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Rotations vs. Translations

Rotations vs. Translations. Quantized Planar Rigid Rotor. Schroedinger’s Wave Equation General Solution: Continuity Condition. Quantized Planar Rigid Rotor(cont.). Wave Function: Orthonormality Condition. Quantized Rigid Rotor. Schroedinger’s Wave Equation: Separation of Variables:

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Rotations vs. Translations

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  1. Rotations vs. Translations

  2. Quantized Planar Rigid Rotor • Schroedinger’s Wave Equation • General Solution: • Continuity Condition

  3. Quantized Planar Rigid Rotor(cont.) • Wave Function: • Orthonormality Condition

  4. Quantized Rigid Rotor • Schroedinger’s Wave Equation: • Separation of Variables: • Results in two equations:

  5. The Phi Equation • This equation is the same as the plane rigid rotor, so it has the same solution:

  6. The Theta Equation • The theta equation can be put into the form of a standard (a.k.a. “already solved”) equation.

  7. Legendre’s equation The theta equation has the form of a famous differential equation called Legendre’s equation: an equation that was solved by Adrien Legendre about 180 years ago

  8. Visualizing Complex Wave Functions Problems involving the quantization of angular momentumproduce wave functions that are complex. We encounter complex wave functions in: Planar Rigid Rotor Rigid Rotor Hydrogen Atom

  9. Complex Wave Functions • Planar Rigid Rotor (a.k.a particle-on-a-ring): • Rigid Rotor: • Hydrogen Atomic Orbital

  10. Spherical Harmonics are Complex

  11. Visualizing the Imaginary • Note that spherical harmonics are real if m=0 and complex otherwise. • A graphical representation of the real function functions is given below. Surfaces of (e.g. Y00, Y10, Y20) the function will only appear green and/or red, depending upon whether the function is positive or negative for those values of • If the function is complex (e.g. Y11, Y21, etc. ) other colors represent complex values. Forexample, if the function is proportional to +i or –i on a surface that can be displayed by yellow/blue.

  12. Complex and Real Spherical Harmonics

  13. Getting Rid of the Imaginary • In most chemistry texts, atomic orbital wave functions are displayed as real functions. This is done by taking linear combinations of complex functions. Using the complex functions… we define the normalized REAL wave functions:

  14. Summary of Rigid Rotor Properties • Energy: • Angular Momentum:

  15. Statistical Thermodynamics of Rotations • Partition Function (assumes EJ<<kBT) • Probability of being in J energy level

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