Question

1 Semiquantitative Results Using Semiclassical Quantization In this problem, you will analyze the consequences of the de Broglie relations (i.e., Bohr-Sommerfeld quantization) on the motion of particles in the same potential as in Problem 3 of Problem Set #3, V(r) = v. ()°. (1) 1.1 Classical orbits Using F = mã, show that for a classical orbit in the potential (1), pº = mav (r), and that the total energy of the particle is E = + V(r) = (;?(r). 1.2 Semiclassical quantization Using your results above, compute n, the number of de Broglie wavelengths l = h/p which fit in the circumference an orbit of radius r. Show that -- (, ()) *. If we assume that to avoid destructive interference that n must be a natural number 1, 2, 3, ..., then this result tells you the allowed radii for the orbits of the mass m. Under the same assumption, determine the allowed momenta p and the allowed energies E for the particle. Hint: You should find that the allowed energies are En = (0:2(12) *** where n = 1, 2, 3, .... 1.3 Comparison with Results from the HUP Show, for both cases a < 0 and a > 0, that the lowest energy states correspond to n = 1 de Broglie wavelengths around the orbit. Compute the ratio of this estimate of the lowest energy with what you obtained using the HUP in Problem 3 of Problem Set 3, and show that the result is near unity for a = -1, a = 2 and a + 0.

Answer 1

0(Vor- 2 ak 十2 2 2

bate: m ( 2万
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