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

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)

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.

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.

Particle in a box

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".

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.

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

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