Back To Solutions of Schrödinger's equa.

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.