Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Quantum Spin Systems with Q3
Wigner is a Mathematica package to help study quantum spin systems.
Frequently used functions from the Wigner package.
Basic Usage
Load the application.
In[114]:=
Needs["QuantumMob`Q3`"]
Consider S=1 spins:
In[118]:=
Let[Spin,S,Spin1]
In[119]:=
v=Ket[S[None]0]v//InputForm
Out[119]=
|〉
0
S
Out[120]//InputForm=
Ket[<|S[$] -> 0|>]
In[121]:=
v=Ket[S[None]0]v//InputForm
Out[121]=
|〉
0
S
Out[122]//InputForm=
Ket[<|S[$] -> 0|>]
In[123]:=
u=S[1]**v
Out[123]=
1
2
2
(-1)
S
2
1
S
In[124]:=
matU=Matrix[u];matU//MatrixForm
Out[125]//MatrixForm=
1 2 |
0 |
1 2 |
In[126]:=
uu=ExpressionFor[matU,S]u-uu//Elaborate
Out[126]=
|〉
(-1)
S
2
|〉
1
S
2
Out[127]=
0
In[128]:=
S[1]**uu
Out[128]=
|〉
0
S
Consider a system of many spins.
In[129]:=
Let[Spin,S]
In[130]:=
v=Ket[S[1]-1/2]
Out[130]=
-
1
2
S
1
You can add additional elements or modifies existing elements.
In[131]:=
v=Ket[v,S[3]1/2]v=Ket[v,S[2]-1/2]v=Ket[v,S[2]-1/2]
Out[131]=
-
1
2
S
1
1
2
S
3
Out[132]=
-
1
2
S
1
-
1
2
S
2
1
2
S
3
Out[133]=
-
1
2
S
1
-
1
2
S
2
1
2
S
3
The spin values are automatically verified:
In[134]:=
Let[Spin,S]
In[135]:=
w=Ket[S[1]1]
::spin
S
1
Out[135]=
|␣〉
Accessing the values
In[136]:=
u=Ket[S[1]1/2,S[2]-1/2]u[S[2]]u[S[3]]u[{S[1],S[3]}]
Out[136]=
1
2
S
1
-
1
2
S
2
Out[137]=
-
1
2
Out[138]=
1
2
Out[139]=
,
1
2
1
2
In[140]:=
Let[Spin,S,T]
In[141]:=
Ket[S[{1,2,3}]1/2]
Out[141]=
1
2
S
1
1
2
S
2
1
2
S
3
In[142]:=
v=(3Ket[S[{1,2,3}]1/2]+4Ket[S[1]-1/2,S[2]-1/2,S[3]-1/2])/5vv=S[1,0]**vuu=T[1,0]**v
Out[142]=
1
5
-
1
2
S
1
-
1
2
S
2
-
1
2
S
3
1
2
S
1
1
2
S
2
1
2
S
3
Out[143]=
4
5
-
1
2
S
1
-
1
2
S
2
-
1
2
S
3
3
5
1
2
S
1
1
2
S
2
1
2
S
3
Out[144]=
4
5
-
1
2
S
1
-
1
2
S
2
-
1
2
S
3
3
5
1
2
S
1
1
2
S
2
1
2
S
3
In[145]:=
Dagger[v]**vDagger[v]**S[1,0]**vDagger[v]**T[1,0]**v
Out[145]=
1
Out[146]=
1
Out[147]=
1
In[148]:=
Let[Spin,J,Spin3/2]
In[149]:=
v=Ket[]v1=J[1,3]**vv2=J[3,1]**J[1,1]**J[2,1]**v1//Simplifyv3=J[1,1]**J[1,2]**v
Out[149]=
|␣〉
Out[150]=
3|␣〉
2
Out[151]=
9
16
3
1
2
J
1
1
2
J
2
1
2
J
3
More Examples
To the column vector form, then back to the Ket form.
To be compared with the Ket form.
Sometime one has to manually put spin values. In such a case, the following form will be useful to avoid tedious typing. Notice the Null input at the end, which will be automatically converted to the default value.