"Consider an infinitely sharp pin of mass M and height H perfectly balanced on its tip. Assume that the mass of the pin is all at the ball on the top of the pin. Classically, we expect the pin to remain in this state forever. Quantum mechanics, however, predicts that the pin will fall over within a finite amount of time. This can be shown as follows:(adsbygoogle = window.adsbygoogle || []).push({});

Show that the total energy of the pin (aka. the Hamiltonian) can be expressed in the

form:

E = Ap^2 − Bx^2

if we assume that x << H. p is the momentum of the pin and x is the lateral displacement of the head of the pin. Find expressions for A and B."

My attempt:

When the pin head moves laterally a distance x, it will have lost some gravitational energy equal to E_g = mg\sqrt{H^2-x^2}.

My first shot at this is to write E = P^2/(2m) + mgy, where y= \sqrt{H^2-x^2}; however, I don't think this can be reduced to the form called for. Suggestions?

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# Homework Help: Trying to write falling rod energies as Hamiltonian

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