Uncertainty Relation between Two Operators ——–————————————— He | 𝑴𝒂𝒙 𝑷𝒍𝒂𝒏𝒄𝒌 𝒏𝒆𝒘𝒔𝒑𝒂𝒑𝒆𝒓

Uncertainty Relation between Two Operators
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Heisenberg was the first to question about assumption of classical physics. He published a paper in 1927 where he presented a detailed new analysis of the nature of experimentation. The most important feature of his paper was the observation that it is not possible to obtain information about the nature of a system without causing a change in the system. Heisenberg's observation became popularly known as the uncertainty principle; it is also referred to as the indeterminacy principle.

Heisenberg summarized his observation at the conclusion of his paper as follows: "In the classical law if we know the presence exactly we can predict the future exactly it is the assumption and not the conclusion that is incorrect."

An interesting application of the commutator algebra is to derive a general relation giving the uncertainties product of two operators, Â and B‌‌. In particular, we want to give a formal derivation of Heisenberg's uncertainty relations.
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Let and denote the expectation values of two Hermition operators A‌‌ and B‌‌ with respect to a normalized state vector |Ψ> :
=<ψ|A‌‌|ψ and =<ψ|B‌‌|> . Introducing the operators ΔA‌‌ and ΔB‌‌,
ΔA‌‌= A‌‌ - , ΔB‌‌=B‌‌-
we have (ΔA‌‌)² = A‌‌²-2A‌‌+²
and (ΔB‌‌)²=B‌‌²-2B‌‌+²
and hence :

<ψ|(ΔA‌‌)²|ψ>=<(ΔA‌‌)²>= - ² , <(ΔB‌‌)²>=-²
Where ²=<ψ|A‌‌²|ψ> and =<ψ|B‌‌²|ψ> The uncertainties ΔA‌‌ and ΔB‌‌ are defined by:

ΔA=√<(ΔA‌‌)²> =√-²
ΔB=√<(ΔB‌‌)²>=√-²
let us write the action of the operators on any state |ψ> as follows:
|Χ> = ΔA‌‌|ψ>=(A‌‌-)|ψ>
|φ>=ΔB‌‌|ψ>=(B‌‌-)|ψ>
The Schwarz inequality for the states |Χ> and |ψ> is given by :<Χ|X><Φ|Φ>≥ |<Χ|Φ>|²
Since A‌‌ and B‌‌ are Hermitian , ΔA‌‌ and ΔB‌‌ must also be Hermitian :

ΔA‌‌† = A‌‌† -=A‌‌-=ΔA‌‌ and ΔB‌‌†=B‌‌-=ΔB‌.Thus , we can show the following three relations:
=<ψ|(ΔA‌‌)²|ψ>
<Φ|Φ>=<ψ|(ΔB‌‌)²|ψ>
=<ψ|ΔA‌‌ΔB‌‌|ψ>
For instance , Since ΔA‌‌†=ΔA‌‌ we have =<ψ|ΔA‌‌†ΔA‌‌|ψ> = <ψ|(ΔA‌‌)²|ψ> Hence , the Schwarz inequality becomes :

<(ΔA‌‌)²> <(ΔB‌‌)²> ≥ |<ΔA‌‌ΔB‌‌>|²
Notice that the last term ΔA‌ΔB‌ of this equation can be written as:

ΔA‌‌ΔB‌‌=1/2 [ΔA‌‌,ΔB‌‌] + 1/2 {ΔA‌‌,ΔB‌‌} =1/2[A‌‌,B‌] +1/2{ΔA‌‌,ΔB‌‌}

Where we have used the fact that [ΔA‌‌,ΔB‌‌]=[A‌‌,B‌‌] . Since [A‌‌,B‌‌] is anti-Hermitian and {ΔA‌‌,ΔB‌‌} is Hermitian and since the expectation value of a Hermitian operator is imaginary , the expectation value <ΔA‌‌ΔB‌‌> becomes equal to the sum of a real part <{ΔA‌‌,ΔB‌‌}>/2 and an imaginary part <[A‌‌,B‌‌]>/2 ; hence:

|<ΔA‌‌ΔB‌‌>|²=1/4|<[A‌‌,B‌‌]>|² +1/4|<{ΔA‌‌,ΔB‌‌}>|²

Since the last term is a positive real number , we can infer the following relation :

|<ΔA‌‌ΔB‌‌>|² ≥ 1/4 |<[A‌‌,B‌‌]>|²

Comparing equations , we conclude that :

<(ΔA‌‌)²> <(ΔB‌‌)²> ≥ 1/4 |<[A‌‌,B‌‌]>|²
Which (by taking its square root) can be reduced to :

ΔA‌‌ΔB‌‌ ≥ 1/2 |<[A‌‌,B‌‌]>|
This uncertainty relation plays an important role in the formal-ism of quantum mechanics ,Its application to position and momentum operators leads to the Heisenberg uncertainty relations, which repre-sent one of the cornerstones of quantum mechanics ....
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Mohanned Qasim
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