20.2. Origins of Matrix and Wave Quantum Mechanics#

20.2.1. Origins of Matrix Mechanics#

Quantization of radiation energy.

  • Planck (1900), Einstein photoelectric effect (1905) and heat capacity of the solids: introduction and evidences of \(h\), \(E = h \nu\)

    (20.1)#\[u(\nu, T) = \dots\]
Light-matter interaction and light dispersion.

  • First experimental evidences: Cauchy; Sellmeier (1872) proposed an empirical law for the refraction index \(n\) taking into account resonances of matter; Lorentz-Drude (1900-1905) provided a theoretical model of Sellmeier formula, using Maxwell equations for electromagentism,

    (20.2)#\[n - 1 = \dfrac{q^2 N}{2 \varepsilon m} \dfrac{1}{\omega_0^2 - \omega^2 + i \gamma \omega} \approx \dfrac{q^2 N}{2 \varepsilon m} \dfrac{1}{\omega_0^2 - \omega^2} \ ,\]

    being \(m\) and \(q\) the mass and the charge of a particle, \(N\) the number density of the charged particles, \(\omega_0\) the natural frequency and \(\gamma\) a damping coefficient of the dynamical equation governing the dynamics of the particle, \(m \ddot{x} + m \gamma \omega \dot{x} + m \omega_0^2 x = q E(t)\).

  • Einstein (1916): quantum theory of the radiation interacting with matter.

    • Statistics of the equilibrium of radiation and matter, with the processes of absorption, spontaneous emission and stimulated emissions. He introduced Einstein coefficiens as proportionality constants

    • E. theory connected Bohr’s atomic with Planck’s radiation formula (20.1): matching Planck’s formula induces relationss between Einstein coefficients,

      \[ \dfrac{A_{jk}}{B_{kj}} \dfrac{g_j}{g_k} = \dfrac{8 \pi \nu^3}{c^3} \qquad , \qquad \dfrac{B_{jk}}{B_{kj}} \dfrac{g_j}{g_k} = 1 \ . \]

      Einstein didn’t know how to compute \(A_{ik}\), \(B_{jk}\), \(B_{kj}\), but he was confident that it would have been possible to do so, once a theory of quanta was established.

  • Ladenburg (1914-24)

    • a material may have more than one resonance. The generalization of the formula (20.2) for the refractive index \(n\) from Lorentz-Drude model reads

      \[n - 1 = \dfrac{q^2}{2 \varepsilon m} \sum_{k} \dfrac{N_k}{\omega_k^2 - \omega^2} \ ,\]

      with \(N_k\) the “number density of particles”1 with natural frequency \(f_k = \dfrac{\omega_k}{2 \pi}\).

    • But what’s the meaning of \(N_k\)? Ladenburg compared the averaged emitted power from Larmor’s formula (20.4) - the classical model - and the emitted power by Planck’s formula, or by Einstein quantum theory of interaction of radiation and matter - the quantum model, for a frequency \(\nu_{ij}\). The power emitted by \(R\) resonators per unit volume is

      …