# Simple proof of oscillation/nodal theorem in quantum mechanics [duplicate]

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?

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.

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• 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. – ZeroTheHero Jan 13 '17 at 0:52