Download
back to solutions of schr dinger s equa n.
Skip this Video
Loading SlideShow in 5 Seconds..
Back To Solutions of Schrödinger's equa. PowerPoint Presentation
Download Presentation
Back To Solutions of Schrödinger's equa.

Back To Solutions of Schrödinger's equa.

103 Views Download Presentation
Download Presentation

Back To Solutions of Schrödinger's equa.

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

  1. Back To Solutions of Schrödinger's equa.

  2. Particle in a Box

  3. Particle in a Box E4 Energy E3=9E1 E2=4E1 E1 E=0

  4. Particle in a Box (Quantization of Momentum) • Using the momentum operator we can determine the avg momentum.

  5. Particle in a Box (Quantization of Momentum) • Using the momentum operator we can determine the avg momentum.

  6. Particle in a Box (Quantization of Momentum) • Using the momentum operator we can determine the avg momentum.

  7. Particle in a Box (Quantization of Momentum)

  8. Particle in a Box (Quantization of Momentum) • Integrating by parts we get

  9. Particle in a Box (Quantization of Momentum) • Integrating by parts we get

  10. Particle in a Box (Quantization of Momentum) • Integrating by parts we get • Hence avg. momentum =0

  11. Particle in a Box (Finite Square Well) • The infinite potential is an oversimplification which can never be realised.

  12. Particle in a Box (Finite Square Well) • The infinite potential is an oversimplification which can never be realised. • A more realistic finite square well is shown U E 0 L

  13. Particle in a Box (Finite Square Well) • The infinite potential is an oversimplification which can never be realised. • A more realistic finite square well is shown Regions of interest are shown (1-3) U E 1 2 3 0 L

  14. Particle in a Box (Finite Square Well) • Practical, given enough energy a particle can escape any well.

  15. Particle in a Box (Finite Square Well) • Practically, given enough energy a particle can escape any well. • A classical particle with E>U can enter the well where it moves freely with a reduced energy (E-U).

  16. Particle in a Box (Finite Square Well) • Practically, given enough energy a particle can escape any well. • A classical particle with E>U can enter the well where it moves freely with a reduced energy (E-U). • A classical particle with E<U is trapped within the well (can not escape).

  17. Particle in a Box (Finite Square Well) • Practically, given enough energy a particle can escape any well. • A classical particle with E>U can enter the well where it moves freely with a reduced energy (E-U). • A classical particle with E<U is trapped within the well (can not escape).

  18. Particle in a Box (Finite Square Well) • In QM there is a probability of it being outside the well.

  19. Particle in a Box (Finite Square Well) • In QM there is a probability of it being outside the well. That is the waveform is nonzero outside the region 0<x<L.

  20. Particle in a Box (Finite Square Well) • In QM there is a probability of it being outside the well. That is the waveform is nonzero outside the region 0<x<L. • For stationary states is found from the time independent Schrödinger’s equation.

  21. Particle in a Box (Finite Square Well) • In QM there is a probability of it being outside the well. That is the waveform is nonzero outside the region 0<x<L. • For stationary states is found from the time independent Schrödinger’s equation.

  22. Particle in a Box (Finite Square Well) • In QM there is a probability of it being outside the well. That is the waveform is nonzero outside the region 0<x<L. • For stationary states is found from the time independent Schrödinger’s equation.

  23. Particle in a Box (Finite Square Well) • Therefore outside the well where U(x)=U,

  24. Particle in a Box (Finite Square Well) • Therefore outside the well where U(x)=U, • where

  25. Particle in a Box (Finite Square Well) • Therefore outside the well where U(x)=U, • where • Since U>E, this term is positive.

  26. Particle in a Box (Finite Square Well) • Therefore outside the well where U(x)=U, • where • Since U>E, this term is positive. • Independent solns to the differential are

  27. Particle in a Box (Finite Square Well) • To keep the waveform finite as and

  28. Particle in a Box (Finite Square Well) • To keep the waveform finite as and • Therefore the exterior wave takes the form

  29. Particle in a Box (Finite Square Well) • To keep the waveform finite as and • Therefore the exterior wave takes the form • The internal wave is given as before by

  30. Particle in a Box (Finite Square Well) • The coefficients are determined by matching the exterior wave smoothly onto the wavefunction for the well interior.

  31. Particle in a Box (Finite Square Well) • The coefficients are determined by matching the exterior wave smoothly onto the wavefunction for the well interior. • That is, and are continuous at the boundaries.

  32. Particle in a Box (Finite Square Well) • The coefficients are determined by matching the exterior wave smoothly onto the wavefunction for the well interior. • That is, and are continuous at the boundaries. • This is obtained for certain values of E.

  33. Particle in a Box (Finite Square Well) • Because is nonzero at the boundaries, the de Broglie wavelength is increase and hence lowers the energy and momentum of the particle.

  34. Particle in a Box (Finite Square Well) • The solution for the finite well is

  35. Particle in a Box (Finite Square Well) • The solution for the finite well is • As long as is small compared to L.

  36. Particle in a Box (Finite Square Well) • The solution for the finite well is • As long as is small compared to L. • where

  37. Particle in a Box (Finite Square Well) • The solution for the finite well is • As long as is small compared to L. • where • The approximation only works for bounds states. And is best for the lowest lying states.

  38. Particle in a Box (Finite Square Well)

  39. The Quantum Oscillator • We now look at the final potential well for which exact results can be obtained.

  40. The Quantum Oscillator • We now look at the final potential well for which exact results can be obtained. • We consider a particle acted on by a linear restoring force .

  41. The Quantum Oscillator • We now look at the final potential well for which exact results can be obtained. • We consider a particle acted on by a linear restoring force . -A A X=0

  42. The Quantum Oscillator • We now look at the final potential well for which exact results can be obtained. • We consider a particle acted on by a linear restoring force . • x rep. its displacement from stable equilibrium(x=0).

  43. The Quantum Oscillator • We now look at the final potential well for which exact results can be obtained. • We consider a particle acted on by a linear restoring force . • x rep. its displacement from stable equilibrium(x=0). • Strictly applies to any object limited to small excursions about equilibrium.

  44. The Quantum Oscillator • The motion of a classical oscillator with mass m is SHM at frequency

  45. The Quantum Oscillator • The motion of a classical oscillator with mass m is SHM at frequency • If the particle is displaced so that it oscillated between x=A and x=-A, with total energy the particle can be given any (nonnegative) energy including zero.

  46. The Quantum Oscillator • The quantum oscillator is described by introducing the potential energy into Schrödinger's equation.

  47. The Quantum Oscillator • The quantum oscillator is described by introducing the potential energy into Schrödinger's equation. • So that,

  48. The Quantum Oscillator • The quantum oscillator is described by introducing the potential energy into Schrödinger's equation. • So that,

  49. The Quantum Oscillator • The quantum oscillator is described by introducing the potential energy into Schrödinger's equation. • So that,

  50. The Quantum Oscillator • We can consider properties which our wavefunction must and can’t have.