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## Schrodinger’s Equation for Three Dimensions

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**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. • However 3-dimensions is needed for application to atomic physics, nuclear physics and other areas.**Schrödinger's Equa 3Dimensions**• For 3-dimensions Schrödinger's equation becomes,**Schrödinger's Equa 3Dimensions**• For 3-dimensions Schrödinger's equation becomes, • Where the Laplacian is**Schrödinger's Equa 3Dimensions**• For 3-dimensions Schrödinger's equation becomes, • Where the Laplacian is • and**Schrödinger's Equa 3Dimensions**• The stationary states are solutions to Schrödinger's equation in separable form,**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,**Particle in a 3 Dimensional Box**• The simplest case is a particle confined to a cube of edge length L.**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.**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.**Particle in a 3 Dimensional Box**• Note: If we consider one coordinate the solution will be the same as the 1-D box.**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):**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,**Particle in a 3 Dimensional Box**• The independent variables are isolated. Each of the terms reduces to a constant:**Particle in a 3 Dimensional Box**• Clearly**Particle in a 3 Dimensional Box**• Clearly • The solution to equations 1,2, 3 are of the form where**Particle in a 3 Dimensional Box**• Clearly • The solution to equations 1,2, 3 are of the form where • Applying boundary conditions we find,**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**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,**Particle in a 3 Dimensional Box**• with and so forth.**Particle in a 3 Dimensional Box**• with and so forth. • Using restrictions on the wave numbers and boundary conditions we obtain,**Particle in a 3 Dimensional Box**• with and so forth. • Using restrictions on the wave numbers and boundary conditions we obtain,**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.**Particle in a 3 Dimensional Box**• Note that three quantum numbers are required to describe the quantum state of the system.**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.**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.**Particle in a 3 Dimensional Box**• The total energy will be quoted in the form**Particle in a 3 Dimensional Box**• The ground state ( ) has energy**Particle in a 3 Dimensional Box**Degeneracy**Particle in a 3 Dimensional Box**• Degeneracy: quantum levels (different quantum numbers) having the same energy.**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).**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.**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.**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**4E0**11/3E0 3E0 2E0 E0 Particle in a 3 Dimensional Box • The 1st five energy levels for a cubic box.**Schrödinger's Equa 3Dimensions**• The formulation in cartesian coordinates is a natural generalization from one to higher dimensions.**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.**Schrödinger's Equa 3Dimensions**• Consider an electron orbiting a central nucleus.**Schrödinger's Equa 3Dimensions**• Consider an electron orbiting a central nucleus. An obvious coordinate choice is a spherical system centred at the nucleus.**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.**Schrödinger's Equa 3Dimensions**• The Laplacian in spherical coordinates is:**Schrödinger's Equa 3Dimensions**• The Laplacian in spherical coordinates is: • Therefore becomes .ie dependent only on the radial component r.**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.**Schrödinger's Equa 3Dimensions**• Solutions to equation can be found by separating the variables in the Schrödinger's equation.**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:**Schrödinger's Equa 3Dimensions**• After some rearranging we find that,**Schrödinger's Equa 3Dimensions**• The terms are grouped so that those involving a single variable appear together surrounded by curly brackets.