Download presentation

Presentation is loading. Please wait.

Published byJayda Basey Modified over 4 years ago

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 lm 00 10 1±1 20 2 2±2

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. Y 00, Y 10, Y 20 ) 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. Y 11, Y 21, 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 E J <<k B T) Probability of being in J energy level

Similar presentations

OK

The Hydrogen Atom Quantum Physics 2002 Recommended Reading: Harris Chapter 6, Sections 3,4 Spherical coordinate system The Coulomb Potential Angular Momentum.

The Hydrogen Atom Quantum Physics 2002 Recommended Reading: Harris Chapter 6, Sections 3,4 Spherical coordinate system The Coulomb Potential Angular Momentum.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google