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Quantum Mechanics in three dimensions.

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Presentation on theme: "Quantum Mechanics in three dimensions."— Presentation transcript:

1 Quantum Mechanics in three dimensions

2 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.

3 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?

4 The Schrodinger Equation in Three Dimensions

5 Particle in a 3-dimensional box
U=0 inside the box Leads to “degenerate” states: unique states with the same energy!

6 A visualization: two dimensional box
First Excited State Ground State Second Excited State

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

8 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:

9 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:

10 The spherical harmonics






16 Quantization of Angular momentum
+1 +2 +3 +4 m -1 -2 -3 -4

17 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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