QM lecture 5
Fourier Spectral Decomposition of the Wave-function
Understanding of the Heisenberg Uncertainty Principle (as a universal properties of waves)
Describing the wavefunction of a free-particle (relativistic and non-relativistic cases) in the context of
the wave-particle duality
5.1 Spectral Decomposition of a function (relative to a basis-set) (09-29-2022)
5.1.A Analogy between the components of a vector V and spectral components of a function Y
5.1.B The scalar product between two periodic functions
5.1.C How to find the spectral components of a function ψ ?
5.1.C.a Case: Periodic Functions.
The Series Fourier Theorem
5.1.C.b Case: Non-periodic Functions
The Fourier Integral
5.1.D Spectral decomposition in complex variable.
The Fourier Transform
5.1.E Correlation between a localized-functions (f ) and its spread-Fourier (spectral) transforms (F)
5.1.F The scalar product in complex variable
5.1.G Notation in Terms of Brackets
5.2 Phase Velocity and Group Velocity (Reading assignment)
5.2.A Planes
5.2.B Traveling Plane Waves and Phase Velocity
Traveling Plane Waves (propagation in one dimension)
Traveling Harmonic Waves
5.2.C Traveling Wavepackage and its Group Velocity
Wavepacket composed of two harmonic waves
Analytical description
Graphical description
Phasor method to analyze a wavepacket
Case: Wavepacket composed of two waves
Case: A wavepacket composed of several harmonic waves
5.3 DESCRIBING the MOTION of a FREE PARTICLE in the context of the WAVE-PARTICLE DUALITY
5.3.A Trial-1: A wavefunction with a definite momentum
5.3.B Trial-2: A wavepacket as a wavefunction
References
R. Eisberg and R. Resnick, “Quantum Physics,” 2nd Edition, Wiley, 1985. Chapter 3. (Section 3.2)
D. Griffiths, "Introduction to Quantum Mechanics"; 2nd Ed., Pearson Prentice Hall. Chapter 2.