6.2. FOUNDATIONS OF QUANTUM PHYSICS 341
In statistics, the error to an average value is expressed by the squared deviation. Namely, the
errors to〈x〉and〈p〉are
〈(∆x)^2 〉=∫
ψ∗(x− 〈x〉)^2 ψdx=〈x^2 〉 − 〈x〉^2 ,〈(∆p)^2 〉=∫
ψ∗(px− 〈px〉)^2 ψdx=〈p^2 x〉 − 〈px〉^2.Assume that〈x〉=0 and〈px〉=0. Then we have
(6.2.39)
〈(∆x)^2 〉=〈x^2 〉=∫
x^2 |ψ|^2 dx,
〈(∆p)^2 〉=〈p^2 x〉=∫
ψ∗(
−i ̄h∂∂x) 2
ψdx.To get the relation (6.2.38) we consider the integral
(6.2.40) I(α) =
∫
(αxψ∗+∂ ψ∗
∂x
)(αxψ+∂ ψ
∂x
)dx,whereαis a real number. The integral (6.2.40) can be written as
I(α) =Aα^2 −Bα+C,where
(6.2.41)
A=
∫
x^2 |ψ|^2 dx=〈(∆x)^2 〉 by (6.2.39),B=−
∫
x∂
∂x|ψ|^2 dx=∫
|ψ|^2 dx= 1 ,C=
∫
∂ ψ∗
∂x∂ ψ
∂xdx=1
̄h^2∫
ψ∗(−ih ̄∂
∂x)^2 ψdx=1
h ̄^2〈(∆p)^2 〉 by (6.2.39).It is clear thatA,B,C>0, and by (6.2.40),
(6.2.42) I(α)≥ 0 ∀α∈R^1.
Letα 0 be the minimal ofI(α). Thenα 0 satisfies
I′(α 0 ) = 2 Aα 0 −B= 0 ⇒α 0 =B
2 A
.
Insertingα 0 =B/ 4 Ain (6.2.42) we get that
AC≥1
4
B^2.
It follows from (6.2.41) that
〈(∆x)^2 〉〈(∆p)^2 〉 ≥h ̄^2
4,
which is the first relation of (6.2.38). The second relation (6.2.38) can be derived in the same
fashion.
The Heisenberg uncertainty relation (6.2.38) has profound physically implications, some
of which are listed as follows: