Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
QuantumMob`Q3mini`
Fermion |
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Details and Options
Examples
(4)
Basic Examples
(3)
Fermions are declared as follows
In[1]:=
Let[Fermion,c,Vacuum"Sea"]Let[Fermion,d]Let[Fermion,f,Spin0,Vacuum"Sea"]
By the above declaration, c[i,j,...] are all bosonic annihilation operators, and Dagger[ci,j,...]] are creation operators.
In[2]:=
c[Down]**c[Up]Dagger[c[1,Up]]**Dagger[c[2,Down]]
Out[2]=
-
c
↑
c
↓
Out[2]=
†
c
1,↑
†
c
2,↓
The defining properties of bosonic operators are the canonical commutation relations
In[3]:=
c[Up]**Dagger[c[Up]]+Dagger[c[Up]]**c[Up]1c[Up]**c[Up]-c[Up]**c[Up]0Dagger[c[Up]]**Dagger[c[Up]]+Dagger[c[Up]]**Dagger[c[Up]]0
Out[3]=
True
Out[3]=
True
Out[3]=
True
These commutators can be conveniently assessed by Commutator[ ] and Anticommutator[ ]
In[4]:=
c[i]**c[j]c[i]**Dagger[c[j]]Dagger[c[i]**c[j]]Anticommutator[c[i],Dagger[c[j]]]
Out[4]=
-
c
j
c
i
Out[4]=
δ
i,j
†
c
j
c
i
Out[4]=
-
†
c
i
†
c
j
Out[4]=
δ
i,j
In[1]:=
Let[Fermion,c,Vacuum"Sea"]Let[Fermion,d]
A list in the Flavor indices of operators are treated specially. It is equivalent to the list of operators
In[2]:=
c[{1,2,3},Up]c[{Up,Down}]c[{1,2},{Up,Down}]c[{1,2},All]
Out[2]=
{,,}
c
1,↑
c
2,↑
c
3,↑
Out[2]=
{,}
c
↑
c
↓
Out[2]=
{,,,}
c
1,↑
c
1,↓
c
2,↑
c
2,↓
Out[2]=
{,,,}
c
1,↑
c
1,↓
c
2,↑
c
2,↓
The final spin index can be All to have all possible spin components :
In[3]:=
d[All]
Out[3]=
{,}
d
↑
d
↓
For c, which has the Sea vacuum, requires another Flavor index; otherwise, All is regarded as a normal symbol.
In[4]:=
c[All]c[1,All]
Out[4]=
c
All
Out[4]=
{,}
c
1,↑
c
1,↓
In[5]:=
c[{1,2,3},All]
Out[5]=
{,,,,,}
c
1,↑
c
1,↓
c
2,↑
c
2,↓
c
3,↑
c
3,↓
The phase convention of Ket involving fermions is such that |␣〉=,,…,.
†
c
1
†
c
2
†
…c
n
1
c
1
1
c
2
1
c
n
In[1]:=
bs=Basis[c@{1,2}]
Out[1]=
,,,
0
c
1
0
c
2
0
c
1
1
c
2
1
c
1
0
c
2
1
c
1
1
c
2
In[2]:=
c[1]**bs
Out[2]=
0,0,,
0
c
1
0
c
2
0
c
1
1
c
2
In[3]:=
c[2]**bs
Out[3]=
0,,0,-
0
c
1
0
c
2
1
c
1
0
c
2
In[4]:=
Dagger[c[1]]**bs
Out[4]=
,,0,0
1
c
1
0
c
2
1
c
1
1
c
2
In[5]:=
Dagger[c[2]]**bs
Out[5]=
,0,-,0
0
c
1
1
c
2
1
c
1
1
c
2
Scope
(1)
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""