Quantum Physics I Problem Set 2: Semi-classical Quantization and Uncertainty Principle - P, Assignments of Quantum Physics

Download Quantum Physics I Problem Set 2: Semi-classical Quantization and Uncertainty Principle - P and more Assignments Quantum Physics in PDF only on Docsity! 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) = α rn . 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 Consider a particle described at some particular instant of time by the wave function ψ(x) = Ae−ax 2 . 1. Determine A so ψ is normalized. 2. Compute 〈x〉, 〈x2〉 and σ2 x = 〈(x− 〈x〉)2〉. 3. Compute 〈p〉, 〈p2〉 and σ2 p = 〈(p− 〈p〉)2〉. 4. Show that by changing a one can make either σ2 x or σ2 p small, but not both at the same time. Compute σxσp. Ehrenfest’s theorem Prove that ∂ ∂t 〈p〉 = ∫ ∞ −∞ Ψ(x, t)∗ ( − ∂V (x) ∂x ) Ψ(x, t). (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.