These days I met one problem and asked a professor for help. But I can not understand his answer. Can you help me explain his answer?(adsbygoogle = window.adsbygoogle || []).push({});

My question is that whether we can assume that a plane wave is orthogonal to the bound state of Hydrogen atom when t->\infty?

Professor answers:<\psi(t)(t)|\psi_0(t)>|^{t \to \infty}.Then the time dependence of this scalar product is \exp i(E_p+I_p)t where E_p =p^2/2m and I_p is the ionization energy. But exp(iEt) goes to zero when t goes to infinity so long as E is different from zero. Since there is no laser field E_p + I_p is always positive.

My questions:As far as I know, when t->\infty, exp(iEt) turns around the central point in the complex plane, but never goes to zero.

Professor answers:Regarding lim_{t -> \pm \infty} \exp{iEt}, you are completely right that |\exp{iEt}|=1 for any t, but that is not the point. Because of the rapid oscillations of both the real and the imaginary part of \exp{iEt} for t -> \pm \infty while E \ne 0, the function goes to zero "for all practical purposes". The latter statement means that \int dE f(E) \exp{iEt} will go to zero for t -> \infty for any sufficiently smooth function f(E) (called a test function). This is a pico summary of what is called theory of "distributions" or "improper functions". The best known example is the delta function \delta(x), which is zero for any value of x except x=0, while \int dx \delta(x) f(x) = f(0). One of many limit relations that can be used to define the delta function is i\pi\delta(x) = \lim_{t ->\infty} \exp{ixt}/x. This implies what I wrote down above.

I cannot follow him.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A plane wave vs the bound state of Hydrogen atom: orthogonal?

**Physics Forums | Science Articles, Homework Help, Discussion**