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The time independent schrödinger equation (tise) for a particle moving in a circular potential well (ring) and a spherical potential well (sphere). The quantization of angular momentum and energy for both systems, and provides the normalized wavefunctions and energy levels. The document also explains the concept of spherical harmonics and their role in describing the wavefunctions of a particle on a sphere.
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Solving the Schrödinger Equation Exactly for Simple Model Systems Harmonic Oscillator Model of Particle Vibrations (2) Vibrational Motion (1) Translational Motion One particle in free space, 1D-box, 3D-box, and 1D-rectangular well Particle on a ring, and sphere (3) Rotational Motion
(3) Rotational Motion (A) Particle on a Ring Total Energy: m p E T V 2 2
z Angular Momentum: Consider a particle of mass m moving in a circle of radius r the xy plane: 2
It follows that: mr I
z 2 2
2 2 2 p r I
Not all values of the angular momentum are allowed in Quantum Mechanics. Thus, both the angular momentum and rotational energy are quantized. I z J pr z
For a wavefunction to be an acceptable solution to the TISE for a particle on a ring, it must map out a path such as: In this case, depends on the azimuthal angle By inspection, this is an acceptable solution: i. values at 0 and 2 are identical (a complete circuit) ii. it is single-valued iii. It reproduces itself upon successive circuits (3) Rotational Motion: Particle on a Ring (continued)
To be consistent with the DeBroglie Relation, only certain wavelengths are acceptable:
This limits the angular momentum: l l z z
ml It follows then that the energy of the particle on a ring is limited to certain values: I m I
z l 2 2 (^2 ) ml ^0 ,^ ^1 ,^2 , The corresponding normalized wavefunctions of the particle on a ring are:
2 1
l^ l im m
l m
7 Note regarding the natural coordinate system for rotational motion:
Our wavefunction is given in spherical polar coordinates: To map between Cartesian coordinates and spherical polar coordinates: 2 2 2 2
2
Variables range as: (^) Volume element:
8 The Time Independent Schrodinger Equation for the Particle on a Ring:
2 2 2 2 2 2
m x y
To map between Cartesian coordinates and spherical polar coordinates: x r sin cos , y r sin sin There is no dependence: and our radius r is fixed for a particle on a ring so we can get rid of these terms: We first need the Hamiltonian Operator: 2 2 2 2 2 2 2 2 2 1 1 x y r r r r
r r r
2 2
2 2 2 2 1 2
The proper Hamiltonian Operator:
Cyclic Boundary Conditions
2 1 2 1
( 2 ) 2
l^ ^ l^ l^ l im im im m
l l l l l l l m m im m im im
2
2
2
ml ml 0 , 1 , 2 ,
Summary of Results for a Particle on a Ring: iii. Angular momentum is also quantized i. Energy is quantized ii. Energy has values of
l
2 2
ml z l
iv. To locate the particle on the ring, need the probability density ^
2 1 2
(^) l l l l im im m m
What does this imply? What does this tell us about the particle’s location and angular momentum?
(B) Particle on a Sphere: For a particle confined of mass m to a sphere of fixed radius r with V = 0 , the Time Independent Schrödinger Equation is: E m 2 2 2 With wavefunctions given by the spherical harmonics: ( , ) ( ) ( ) ( , ) , l ml Y ( , ) , l ml Y The spherical harmonics depend on two quantum numbers: m l l l l l , 1 , 2 , l 0 , 1 , 2 , orbital angular momentum quantum number magnetic quantum number
i m l l e l m Y l sin 8 3 1 cos 4 3 1 0 4 1 0 0 ( , ) 2 1 2 1 2 1 The Spherical Harmonics: (see 18.33 in text) Plots of some of the wavefunctions:
The Angular Momentum of the Particle on a Sphere:
2
l
2 2
ml Recall for a particle on a ring: Recall for a particle on a sphere: By comparison the angular momentum must be: 2 1
, ( 1 ) ,
l m l
2 l
What is the physical meaning of ml? 2 , 1 , 0 , 1 , 2 2 l m l For a given l , the position of the particle is confined to specific regions as determined by ml.
