Chapter 8 Vector Spaces in Quantum Mechanics

Chapter 8 Vector Spaces in Quantum Mechanics 104The probability amplitude 〈x|ψ〉 appearing in Eq. (8.80) are functions of the continuousvariable x, and is often written 〈x|ψ〉 = ψ(x), which we identify as the wave function ofthe particle. If we now consider the inner productor〈x ′ |ψ〉 =ψ(x ′ ) =∫ +∞−∞∫ +∞−∞〈x ′ |x〉〈x|ψ〉dx (8.82)〈x ′ |x〉ψ(x)dx (8.83)we now see that we have an interesting difficulty. We know that 〈x ′ |x〉 = 0 if x ′ x, soif 〈x|x〉 is assigned a finite value, the integral on the right hand side will vanish, so thatψ(x) = 0 for all x!! But if ψ(x) is to be a non-trivial quantity, i.e. if it is not to be zero forall x, then it cannot be the case that 〈x|x〉 is finite. In other words, 〈x ′ |x〉 must be infinitefor x = x ′ in some sense in order to guarantee a non-zero integral. The way in which thiscan be done involves introducing a new ‘function’, the Dirac delta function, which hassome rather unusual properties.What we are after is a ‘function’ δ(x − x 0 ) with the property thatf (x 0 ) =for all (reasonable) functions f (x).∫ +∞−∞δ(x − x 0 ) f (x)dx (8.84)So what is δ(x − x 0 )? Perhaps the simplest way to get at what this function looks like isto examine beforehand a sequence of functions defined byD(x, ɛ) = ɛ −1 −ɛ/2 < x < ɛ/2= 0 x < −ɛ/2, x > ɛ/2. (8.85)