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QUANTUM PHYSICS I PROBLEM SET 2 due October 3rd, before class Semi-classical quantization for a polynomial potential In class we discussed the quantization of circular orbits of an electron around a nucleus considering only the Coulomb electrostatic force between the electron and the nucleus. Repeat the argument in the case of a modified interaction between electron and nucleus: assume the potential between them is V (r) = rαn . 1. Write them the quantization of angular momentum rule (which is independent of the potential) and the “F = ma”’ equation. 2. Determine the radii and energies of the allowed orbits A first look at the Uncertainty Principle 2 Consider a particle described at some particular instant of time by the wave function ψ(x) = Ae−ax . 1. Determine A so ψ is normalized. 2. Compute hxi, hx2 i and σx2 = h(x − hxi)2 i. 3. Compute hpi, hp2 i and σp2 = h(p − hpi)2 i. 4. Show that by changing a one can make either σx2 or σp2 small, but not both at the same time. Compute σx σp . Ehrenfest’s theorem Prove that ∂ hpi = ∂t ∂V (x) Ψ(x, t) − Ψ(x, t). ∂x −∞ Z ∞ ∗ (1) This result is one way to show that, under certain circumstances, macroscopic objects obey Newton’s law F = ma. Describe in words the connection of the formula above with Newton’s law.