Rotations vs. Translations

1
Rotations vs. Translations
Translations Rotations





2
Quantized Planar Rigid Rotor

• Schroedingers Wave Equation
• General Solution
• Continuity Condition

3
Quantized Planar Rigid Rotor(cont.)

• Wave Function
• Orthonormality Condition

4
Quantized Rigid Rotor

• Schroedingers 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
Legendres equation

• The theta equation has the form of a famous
  differential equation called Legendres equation
  
• an equation that was solved by Adrien Legendre
  about 180 years ago

8
Visualizing Complex Wave Functions

• Problems involving the quantization of angular
  momentum produce 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
l m
0 0
1 0
1 1
2 0
2 1
2 2
11
Visualizing the Imaginary

• Note that spherical harmonics are real if m0 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. For
  example, 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 EJltltkBT)
• Probability of being in J energy level