Schrodinger’s Equation for Three Dimensions

1. Schrodinger’s Equation for Three Dimensions

2. QM in Three Dimensions • The one dimensional case was good for illustrating basic features such as quantization of energy.

3. QM in Three Dimensions • The one dimensional case was good for illustrating basic features such as quantization of energy. • However 3-dimensions is needed for application to atomic physics, nuclear physics and other areas.

4. Schrödinger's Equa 3Dimensions • For 3-dimensions Schrödinger's equation becomes,

5. Schrödinger's Equa 3Dimensions • For 3-dimensions Schrödinger's equation becomes, • Where the Laplacian is

6. Schrödinger's Equa 3Dimensions • For 3-dimensions Schrödinger's equation becomes, • Where the Laplacian is • and

7. Schrödinger's Equa 3Dimensions • The stationary states are solutions to Schrödinger's equation in separable form,

8. Schrödinger's Equa 3Dimensions • The stationary states are solutions to Schrödinger's equation in separable form, • The TISE for a particle whose energy is sharp at is,

9. Particle in a 3 Dimensional Box

10. Particle in a 3 Dimensional Box • The simplest case is a particle confined to a cube of edge length L.

11. Particle in a 3 Dimensional Box

12. Particle in a 3 Dimensional Box • The simplest case is a particle confined to a cube of edge length L. • The potential energy function is for • That is, the particle is free within the box.

13. Particle in a 3 Dimensional Box • The simplest case is a particle confined to a cube of edge length L. • The potential energy function is for • That is, the particle is free within the box. • otherwise.

14. Particle in a 3 Dimensional Box • Note: If we consider one coordinate the solution will be the same as the 1-D box.

15. Particle in a 3 Dimensional Box • Note: If we consider one coordinate the solution will be the same as the 1-D box. • The spatial waveform is separable (ie. can be written in product form):

16. Particle in a 3 Dimensional Box • Note: If we consider one coordinate the solution will be the same as the 1-D box. • The spatial waveform is separable (ie. can be written in product form): • Substituting into the TISE and dividing by we get,

17. Particle in a 3 Dimensional Box • The independent variables are isolated. Each of the terms reduces to a constant:

18. Particle in a 3 Dimensional Box • Clearly

19. Particle in a 3 Dimensional Box • Clearly • The solution to equations 1,2, 3 are of the form where

20. Particle in a 3 Dimensional Box • Clearly • The solution to equations 1,2, 3 are of the form where • Applying boundary conditions we find,

21. Particle in a 3 Dimensional Box • Clearly • The solution to equations 1,2, 3 are of the form where • Applying boundary conditions we find, • where

22. Particle in a 3 Dimensional Box • Clearly • The solution to equations 1,2, 3 are of the form where • Applying boundary conditions we find, • where • Therefore,

23. Particle in a 3 Dimensional Box • with and so forth.

24. Particle in a 3 Dimensional Box • with and so forth. • Using restrictions on the wave numbers and boundary conditions we obtain,

25. Particle in a 3 Dimensional Box • with and so forth. • Using restrictions on the wave numbers and boundary conditions we obtain,

26. Particle in a 3 Dimensional Box • with and so forth. • Using restrictions on the wave numbers and boundary conditions we obtain, • Thus confining a particle to a box acts to quantize its momentum and energy.

27. Particle in a 3 Dimensional Box • Note that three quantum numbers are required to describe the quantum state of the system.

28. Particle in a 3 Dimensional Box • Note that three quantum numbers are required to describe the quantum state of the system. • These correspond to the three independent degrees of freedom for a particle.

29. Particle in a 3 Dimensional Box • Note that three quantum numbers are required to describe the quantum state of the system. • These correspond to the three independent degrees of freedom for a particle. • The quantum numbers specify values taken by the sharp observables.

30. Particle in a 3 Dimensional Box • The total energy will be quoted in the form

31. Particle in a 3 Dimensional Box • The ground state ( ) has energy

32. Particle in a 3 Dimensional Box Degeneracy

33. Particle in a 3 Dimensional Box • Degeneracy: quantum levels (different quantum numbers) having the same energy.

34. Particle in a 3 Dimensional Box • Degeneracy: quantum levels (different quantum numbers) having the same energy. • Degeneracy is a natural phenomena which occurs because of the same in the system described (cubic box).

35. Particle in a 3 Dimensional Box • Degeneracy: quantum levels (different quantum numbers) having the same energy. • Degeneracy is a natural phenomena which occurs because of the same in the system described (cubic box). • For excited states we have degeneracy.

36. Particle in a 3 Dimensional Box • There are three 1st excited states having the same energy. They correspond to combinations of the quantum numbers whose squares sum to 6.

37. Particle in a 3 Dimensional Box • There are three 1st excited states having the same energy. They correspond to combinations of the quantum numbers whose squares sum to 6. • That is

38. 4E0 11/3E0 3E0 2E0 E0 Particle in a 3 Dimensional Box • The 1st five energy levels for a cubic box.

39. Schrödinger's Equa 3Dimensions • The formulation in cartesian coordinates is a natural generalization from one to higher dimensions.

40. Schrödinger's Equa 3Dimensions • The formulation in cartesian coordinates is a natural generalization from one to higher dimensions. • However it not often best suited to a given problem. Thus it may be necessary to convert to another coordinate system.

41. Schrödinger's Equa 3Dimensions • Consider an electron orbiting a central nucleus.

42. Schrödinger's Equa 3Dimensions • Consider an electron orbiting a central nucleus. An obvious coordinate choice is a spherical system centred at the nucleus.

43. Schrödinger's Equa 3Dimensions • Consider an electron orbiting a central nucleus. An obvious coordinate choice is a spherical system centred at the nucleus. This is an example of a central force.

44. Schrödinger's Equa 3Dimensions • The Laplacian in spherical coordinates is:

45. Schrödinger's Equa 3Dimensions • The Laplacian in spherical coordinates is: • Therefore becomes .ie dependent only on the radial component r.

46. Schrödinger's Equa 3Dimensions • The Laplacian in spherical coordinates is: • Therefore becomes .ie dependent only on the radial component r. • Substituting into the time TISE leads to Schrödinger's equation for a central force.

47. Schrödinger's Equa 3Dimensions • Solutions to equation can be found by separating the variables in the Schrödinger's equation.

48. Schrödinger's Equa 3Dimensions • Solutions to equation can be found by separating the variables in the Schrödinger's equation. • The stationary states for the waveform are:

49. Schrödinger's Equa 3Dimensions • After some rearranging we find that,

50. Schrödinger's Equa 3Dimensions • The terms are grouped so that those involving a single variable appear together surrounded by curly brackets.