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QuantumMob`Q3mini`
Boson |
Details and Options
Examples
(2)
Basic Examples
(1)
Bosons are declared as follows
In[1]:=
Let[Boson,a,b]
By the above declaration, a[i,j,...] are all bosonic annihilation operators, and Dagger[a[i,j,...]] are creation operators.
This form is equivalent to the above form.
In[2]:=
a[2]**a[1]Dagger[a[1]]**a[2]a[i]**Dagger[a[j]]
Out[2]=
a
2
a
1
Out[2]=
†
a
1
a
2
Out[2]=
δ
i,j
†
a
j
a
i
The defining properties of bosonic operators are the canonical commutation relations
In[3]:=
a[1]**Dagger[a[1]]-Dagger[a[1]]**a[1]1a[1]**a[1]-a[1]**a[1]0Dagger[a[1]]**Dagger[a[1]]-Dagger[a[1]]**Dagger[a[1]]0
Out[3]=
True
Out[3]=
True
Out[3]=
True
These commutators can be conveniently assessed by Commutator[ ] and Anticommutator[ ]
In[4]:=
a[i]**Dagger[a[j]]**a[k]Commutator[a[i],Dagger[a[j]]]
Out[4]=
a
k
δ
i,j
†
a
j
a
k
a
i
Out[4]=
δ
i,j
Note that operators with different Heads are assumed to be different, regardless of their Flavor indices:
In[5]:=
a[i]**Dagger[b[j]]Commutator[a[i],Dagger[b[j]]]
Out[5]=
†
b
j
a
i
Out[5]=
0
Scope
(1)
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