GROUP THEORY TUTORIAL

Angular Momentum Operations

GTPack [1,2] contains various modules to handle angular momentum operators and representations.

• [1] W. Hergert, R. M. Geilhufe, Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica, Wiley-VCH, 2018 [2] R. M. Geilhufe, W. Hergert, GTPack: A Mathematica group theory package for applications in solid-state physics and photonics, Frontiers in Physics, 6:86, 2018
 GTJx gives the x component of the total angular momentum operator GTJy gives the y component of the total angular momentum operatorXXXX GTJz gives the z component of the total angular momentum operator

The components of the total angular momentum operator in terms of matrix representations acting on a finite sub space indexed by the total angular momentum quantum number J.

Components of the total angular momentum operator
 In[1]:=
 In[2]:=
 In[3]:=
 Out[3]//MatrixForm=
 Out[3]//MatrixForm=
 Out[3]//MatrixForm=
Verify commutation relations.
 In[4]:=
 Out[4]=
 In[5]:=
 Out[5]=
 In[6]:=
 Out[6]=
 GTJplus gives the raising operator GTJminus gives the lowering operator

The raising operator acts as J+|j; m> = |j; m+1>

The lowering operator acts as J-|j; m> = |j; m-1>

J+ and J- are related to Jx and Jy:

J+= Jx + i Jy

J-= Jx - i Jy

Matrix representations
 In[7]:=
 In[8]:=
 Out[8]//MatrixForm=
 Out[8]//MatrixForm=
Define a state vector
 In[9]:=
Verify action of J+ and normalization.
 In[10]:=
 Out[10]=
 In[11]:=
 Out[11]=
Verify action of J+ as a raising operator:
 In[12]:=
 In[13]:=
 Out[13]=
Verify action of J- as a lowering operator:
 In[14]:=
 In[15]:=
 Out[15]=
Verify commutation relations
 In[16]:=
 Out[16]=
 In[17]:=
 Out[17]=
 In[18]:=
 Out[18]=
Verify relation to Jx and Jy
 In[19]:=
 Out[19]=
 In[20]:=
 Out[20]=
 GTJTransform applies a symmetry transformation to the basis functions of an irreducible representation of O(3) GTJMatrix gives the representation matrix of a symmetry element for an irreducible representation of O(3)
GTJTransform and GTJMatrix are closely related. While GTJTransform gives the action of a symmetry element on one specific basis function, GTJMatrix gives the transformation matrix of the entire subspace.
 In[21]:=
 Out[21]=
 In[22]:=
 Out[22]=
 In[23]:=
 Out[23]//MatrixForm=
 GTAngularMomentumRep applies a symmetry transformation to the basis functions of an irreducible representation of O(3) GTAngularMomentumChars gives the representation matrix of a symmetry element for an irreducible representation of O(3)

For the implementation of irreducible representations of O(3), SO(3) and SU(2) we follow [1].

[1] Altman, S. L., Rotations, quaternions, and double groups. Chapter 14. Clarendon, 1986

Install the point group Oh (GTInstallGroup)
 In[24]:=
 Out[24]=
Calculate the character Table (GTCharacterTable)
 In[25]:=
Calculate the character system of a single s, p, and d electron
 In[26]:=
 Out[26]=
 In[27]:=
 Out[27]=
 In[28]:=
 Out[28]=
Calculate the qualitative splitting in a cubic crystal field
 In[29]:=
 Out[29]=
 In[30]:=
 Out[30]=
 In[31]:=
 Out[31]=
Calculate a matrix representation for d-electrons
 In[32]:=
 In[33]:=
 Out[33]//MatrixForm=
 Out[33]//MatrixForm=
 Out[33]//MatrixForm=