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Schrodinger’s Equation for Three Dimensions. QM in Three Dimensions. The one dimensional case was good for illustrating basic features such as quantization of energy. QM in Three Dimensions. The one dimensional case was good for illustrating basic features such as quantization of energy.

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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.