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define the intergral：I=∫dτ|ξAφ+iBφ|^2；
Where A ,B are Hermitian Operators；ξ is a C-Number,φ is the probability wave.Obviously:
I≥0 and
I=[<φ|(ξA+iB)][|(ξA+iB)|φ>]
=ξ^2(<φ|A)(A|φ>)+iξ(<φ|A)(B|φ>)-(<φ|B)(A|φ>)+(<φ|B)(B|φ>)
=ξ^2<A^2>-ξ<C>+<B^2>
Where we define another Hermitian Operator C,C=-i[A,B].Then we treat I=I(ξ) as the quadratic function of ξ.Because <A^2> would always be positive,therefore the condition which makes I(ξ)≥0 should be：
<C>^2-4<A^2><B^2>≤0
That means：<A^2><B^2>≥<C>^2/4
However if we define：ΔA=A-<A>；ΔB=B-<B>；we would find：[ΔA,ΔB]=iC；and
I(ξ)=ξ^2<A>^2-ξ<C>+<ΔB>^2≥0
It makes：<ΔA^2><ΔB^2>≥<C>^2/4
That is：<ΔA><ΔB>≥|<C>|/2
Therefore if A→x(coordinate)；B→p(momentum),we would have：C=-i[x,p]=h/2π；
So the uncertainty relations can be usually write as：<Δx><Δp>≥h/4π
Where A ,B are Hermitian Operators；ξ is a C-Number,φ is the probability wave.Obviously:
I≥0 and
I=[<φ|(ξA+iB)][|(ξA+iB)|φ>]
=ξ^2(<φ|A)(A|φ>)+iξ(<φ|A)(B|φ>)-(<φ|B)(A|φ>)+(<φ|B)(B|φ>)
=ξ^2<A^2>-ξ<C>+<B^2>
Where we define another Hermitian Operator C,C=-i[A,B].Then we treat I=I(ξ) as the quadratic function of ξ.Because <A^2> would always be positive,therefore the condition which makes I(ξ)≥0 should be：
<C>^2-4<A^2><B^2>≤0
That means：<A^2><B^2>≥<C>^2/4
However if we define：ΔA=A-<A>；ΔB=B-<B>；we would find：[ΔA,ΔB]=iC；and
I(ξ)=ξ^2<A>^2-ξ<C>+<ΔB>^2≥0
It makes：<ΔA^2><ΔB^2>≥<C>^2/4
That is：<ΔA><ΔB>≥|<C>|/2
Therefore if A→x(coordinate)；B→p(momentum),we would have：C=-i[x,p]=h/2π；
So the uncertainty relations can be usually write as：<Δx><Δp>≥h/4π
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