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Schrodinger’s Equation for Three Dimensions

Schrodinger’s Equation for Three Dimensions

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Schrodinger’s Equation for Three Dimensions

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  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.