Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Quantum Many-Body Systems with Q3
Mathematica(R) package Q3 helps study quantum many-body systems. It provides various tools and utilities for symbolic and numerical calculations to simulate quantum many-body systems.
Represents fermionic operators | |
Represents operators for Majorana fermions | |
Represents bosonic operators | |
Represents operators satisfying the Heisenberg canonical commutation relations |
Species involved in the study of quantum many-body systems.
You first need to load the package.
In[1]:=
Needs["QuantumMob`Q3`"]
Let us consider a system of fermions. Choose a symbol, here c, to denote the fermions:
In[4]:=
Let[Fermion,c]
Here is an example of the non-commutative multiplication of fermion operators:
In[3]:=
c[1]**c[2]
Out[3]=
-
c
2
c
1
This returns the state:
Vacuum
In[4]:=
v1=Ket[]
Out[4]=
|␣〉
In[5]:=
v1//InputForm
Out[5]//InputForm=
Ket[<||>]
The fermion operators act on the state vectors:
In[6]:=
v2=Dagger[c[1]]**Dagger[c[2]]**Ket[]
Out[6]=
|
1
c
1
1
c
2
Hamiltonians are written in terms of Fermion operators.
In[7]:=
op=Q[c[1],c[2]]+tPlusDagger@FockHopping[c[1],c[2]]
Out[7]=
†
c
1
c
1
†
c
1
c
2
†
c
2
c
1
†
c
2
c
2
Calculate the matrix representation in the computational basis:
In[8]:=
mat=Matrix[op];mat//MatrixForm
Out[9]//MatrixForm=
0 | 0 | 0 | 0 |
0 | 1 | t | 0 |
0 | t | 1 | 0 |
0 | 0 | 0 | 2 |
Now, consider a Boson operator:
In[115]:=
Let[Boson,a]
In[140]:=
op=Hop[a@{1,2,3}];op=PlusDagger[op]
Out[141]=
†
a
1
a
2
†
a
2
a
1
†
a
2
a
3
†
a
3
a
2
In[142]:=
ket=Dagger[a[1]]**Ket[]
Out[142]=
|
1
a
1
In[143]:=
op**ket
Out[143]=
|
1
a
2
 |
|
 |
|
""