WOLFRAM|DEMONSTRATIONS PROJECT

Fundamental Commutation Relations in Quantum Mechanics

​
operator 1
x
y
z
p
x
p
y
p
z
L
x
L
y
L
z
operator 2
x
y
z
p
x
p
y
p
z
L
x
L
y
L
z
reverse order
x  x
p
x
 -iℏ
∂
∂x
[x, ​
p
x
] = iℏ
All the fundamental quantum-mechanical commutators involving the Cartesian components of position, momentum, and angular momentum are enumerated. Commutators of sums and products can be derived using relations such as
[A,B+C]=[A,B]+[A,C]
and
[A,BC]=B[A,C]+[A,B]C
. For example, the operator
2
L
=
2
L
x
+
2
L
y
+
2
L
z
obeys the commutation relations

2
L
,
L
x
=
2
L
,
L
y
=
2
L
,
L
z
=0
.