l13 - Review Finite square well Tunneling Approximate...

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Unformatted text preview: Review Finite square well Tunneling Approximate energy solution Review We have looked at several examples of the time-independent Schrdinger equation: planckover2pi1 2 2 m 2 x 2 + U = E We have solved the infinite quantum well. We have looked at the quantum mechanical harmonic oscillator: The ground state looks like = A exp [- ax 2 ] . The length scale is a = km / 2 planckover2pi1 , but particles can go farther than the classical limit. However, larger excursions are exponentially killed . . . The state energies go like E n = ( n + 1 2 planckover2pi1 ) with n = , 1 , 2 , . . . At zero temperature, motion does not cease! Zero temperature means all particles are in their ground state. We also talked about the meaning of (probability amplitude) and 2 (probability). Review Finite square well Tunneling Approximate energy solution Finite square well (Serway 6.5) Weve done a particle in a restoring force potential (the harmonic oscillator), and also in an infinite square well. Lets now consider a finite square well! x =0 x = L U =0 U = U I II III Based on our solution to the harmonic oscillator, we expect that a particle should be sort-of confined but that it might extend out past its classical limit. Outside the well (regions I and III ), Schrdingers equation for a particle traveling in a constant finite potential is planckover2pi1 2 2 m d 2 dx 2 + U = E planckover2pi1 2 2 m d 2 dx 2 = ( U E ) d 2 dx 2 = 2 with 2 2 m ( U E ) planckover2pi1 2 Review Finite square well Tunneling Approximate energy solution Finite square well II Again, for outside the finite square well (regions I and III ) we had d 2 dx 2 = 2 with 2 2 m ( U E ) planckover2pi1 2 Now in the case where E > U we have a particle which happily travels along with a different net energy and thus a different de...
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l13 - Review Finite square well Tunneling Approximate...

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