Quantum Mechanics - Wave Functions

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last changes: 09.05.15
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References
http://en.wikipedia.org/wiki/ [Wikipedia-E] - in Englisch
http://de.wikipedia.org/wiki [Wikipedia-G] - in German

Lars V. Ahlfors, Complex Analysis (Funktionentheorie)

Mini Dictionary Physics

T. Fliessbach, Quantenmechanik (in German)

Funktionentheorie (in German)

K.E. Hellwig, Quantenmechanik, Studienhilfe der Studentenschaft e.V. , Technische Universität Berlin, WS 1977/78 (no official release, from students for students)
(in German)
and other not always referenced sources found in the internet and literature

Contents

01: particles and elementary particles, photons, Planck Constant, energy and momentum
02: plane waves, complex numbers
03: electron, positron

04: wave functions, operators, uncertainty relation, Fourier transformation
04a: operators, Schrödinger equation

Between other themes Quantum Mechanics investigates 
particles. There exists two major classes of particles, particles with a rest mass of zero and particles with a rest mass different from zero.

Elementary particles are not composed of smaller particles.

Particles with a rest mass zero are elementary particles.
To explain the rest mass of elementary particles the Higgs field has been introduced.

Elementary particles with a rest mass different from zero are W-Bosons and Z-Bosons, quarks, leptons.

An example of an elementary particle with a rest mass of zero is the photon. This does not mean, a photon has no mass. A photon has an energy and after the equation E = m * c2 it has a mass. This mass can be observed, if a photon is influenced by a strong gravitation field. The photon is in movement ever, it does not rest. It moves with the velocity of light. Any particle with a rest mass different from zero can not reach this velocity, because it's relativistic mass would extend all limits otherwise.

An example of an elementary particle with a rest mass different from zero is the electron.

There exists other classifications of particles, one distinguishes between bosons and fermions.
The classification depends from the spin of the particle. The electron is a fermion with spin 1/2, the photon is a boson with spin 1. These spins are expressed in units of .

Other fermions are the proton and the neutron. Both are particles with spin 1/2.

In this terminology the proton is not an elementary particle, because it is composed of quarks.
Quarks are fermions with spin 1/2.
Conclusion: fermions can be composed particles or they can be elementary particles.

In general fermions are defined as particles with half-integer spin, bosons as particles with integer spin.
That means, fermions can have spins 1/2, 3/2 ... und bosons can have spins 0,1,2,...

An example for a boson with spin 2 is the graviton (if it exists).


The following sections of this website are dedicated to the photon.

Refraction of light can be explained if properties of a wave are assigned to light. Following this idea light is composed of waves with different wave lengths. In the empty space all these waves expand with the same constant speed c, the velocity of light in the vacuum. If light moves through matter this speed can be different for the waves with different wave lengths. This model is used to explain the refrection of light e.g. in a prism.

In the range of visible (light)waves each wave is assigned a color, raising the spectrum of light. White light occurs if all these colors overlap.

The visible spectrum of light comprises only a small range of all possible wave lengths.

In this view a photon is considered to be a (light) wave characterized by it's constant wave length expanding with the constant velocity of light in the vacuum. It is assigned an energy which is invers proportional to its wave length, that means, if the wave length increases the energy decreases. Light colors are attached to photons with certain wave lengths.
 
Green light has a wave length of 500 nm (500 Nanometer = 500 * 10-9 m)

X-Rays have a greater energy (a smaller wave length) then visible photons, microwaves have a smaller energy (a greater wave length) then visible photons.

Another term which characterizes waves is the frequency. The frequency is defined as the number of oscillations during a second. The physical unit is defined in Hertz: 1 Hz = 1/s.
(one Hz is one oscillation in one second)

The oscillations considered here are periodic movements with a defined oscillation time. An example for harmonic oscillations is the the sinus function.

Harmonic oscillations will be considered in the following.

From the oscillation time T (in seconds) the frequency f can be calculated as 1/T (in Hz).

From experimental data it is known that the energy E of photons is proportional to the frequency:

E = h * f

with the Planck constant h.

Another term is often in use (for historical reasons), the constant  .

It is h = * 2

h is defined as the Plancksche Konstante (Planck constant)
 h = 6,626 * 10-34 Js

Instead of f as symbol for the frequency often  is used.



Light is radiated by an atom in a finite time, e.g. 10-8 s, this could be interpreted as a finite length of a photon of about 3 meters, the coherence length.

Insite this range the wave length is considered to be constant. It is the length inside which interference of photons can occure.


Link: << Questions and speculations >>

For harmonic oscillations a circle frequency  is defined, .
T denotes the time for an oscillation.

With  = 1 / T it follows T = 1 /  and therefore  = 2 (resp.  = 2f)

Examples for harmonic oscillations are given by the functions sin and cos.

Circle frequency  and energy E of a Photon follow the equation:

With increasing circle frequency the energy increases.

A Photon is assigned also a momentum p

The momentum is invers proportional to the wave length

p = h * 1/
resp.
   with 

k is the symbol for the absolute value of the wave number vector, h is die Planck constant,  the wave length.

For the energy of a photon the following equation is valid: E = p * c = h * c /.
 
Consider the physical units for h, E and p:

h = 6,626 * 10-34 Js 
p = h * 1/
E = h * f 

J is the SI - Unit for the energy: [J] = N * m 
m : Meter,
N : Newton, N = kg * m / s2 ,  N is the SI - Unit of the force F, [F] = N
s : second

  • => J = kg * m2 / s2
  • J * s : Wirkung (dimension of action- [Wikipedia-E: Article Planck constant])
  • denotes a length, [] = m, f denotes the frequency, [f] = 1/s
  • p is the momentum, defined as the temporary change of a force 
that means [p] = N/s = (kg * m / s2) / s = kg * m / s = "Mass * velocity

 p = h * 1/, in Units: [p] = [h] / [] resp.
[p] = kg * m / s = [h] / [] = (J * s) / m,  J * s / m = ( kg * m2 / s) / m = kg * m / s 
(this proves the correctness of the equation [p] = [h] / [] expressed in units) 

For E = h * f it follows [E] = [h] * [f] resp. [E] = J =  [h] * [f] = (J * s) * (1/s) = J 
(this proves the correctness of the equation expressed in units)
 

The momentum of light can be observed, it causes the tail of comets.

The number of photons of an infalling energy is calculated in the following exercise.


Exercise (after Höfling, Physik Band II, Quanten und Atome)

For perceiving green light with a  wave length  = 500 nm (1 nm = 1 Nanometer = 10-9 m) the human retina needs a "Lichtleistung" of P = 2 * 10-18

(W = Watt = J/s, J = Joule, s = Sekunde, 1 Watt denotes the infalling energy "1 Joule in 1 second")

How many photons have to met the retina in 1 s to see this light ?
-------------------------------------------------------------------------
Energie of 1 photon:

Die Wellenlänge hängt bei einer Lichtwelle mit der Ausbreitungsgeschwindigkeit c folgendermaßen zusammen:
 

A bigger representation of the image


Translations:

die Wellenlänge hängt bei einer Lichtwelle mit der Ausbreitungsgeschwindigkeit c folgendermaßen zusammen:
die wave lenght of a (light) wave is related to the (expansion) velocity c in the following sense

n Photonen haben die Energie: n photons have the energy

und ergeben in der Zeit t die Lichtleistung:
and yield a "Lichtleistung" (measured in W - Watt) during the time t


mit:
with

und
: and

erhält man
: it follows

Result: 5 Photons have to met the retina in 1 second to perceive the green ligh.

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