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Quantum Mechanics in three dimensions

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**The Schrodinger Equation**

The time dependent Schrodinger equation: can be “separated” to get the time-independent Schrodinger equation which can be used to find the “stationary states” or standing waves in a potential.

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**The time-independent Schrodinger equation in 3 dimensions**

“Laplacian” Can we use our previous knowledge to guess some of the characteristics of a particle in a 3 dimensional “box”? What are the boundary conditions? What is the form of the wave function? Can you deduce anything about the ground state? Higher states?

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**The Schrodinger Equation in Three Dimensions**

“Laplacian”

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**Particle in a 3-dimensional box**

U=0 inside the box Leads to “degenerate” states: unique states with the same energy!

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**A visualization: two dimensional box**

First Excited State Ground State Second Excited State

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**Spherical coordinates**

…make the most sense when describing atoms. f r q

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**The Schrodinger Equation in Spherical Coordinates**

conversion from cartesian coordinates to spherical polar coordinates Laplacian in spherical polar coordinates: The Schrodinger equation in spherical polar coordinates:

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**The polar solution The polar part of the Schrodinger equation is:**

With some rearrangement, this can be recognized as the associated Legendre equation: Luckily, someone has already solved this equation, so we don’t have to:

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**The spherical harmonics**

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**Quantization of Angular momentum**

+1 +2 +3 +4 m -1 -2 -3 -4

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**The Bohr Atom Revisited**

Classically: Bohr figured out that angular momentum was actually quantized: The Schrodinger equation in three dimensions gives us another insight as to why that is:

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