin spaces. The length of this must be N-1, where N is the number of states in the tensor product to be coupled. The elements of this sublist are the same as the first two elements of each sublist in the jcoupling parameter defined for JzKetCoupled. If this parameter is not specified, the default value is taken, which couples the first and second product basis spaces, then couples this new coupled space to the third product space, etc

Examples

Couple a tensor product of numerical states for two spaces:

>>> from sympy.physics.quantum.spin import JzKet, couple
>>> from sympy.physics.quantum.tensorproduct import TensorProduct
>>> couple(TensorProduct(JzKet(1,0), JzKet(1,1)))
-sqrt(2)*|1,1,j1=1,j2=1>/2 + sqrt(2)*|2,1,j1=1,j2=1>/2

Numerical coupling of three spaces using the default coupling method, i.e. first and second spaces couple, then this couples to the third space:

>>> couple(TensorProduct(JzKet(1,1), JzKet(1,1), JzKet(1,0)))
sqrt(6)*|2,2,j1=1,j2=1,j3=1,j(1,2)=2>/3 + sqrt(3)*|3,2,j1=1,j2=1,j3=1,j(1,2)=2>/3

Perform this same coupling, but we define the coupling to first couple the first and third spaces:

>>> couple(TensorProduct(JzKet(1,1), JzKet(1,1), JzKet(1,0)), ((1,3),(1,2)) )
sqrt(2)*|2,2,j1=1,j2=1,j3=1,j(1,3)=1>/2 - sqrt(6)*|2,2,j1=1,j2=1,j3=1,j(1,3)=2>/6 + sqrt(3)*|3,2,j1=1,j2=1,j3=1,j(1,3)=2>/3

Couple a tensor product of symbolic states:

>>> from sympy import symbols
>>> j1,m1,j2,m2 = symbols('j1 m1 j2 m2')
>>> couple(TensorProduct(JzKet(j1,m1), JzKet(j2,m2)))
Sum(CG(j1, m1, j2, m2, j, m1 + m2)*|j,m1 + m2,j1=j1,j2=j2>, (j, m1 + m2, j1 + j2))

sympy.physics.quantum.spin.uncouple(expr, jn=None, jcoupling_list=None)

Uncouple a coupled spin state

Gives the uncoupled representation of a coupled spin state. Arguments must be either a spin state that is a subclass of CoupledSpinState or a spin state that is a subclass of SpinState and an array giving the j values of the spaces that are to be coupled

Parameters :

expr : Expr

The expression containing states that are to be coupled. If the states are a subclass of SpinState, the jn and jcoupling parameters must be defined. If the states are a subclass of CoupledSpinState, jn and jcoupling will be taken from the state.

jn : list or tuple

The list of the j-values that are coupled. If state is a CoupledSpinState, this parameter is ignored. This must be defined if state is not a subclass of CoupledSpinState. The syntax of this parameter is the same as the jn parameter of JzKetCoupled.

jcoupling_list : list or tuple

The list defining how the j-values are coupled together. If state is a CoupledSpinState, this parameter is ignored. This must be defined if state is not a subclass of CoupledSpinState. The syntax of this parameter is the same as the jcoupling parameter of JzKetCoupled.

Examples

Uncouple a numerical state using a CoupledSpinState state:

>>> from sympy.physics.quantum.spin import JzKetCoupled, uncouple
>>> from sympy import S
>>> uncouple(JzKetCoupled(1, 0, (S(1)/2, S(1)/2)))
sqrt(2)*|1/2,-1/2>x|1/2,1/2>/2 + sqrt(2)*|1/2,1/2>x|1/2,-1/2>/2

Perform the same calculation using a SpinState state:

>>> from sympy.physics.quantum.spin import JzKet
>>> uncouple(JzKet(1, 0), (S(1)/2, S(1)/2))
sqrt(2)*|1/2,-1/2>x|1/2,1/2>/2 + sqrt(2)*|1/2,1/2>x|1/2,-1/2>/2

Uncouple a numerical state of three coupled spaces using a CoupledSpinState state:

>>> uncouple(JzKetCoupled(1, 1, (1, 1, 1), ((1,3,1),(1,2,1)) ))
|1,-1>x|1,1>x|1,1>/2 - |1,0>x|1,0>x|1,1>/2 + |1,1>x|1,0>x|1,0>/2 - |1,1>x|1,1>x|1,-1>/2

Perform the same calculation using a SpinState state:

>>> uncouple(JzKet(1, 1), (1, 1, 1), ((1,3,1),(1,2,1)) )
|1,-1>x|1,1>x|1,1>/2 - |1,0>x|1,0>x|1,1>/2 + |1,1>x|1,0>x|1,0>/2 - |1,1>x|1,1>x|1,-1>/2

Uncouple a symbolic state using a CoupledSpinState state:

>>> from sympy import symbols
>>> j,m,j1,j2 = symbols('j m j1 j2')
>>> uncouple(JzKetCoupled(j, m, (j1, j2)))
Sum(CG(j1, m1, j2, m2, j, m)*|j1,m1>x|j2,m2>, (m1, -j1, j1), (m2, -j2, j2))

Perform the same calculation using a SpinState state

>>> uncouple(JzKet(j, m), (j1, j2))
Sum(CG(j1, m1, j2, m2, j, m)*|j1,m1>x|j2,m2>, (m1, -j1, j1), (m2, -j2, j2))