In this blog I take you to an original argument in the book, by Landau Lifshitz's on Qunatum Mechanics, about using uncertainty principle to determine if the number of bound states for a potential will be finite or infinite.

I am a fan of two books, not just books but masterpieces, on quantum mechanics. Dirac's "Principles of Quantum Mechanics" and "Quantum Mechanics by 'Landau Lifshitz. You have to read them to see what you have missed. Here I reproduce a few pages from Landau Lifshitz on Uncertainty Principle. One of the earliest result I learnt from "Dau" was that bound states in one dimension are nondegenerate. I was thrilled that I went around telling everyone about it. Let alone the proof, I had not seen this even as a statement in any other book.

Heisenberg uncertainty principle is applied to get an understanding of a wide range of qunatum phenomena. The most common text book application, and a frequently ask questions in various interviews is its applicaton to zero point energy of the harmonic oscillator or to the ground state of hydrogen atom. There are many other well known applications.

While going through the ``NonRelativistic Quantum Mechanics'' by Landau Lifshitz, I came across an argument about number of bound states that a potential can hold using uncertainty principle. This is masterpiece and not found elsehwere. The relevant paragraph is reproduced below for the the benefit of younger generation and to motivate them to learn from masters.

The equation (18.2) referred to in the above is \begin{equation*} U(r) = -\alpha r^{-s}, \qquad \qquad (\alpha>0).\qquad\quad\qquad (18.2)\end{equation*}

There is a lot more to be learnt from Section 18, where the above discussion appears. The title of the section is ``Fundamnetal prperties of the

Schrodinger equation`` and several general properties of the solutions are discussed. This is strongly recommended for every student and teacher of quantum mechanics.

L. D. Landau and E. M. Lifshitz,"Quantum Mechanics- Non relativistic" Pergamon Press Ldt, Second revised edition (1965)