Question

QM 30 Relating classical and quantum mechanics IV. Supplement: Highly-excited energy eigenstates A particle is in the potential well shown at right A. First, treat this problem from a purely elassical standpoint (assume the particle has enough energy to reach both regions) Give an example of a real physical situation that corresponds to this potential well. 1. region I | region II 2. In which region of the well would the particle have greater kinetic energy? Explain. 3. In which region of the well would the particle spend more time? Explain. Now treat the problem from a quantum mechanical point of view. Consider specifically the case in which the particle occupies a highly-excited energy eigenstate. B. What general shape do you expect for the function yx) in each region of the well? (Hint: Is the wave function linear, exponential, or sinusoidal) 1. 2. In which region is the wave number of the function w.(x) larger? Explain. 3. In which region is the amplitude of the function v.o) larger? Explain. Hint: Recall the correspondence principle.) ptx) Sketch a graph of this highly-excited eigenstate 4. Tutorials in Physics Qwuanium Mechanics CMcDermott,Heron, Shaffer, and P.E.G., U. Wash. Preliminary First Edition, 2015

Answer 1

Given Qua Patictes ate Scoton that v the Poknta stpe ute constant momentum. Jane. As the Pat kleJ Enlq Clasi the vegion So the Moon

Patide I More ti 3 aiaimun) y state ampstede) Since าง.ave num Xy As loop IJhn "he IS

the wave Ateanter the funcfien s 3, the ampis tide. Nave in 4