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

8 v7defope.nb
SetQuantumObject@qoD;
ColumnA9
CommutatorExpandAHope2 +qoL 2 E,
CommutatorExpandAHdirac2 +qoL 2 E
=, Dividers → AllE
ope2 2 +qo 2 − Pope2, qoT −
+2ope2 ⋅qo
qo 2 ∞
+ ⁄ j3=−∞
H2 +j3L ˆ q1
^ ⋅ Zj3 ˆ q1
+
∞
⁄ j=−∞
H1 +jL ˆ q1
^ ⋅ Zj ˆ q1
∞
⋅qo + ⁄ j=−∞ qo ⋅ H1 +jL ˆ q1
^ ⋅ Zj ˆ q1
The two expressions above are equivalent, as can be seen when they are applied to the same ket:
SetQuantumObject@qoD;
ColumnB:
CommutatorExpandAHope2 +qoL 2 E ⋅
CommutatorExpandAHdirac2 +qoL 2 E ⋅
100 ˆ
q1
^,
100 ˆ
q1
^
>, Dividers → AllF
qo 2 ⋅
100 ˆ q1
^ +qo ⋅
101 ˆ q1
^ +ope2 ⋅qo ⋅
100 ˆ q1
^ +
102 ˆ q1
^
∞
⁄ j=−∞
H1 +jL ˆ
q1
^ ⋅ Zj ˆ
q1
⋅qo ⋅ 100 ˆ
q1
^ +qo 2 ⋅
100 ˆ
q1
^ +qo ⋅
101 ˆ
q1
^ +
102 ˆ
q1
^
by José Luis Gómez-Muñoz
http://homepage.cem.itesm.mx/lgomez/quantum/
jose.luis.gomez@itesm.mx