1. Particle in a Box - 1D (14.3)
A simple problem that demonstrates the principles of quantum mechanics is the particle in a box (PIB)
Inside the box of a given length (a), the potential is zero; outside the box the potential is infinite
The wavefunction must be zero outside the box since the PE can’t be infinite
The solution to the SE for the PIB is an oscillating function
It is sinusoidal since the wavefunction must be zero at the ends of the box
An arbitrary integer (n) is in the solution and is referred to as a quantum number
The energy of the PIB is also dependent on the quantum number
Quantum number means the energy is quantized (only has certain values)
Energy also depends on the length of the box

2. Properties of PIB Wavefunctions (14.3-14.4)
Each solution of the PIB SE represents a state of the system
The lower the state (smaller n) the lower the energy of the system
First state is referred to as the ground state; higher energy states are called excited states
Some interesting features arise from the solution of the PIB SE
As the quantum number increases, so does the number of values of x for which the PIB wavefunction equals zero (nodes)
The PIB probability densities show a number of interesting characteristics (non-uniform probabilities, regions of zero probability within the box)
PIB for higher dimensions (2D, 3D) show very similar behavior
Wavefunction is a product of 1D PIB wavefunctions
Energy is a sum of 1D PIB energies
Some energy levels are degenerate (have same energies)

3. Excited States and Spectroscopy (18.1-18.2)
In order for a system to go from one state to another, it must absorb or emit a quantized amount of energy (radiation)
Light is often used to promote particles to higher states (absorption)
Light is often emitted when excited states go back to lower lying states (emission)
Spectroscopy is the use of light to probe states in matter
Type of light needed depends on the physical problem one is interested in (i.e., the potential energy function)
Selection rules dictate whether two states can be connected by shining light on it (depends on the nature of the wavefunctions of the two states)
Does PIB model any real physical phenomena?
The spectroscopy of highly conjugated molecules (e.g., polyenes) can be explained quite well by PIB model

4. Particle in a Box

5. PIB Wavefunctions

6. PIB Probability Densities

7. Absorption and Emission of Radiation