l13 - Review Finite square well Tunneling Approximate...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Review Finite square well Tunneling Approximate energy solution Review • We have looked at several examples of the time-independent Schrödinger equation: − planckover2pi1 2 2 m ∂ 2 ∂ x 2 ψ + U ψ = E ψ • We have solved the infinite quantum well. • We have looked at the quantum mechanical harmonic oscillator: • The ground state looks like ψ = 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 ω ) 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 ψ (probability amplitude) and ψ 2 (probability). Review Finite square well Tunneling Approximate energy solution Finite square well (Serway 6.5) • We’ve done a particle in a restoring force potential (the harmonic oscillator), and also in an infinite square well. Let’s 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 ), Schrödinger’s equation for a particle traveling in a constant finite potential is − planckover2pi1 2 2 m d 2 ψ dx 2 + U ψ = E ψ planckover2pi1 2 2 m d 2 ψ dx 2 = ( U − E ) ψ d 2 ψ dx 2 = α 2 ψ with α 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 ψ dx 2 = α 2 ψ with α 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...
View Full Document

{[ snackBarMessage ]}

Page1 / 12

l13 - Review Finite square well Tunneling Approximate...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online