5.61 Fall 2007
Lecture #11
page 1
PRINCIPLES OF QUANTUM MECHANICS (cont’d)
ˆˆˆ ˆˆˆ ,ABABBA⎡⎤=−⎣⎦
COMMUTATORS
Order counts when applying multiple operators!
e.g.
 = p̂x̂ = ?
⇒
and can we write p̂x̂ = x̂p̂ ?
operate on function to obtain
()
()
ˆ x =g x
Af
= −i!x
∴
⇒
ˆA
.
⎛
⎞
( p̂x̂ ) f ( x ) = ⎜⎝ −i! dxd ⎟⎠ ( x ) f ( x )
⎛
⎞
d
d
f x − i!f x =
⎜ −i!x − i!⎟ f x
dx
dx
⎝
⎠
()
()
()
⎛
⎞
d
 = ⎜−i!x − i!⎟ = p̂x̂
dx
⎝
⎠
( )
Now try B̂ = x̂p̂
ˆ x = x ⎛ −i! d ⎞ f x = ⎛ −i!x d ⎞ f x
Bf
⎜⎝
⎜⎝
dx ⎟⎠
dx ⎟⎠
() ()
()
∴
B̂ = −i!x
∴
x̂p̂ ≠ p̂x̂
()
d
= x̂p̂
dx
Define commutator
For two operators  and B̂ ,
⎡ Â, B̂ ⎤ = ÂB̂ − B̂Â = Ĉ
⎣
⎦
e.g.
⎡⎣ x̂, p̂⎤⎦ = x̂p̂ − p̂x̂ = i! ≠ 0 !
need not be zero!
5.61 Fall 2007
Lecture #11
page 2
Important general statements about commutators:
1)
For operators that commute
⎡ Â, B̂ ⎤ = 0
⎣
⎦
• it is possible to find a set of wavefunctions that are eigenfunctions
of both operators simultaneously.
e.g. can find wavefunctions ψ n such that
Âψ n = anψ n
and
B̂ψ n = bnψ n
• This means that we can know the exact values of both observables
A and B simultaneously (no uncertainty limitation).
2)
For operators that do not commute
⎡ Â, B̂ ⎤ ≠ 0
⎣
⎦
• it is not possible to find a set of wavefunctions that are
simultaneous eigenfunctions of both operators.
• This means that we cannot know the exact values of both
observables A and B simultaneously
⇒
uncertainty !
e.g.
⎡⎣ x̂, p̂⎤⎦ = i! ≠ 0
⇒
ΔxΔp ≥
!
2