Quantum mechanics

For the interferences not to be destructive, the wavelength of the orbit must be a multiple integer of the wavelength of the electron.

If \(R\) is the radius of the orbit, then :

\(2\pi R = n\lambda\)

Hence :

\(R=\frac {n \lambda}{2 \pi}\)

Only certain orbits can allow the existence of a wave (stationary wave) for the electron spinning around the nucleus.

Moreover, according to wave mechanics :

\(\lambda = \frac {h}{p}=\frac {h}{mv}\)

Hence :

\(mvR = n\frac{h}{{2\pi }}\)

It can be interpreted by saying that the kinetic moment of the electron regarding the proton is a multiple integer of the quantity \(\hbar = h/2\pi\). Naturally, Bohr's results are found.

• About the watch : the kinetic moment of the needle is around \(mL^2\Omega\).

  \(m\) is around \(1\) gram, \(L\) is around \(1\) \(cm\) and is one revolution per minute, so the kinetic moment is equal to \(10^{-10}J.s\).

  This is much bigger than \(h\) : the watch is not a quantum system.

• About the oscillating circuit, the energy of the coil and an order of time, inverse of the frequency can be calculated.

  A quantity homogenous to an action is found.

  It is equal to \(3.10^{-17}J.s>>h\).

  The circuit is not a quantum system.

When a hydrogen atom goes from a level defined by the quantum number \(n\) to a level corresponding to a quantum level \(2\), it emits a photon which energy is :

\(h\nu = \frac{{hc}}{\lambda } = {E_n} - {E_2}\)

Consequently, the wave number (defined as the inverse of the wave length of the emitted radiation) is written :

\(\frac{1}{\lambda } = \frac{1}{{hc}}\left( {{E_n} - {E_2}} \right) = \frac{{{E_0}}}{{hc}}\left( {\frac{1}{4} - \frac{1}{{{n^2}}}} \right)\)

Rydberg's constant can be interpreted like the quantity :

\({R_H} = \frac{{{E_0}}}{{hc}} = {1,09.10^7}{m^{ - 1}} = {1,09.10^5}c{m^{ - 1}}\)

The theoretical value matches very well the experimental value given by J. Balmer.

• The acceleration of the electron is \(a=v_0^2/r_0\). The trajectory is circular :

  \(ma = m\frac{{v_0^2}}{{{r_0}}} = \frac{{{e^2}}}{{4\pi {\varepsilon _0}}}\frac{1}{{r_0^2}}\;\;\;\;\;\;\;so\;\;\;\;\;\;\;a = \frac{{{e^2}}}{{4\pi {\varepsilon _0}m}}\frac{1}{{r_0^2}}\;\;\;\;\;and\;\;\;\;\;v_0^{} = \sqrt {\frac{{{e^2}}}{{4\pi {\varepsilon _0}m}}\frac{1}{{r_0^{}}}}\)

  The power is :

  \(P = \frac{2}{3}\frac{{{e^2}}}{{4\pi {\varepsilon _0}{c^3}}}{\left( {\frac{{{e^2}}}{{4\pi {\varepsilon _0}m}}\frac{1}{{r_0^2}}} \right)^2}\)

  In order of magnitude :

  \({v_0} \approx {10^6}m.{s^{ - 1}}\;;\;{T_0} \approx {3.10^{ - 16}}s\;;\;P \approx {10^{ - 7}}W\;and\;\delta E = P{T_0} \approx {3.10^{ - 23}}J\)

  The relative variation of energy is :

  \(\frac {\delta E}{E_0} \approx 10^{-5}\)

• The trajectory is a spiral : the radius \(r(t)\) diminishes very slowly in function of time.

  The trajectory can be supposed as almost circular.

  The energy on the near-circular trajectory is :

  \(E(t) = - \frac{1}{2}\frac{{{e^2}}}{{4\pi {\varepsilon _0}}}\frac{1}{{r(t)}}\)

  And :

  \(\frac{{dE(t)}}{{dt}} = \frac{1}{2}\frac{{{e^2}}}{{4\pi {\varepsilon _0}}}\frac{{\dot r(t)}}{{r{{(t)}^2}}} = - P = - \frac{2}{3}\frac{{{e^2}}}{{4\pi {\varepsilon _0}{c^3}}}{\left( {\frac{{{e^2}}}{{4\pi {\varepsilon _0}m}}\frac{1}{{r{{(t)}^2}}}} \right)^2}\)

  Hence the differential equation :

  \(\frac{{dr(t)}}{{dt}} = - \frac{4}{3}\frac{1}{{{m^2}{c^3}}}{\left( {\frac{{{e^2}}}{{4\pi {\varepsilon _0}}}} \right)^2}\frac{1}{{r{{(t)}^2}}}\)

  After integration :

  \({r^3} - r_0^3 = - {\left( {\frac{{{e^2}}}{{4\pi {\varepsilon _0}}}} \right)^2}\frac{4}{{{m^2}{c^3}}}t\)

• The duration is :

  \({t_f} = \frac{{{m^2}{c^3}}}{4}{\left( {\frac{{4\pi {\varepsilon _0}}}{{{e^2}}}} \right)^2}r_0^3 \approx 0,5\;ns\)

  The radius of the trajectory of the electron around the nucleus inexorably diminishes.

  The model proposed for the hydrogen atom does not lead to a stable system.

  Only the quantum mechanics theory can overcome this difficulty.