QUANTUM MECHANICS II
The Schrodinger wave equation 
General form 
Normalization & Probability 
Properties of valid wave functions
Boundary conditions 
Time-Independent Schrödinger Wave
Equation 
Stationary State 
Comparison of Classical and Quantum
Mechanics
Newton’s second law can be derived from the Schrödinger
wave equation, so the latter is the more fundamental.
Classical mechanics only appears to be more precise because
it deals with macroscopic phenomena.
Newton’s second law (𝐹 ⃗ = 𝑑𝑝⃗/𝑑𝑡) and Schrödinger’s wave
equation are both differential equations.
Expectation Values
The expectation value is the expected result of the average of
many measurements of a given quantity.The expectation value
of x is denoted by <x> 
Continuous Expectation Values 
Momentum Operator 
Position and Energy Operators 
Infinite Square-Well Potential 
Quantization 
Quantized Energy 
Finite Square-Well Potential 
Penetration Depth 
Three-Dimensional Infinite-Potential Well
Schrödinger wave equation 
Degeneracy
A quantum state is degenerate when there is more than one wave
function for a given energy
Degeneracy results from particular properties of the potential energy
function that describes the system
A perturbation of the potential
energy can remove the degeneracy.
Simple Harmonic Oscillator 
Parabolic Potential Well 
Analysis of the Parabolic Potential Well 
Barriers and Tunneling 
Reflection and Transmission
The potentials and the Schrödinger wave equation for the three regions 
The corresponding solutions are 
Probability of Reflection and Transmission 
Tunneling 
