This made me happy :)

Given the wave function

\[ \Psi(x) = \frac{N}{x^2 + a^2} \]

we are to find the values of n{0}Nn \in \left \{ 0 \right \} \bigcup \mathbb{N} such that xn\left \langle x^n \right \rangle has a meaningful physical value. And by xn\left \langle x^n \right \rangle we mean the following integral

+ΨxnΨ  dx \int_{-\infty}^{+\infty} \Psi^{*} x^{n} \Psi \; \mathrm{d}x

First, normalization will result in

+ΨΨ  dx=N2limt(axa2+x2+arctan(xa)2a3)tt=N2π2a3=1        N=2a3π \int_{-\infty}^{+\infty} \Psi^{*} \Psi \; \mathrm{d}x = N^{2} \lim _{t\rightarrow \infty} \left (\frac{\frac{ax}{a^2 + x^2} + \arctan \left ( \frac{x}{a} \right )}{2a^3} \right )_{-t}^{t} = \frac{N^{2}\pi}{2a^{3}} = 1 \;\; \therefore \;\; N = \sqrt{\frac{2a^{3}}{\pi}}

We can find xn\left \langle x^n \right \rangle taking this integral to the complex plane, and observing that if we allow the function Ψ \Psi to take complex values and we define

f(z)=zn(z2+a2)2 f(z) = \frac{z^{n}}{(z^{2} + a^{2})^{2}}

Then the residue of f f at iaia will be

Res(f,ia)=limziaddzf(z)=(ia)n1  1n4a2 \mathrm{Res}(f, ia) = \lim_{z\rightarrow ia} \frac{\mathrm{d} }{\mathrm{d} z} f(z) = (ia)^{n-1} \; \frac{1-n}{4a^{2}}

And according to a suitable theorem, we are able to write

+ΨxnΨ  dx=2πi×N×Res(f,ia)=2π2in(1n)an32 \int_{-\infty}^{+\infty} \Psi^{*} x^{n} \Psi \; \mathrm{d}x = 2 \pi i \times N \times \mathrm{Res}(f, ia) = \frac{\sqrt{2\pi}}{2} i^{n} (1-n) a^{n-\frac{3}{2}}

And then we are allowed to say that only when nn takes the values n=2kn = 2k with k{0}Nk \in \left \{ 0 \right \} \bigcup \mathbb{N} the integral has a physical meaning.

Note by Lucas Tell Marchi
6 years, 3 months ago

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