Schrödinger Equations

1. Black body radiation In the late 19th century, it was seen that every body is radiating, where its intensity is proportional to the fourth power of its temperature (see also graph below). Furthermore, one was able to measure the different frequencies a black body with different temperatures was radiating.
1.1. Rayleigh–Jeans law The Rayleigh–Jeans law is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature through classical arguments. The idea behind this law was to find all possible standing waves within a black body.
1.1.1. Mathematics When we want to find out how many standing waves exists in cube with length $l_c$ we must consider that at each corner of the cube the wave must have a zero crossing so the longest wavelength would be
$$l_c = \frac{\lambda_{max}}{2} \Rightarrow \lambda_{max}=2 \cdot l_c$$
and the other wavelengths would be:
$$l_c = \frac{n_x \cdot \lambda}{2} \Rightarrow \lambda=\frac{2 \cdot l_c}{n_x }$$
with $n_x = 1,2,3,4,…$
The same is true for the x, y and z-direction, but one must consider all possible waves not only those on the coordinate axes. The other waves (arbitrary ones) can be created from the original waves on the coordinate system (superposition). Therefore the following applies:
$$\lambda_x = \frac{\lambda}{\cos \alpha}$$
$$\lambda_y = \frac{\lambda}{\cos \beta}$$
$$\lambda_z = \frac{\lambda}{\cos \gamma}$$
Where $\alpha, \beta$ and $\gamma$ are the angles from the wave to the coordinate system. When adding the result in the formula above we get the following:
$$n_x = \frac{2 \cdot l_c \cdot \cos \alpha}{\lambda}$$
$$n_y = \frac{2 \cdot l_c \cdot \cos \beta}{\lambda}$$
$$n_z = \frac{2 \cdot l_c \cdot \cos \gamma}{\lambda}$$
With those there formulas we have the wave in all three directions. When we now apply Pythagoras we get a formula for all possible waves in all three directions (squaring and adding):
$$n_x^2+n_y^2+n_z^2 = (\frac{2 \cdot l_c}{\lambda})^2 \cdot (\cos^2 \alpha + \cos^2 \beta+ \cos^2 \gamma)$$
$$\lambda = \frac{c}{f}$$
$$f=\frac{c}{2 \cdot l_c} \sqrt{n_x^2+n_y^2+n_z^2}$$
$$N(f)df=\frac{4 \cdot \pi}{8}\frac{4 \cdot l_c^2 \cdot f^2}{c^2}\cdot 2 \cdot \frac{2 \cdot l_c}{c}df=\frac{8\cdot \pi \cdot l_c^3 \cdot f^2}{c^3}df$$
$$u_f=\frac{8\cdot \pi \cdot f^2 \cdot k \cdot T}{c^3}df$$
1.2. Plancks law Max Planck was the one who found the formula which describes the seen behaviour. For the formula he introduced a new constant named after him, the Plack constant which has a value of $6.62607015e-34 J\cdot s$. Max Planck used the number of modes in vibration from Rayleigh–Jeans law which were absolutely correct ($N= \frac{8 \cdot \pi \cdot f^2}{c^3}$). But changed the energy $k \cdot T$ which Rayleigh–Jeans used. He said the energy is discrete $E=n \cdot h \cdot f$

Theorem 1 (Planck’s Law). \begin{equation}\label{eq:NL}
u_{\lambda(\lambda,T)}= \frac{8 \cdot \pi \cdot h \cdot c}{\lambda^5} \cdot \frac{1}{e^{\frac{h \cdot c}{\lambda \cdot k \cdot T}}}-1
  • $h = 6.62607015e-34 J\cdot s$
  • $k = 1.380649e-23 \frac{J}{K}$
  • $c = 299792458 \frac{m}{s}$
  • $\lambda = $ wavelength in $m$
  • $T = $ temperature in $K$
In Theorem 1 the main part is \eqref{eq:NL}.

2. Wave Function
2.1. Broglie-Einstein postulates $$\lambda=\frac{h}{p}$$
$$ f=\frac{E}{h}$$

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