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QuantumMob`Q3mini`
Heisenberg |
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Examples
(2)
Basic Examples
(2)
Consider pairs of operators that satisfy Heisenberg’s canonical commutation relations.
In[1]:=
Let[Heisenberg,x]
In[2]:=
x[1]Canon[x[1]]
Out[2]=
x
1
Out[2]=
c
x
1
They satisfy the Heisenberg canonical commutation relation.
In[3]:=
Commutator[x[1],Canon[x[1]]]
Out[3]=
In[4]:=
x[1]**Canon[x[1]]
Out[4]=
+
c
x
1
x
1
In[5]:=
Dagger[x[1]]
Out[5]=
x
1
In[6]:=
Canon@x[2]**x[1]**Canon[x[1]]**x[2]
Out[6]=
+
c
x
2
x
2
c
x
1
c
x
2
x
2
x
1
In[7]:=
AnyHeisenbergQ@Canon@x[1]AnySpeciesQ@Canon@x[1]
Out[7]=
True
Out[7]=
True
In[8]:=
AnySpeciesQ[Canon[q[1]]]AnySpeciesQ[q[1]]
Out[8]=
False
Out[8]=
False
Consider the state vector.
In[1]:=
in=Ket[x[1]3]
Out[1]=
|
3
x
1
In[2]:=
x[1]**inCanon[x[1]]**in
Out[2]=
3
2
2
x
1
2
|4
x
1
Out[2]=
-|+
3
2
2
x
1
2
|4
x
1
In[3]:=
Dagger[in]**x[1]
Out[3]=
3
2
2
x
1
2
4
x
1
In[4]:=
Dagger[in]**Canon[x[1]]
Out[4]=
|-|
3
2
2
x
1
2
4
x
1
Compare the above results with the following.
In[5]:=
Let[Boson,a]
In[6]:=
in=Ket[a[1]3]
Out[6]=
|
3
a
1
In[7]:=
X=(a[1]+Dagger[a[1]])/Sqrt[2]P=(a[1]-Dagger[a[1]])/(I*Sqrt[2])
Out[7]=
a
1
†
a
1
2
Out[7]=
-
-
a
1
†
a
1
2
In[8]:=
X**inP**in
Out[8]=
3
2
2
a
1
2
|4
a
1
Out[8]=
-|+
3
2
2
a
1
2
|4
a
1
In[9]:=
Dagger[in]**XDagger[in]**P
Out[9]=
3
2
2
a
1
2
4
a
1
Out[9]=
|-|
3
2
2
a
1
2
4
a
1
 |
|
|
""