1. Something more about…. Standing Waves Wave Function
Differential Wave Equation
Standing Waves
Boundary Conditions:
X=0
X=L
Separation of variables:
Wave Function:

2. Space: f(x) TIme: f(t) Space: X(x) Time: T(t)
Equivalent to two ordinary (not partial) differential equations:
Space: X(x)
Time: T(t)

3. All material particles are associated with Waves
A central concept of Quantics: wave–particle duality is the concept that all matter and energy exhibits both wave -like and particle -like properties.
The Wave Nature of Matter
All material particles are associated with Waves
(„Matter waves“
E = hn
E = mc2
mc2 = hn = hc/l
or: mc = h/l
De Broglie
A normal particle with nonzero rest mass m travelling at velocity v
mv = p = h/l
Then, every particle with nonzero rest mass m travelling at velocity v has an related wave l
l = h/ mv
The particle property is caused by their mass.
The wave property is related with particles' electrical charges.
Particle-wave duality is the combination of classical mechanics and electromagnetic field theory.

4. 2p sin (q) = Dpy=2pl/Dy Dpy . Dy ≈ 2pl = 2h sin (q) = ±l/Dy
The Waves and the Incertainty Principle of Heisenberger
„The measurement of particle position leads to loss of knowledge about particle momentum and visceversa.“ p y Dy q
2p sin (q) = Dpy=2pl/Dy
Dpy . Dy ≈ 2pl = 2h v m x
sin (q) = ±l/Dy
The momentum of the incoming beam is all in the x direction. But as a result of diffraction at the slit, the diffracted beam has momentum p with components on both x and y directions.

5. Particle in a box

7. two waves can interfere to form another, more complex wave
two waves can interfere to form another, more complex wave. If we add enough waves together, we can make a wave packet with a range of locations that's as small as we want. This doesn't give a particle an exact location, but it looks like an exact location and so explains why we see it that way.
The superposition of a group of waves differing from each other in wavelength yields a Wave Packet
Wave Packet
two waves can interfere to form another, more complex wave. If we add enough waves together, we can make a wave packet with a range of locations that's as small as we want. This doesn't give a particle an exact location, but it looks like an exact location and so explains why we see it that way.
Since a particle is a wave, it simultaneously exists over a range of locations. This range is called the uncertainty of the particle's position. In the picture to the right, "x" stands for the particle's position, then "delta x" represents the range of possible locations for the particle, or the uncertainty of "x".

8. The Waves Packets and the Incertainty Principle of Heisenberger
The narrower you want to make the range of positions, the more waves you have to add together. This creates a wider range of wavelengths for the wave packet. According to de Broglie, a particle's momentum (its speed times its mass) is related to its wavelength. Therefore a narrow range, or small uncertainty, in position means a wide range, or large uncertainty, in momentum.
A wave packet is a localized disturbance that results from the sum of many different wave forms. If the packet is strongly localized, more frequencies are needed to allow the constructive superposition in the region of localization and destructive superposition outside the region.

9. General solution: Principle of superposition
Eigenvalue Condition:
n=0, ±1, ±2, ±3……
Eigenfunctions:
Since any linear Combination of the Eigenfunctions would also be a solution
General solution: Principle of superposition
Fourier Series

10. Fourier Series REAL Fourier Series COMPLEX Fourier Series
Any arbitrary function f(x) of period L can be expressed as a Fourier Series
REAL
Fourier Series
COMPLEX
Fourier Series