In L. D. Landau and E. M. Lifshitz Quantum Mechanics it is said about oscillation theorem, is there any short and simple way to prove it, which would be understandable for the 2nd year student of math/physics department?

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As one can see it states that in bounded (let it be one dimensional) space $\psi$ function of the ground energy state have zero "zeroes" and in $n$th excited state have exactly $n$ "zeroes". How one can prove it in short and easy way?

Note: this is not a hometasklike question. This is a try to deeper understand the general physics course.


marked as duplicate by Qmechanic quantum-mechanics Jan 29 '17 at 16:23

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    $\begingroup$ Related: physics.stackexchange.com/q/186140/50583, physics.stackexchange.com/q/75205/50583 Also, please type out quotes instead of adding a picture of text - the pictures are not searchable for the text they contain, hence less useful than actual text. $\endgroup$ – ACuriousMind Jan 12 '17 at 14:20
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    $\begingroup$ I sketched a proof of the nodal theorem in my Phys.SE answer here. $\endgroup$ – Qmechanic Jan 12 '17 at 14:35
  • $\begingroup$ Thx, @Qmechanic, it is very helpful. $\endgroup$ – blitzar Jan 12 '17 at 14:48
  • $\begingroup$ Also look up "Sturm-Liouville problem". $\endgroup$ – Blazej Jan 12 '17 at 15:56
  • $\begingroup$ There is a nice proof of a weaker result in the textbook by Albert Messiah. He shows that, if $E_2> E_1$, then $\psi_2$ must have more zeroes than $\psi_1$. The stronger result depends on technical properties of solutions to Sturm-Liouville systems. $\endgroup$ – ZeroTheHero Jan 13 '17 at 0:52