Comparison of Symbolic Quantum Operators Versus Dirac-Notation ...

6 v7defope.nb
ColumnB:
ope2 4 ⋅ Jb 100 ˆ
q1
^ +c 200 ˆ
q1
^ +d 1020 ˆ
q1
^N,
dirac2 4 ⋅ Jb 100 q1ˆ ^ +c 200 ˆ
q1
^ +d 1020 ˆ
q1
^N
>, Dividers → AllF
b 104 ˆ q1
^ +c 204 ˆ q1
^ +d 1024 ˆ q1
^
b 104 ˆ
q1
^ +c 204 ˆ
q1
^ +d 1024 ˆ
q1
^
Differences between an operator created with DefineOperatorOnKets[] and an operator
created with kets and bras
These are the definitions of the two operators we are working with. The operator dirac2 is created from kets and bras, while
the symbolic operator ope2 is created with the Quantum Mathematica command DefineOperatorOnKets[]:
Needs@"Quantum`Notation`"D;
∞
dirac2 = ‚
j=−∞
Hj +1L ˆ
q1
^ ⋅ Zj ˆ
q1
;
DefineOperatorOnKetsBope2, :
n_ ˆ
q1
^ ⧴
Hn +1L ˆ
q1
^>F;
Both have the same effect on a superposition of kets:
ColumnB:
ope2 ⋅ Jb 100 ˆ
q1
^ +c 200 ˆ
q1
^ +d 1020 ˆ
q1
^N,
dirac2 ⋅ Jb 100 ˆ
q1
^ +c 200 ˆ
q1
^ +d 1020 ˆ
q1
^N
>, Dividers → AllF
b 101 ˆ q1
^ +c 201 ˆ q1
^ +d 1021 ˆ q1
^
b 101 ˆ q1
^ +c 201 ˆ q1
^ +d 1021 ˆ q1
^
However the hermitian conjugate of ope2 has no action on kets, while the hermitian conjugate of dirac2 has a very specific
effect:
ColumnB:
Hope2L † ⋅ Jb 100 ˆ
q1
^ +c 200 ˆ
q1
^ +d 1020 ˆ
q1
^N,
Hdirac2L † ⋅ Jb 100 q1ˆ ^ +c 200 ˆ
q1
^ +d 1020 ˆ
q1
^N
>, Dividers → AllF
ope2 † ⋅ Jb 100 ˆ q1
^ +c 200 ˆ q1
^ +d 1020 ˆ q1
^N
b 99 ˆ q1
^ +c 199 ˆ q1
^ +d 1019 ˆ q1
^