(quantum-mechanics:history:topics)= # Topics **Atomic models.** - 1891, J.J.Thomson find electrons in cathodic rays. He found out that cathodic rays are made of negatively charged particles[^thomson-electron], deflected by a transverse electromagnetic field. He: - evaluated the ratio $\frac{\text{mass}}{\text{el.charge}}$ of these particles. See [Scoperta dell'elettrone - J.J.Thomson](https://basics2022.github.io/bbooks-physics-hs/ch/modern/experiments-matter.html#scoperta-dell-elettrone-j-j-thomson) - proposed the pudding atomic model, with negatively charged particles in a positive cloud. [^thomson-electron]: While J.J.Thomson won the Nobel prize in Physics (1906) for having found the first sub-atomic particle, the electron, his son G.P.Thomson won the Noble prize (1937) for its discovery of the diffraction of electrons by crystals in 1927, thus showing wave behavior. Particle-wave duality never ends... - 1909, Rutherford scattering experiment and Rutherford atomic model: electrons around a compact nucleus - 1913, Bohr... - 1916, Sommerfeld - relativistic model - elliptial orbits - introduced another quantum number and degeneracy - describe energy level split by magnetic field (Zeeman effect) - 1925, Heisenberg and Schrodinger first theories about QM; Pauli solution of the $\text{H}$ atom system **Spectroscopy.** - 1707, Newton "Optics" described his experiments about diffraction of light with glass prisms - In the second half of the XIX century, experimental studies on the emission/absorption spectra in chemistry. Each element showed characteristics black lines in otherwise continuous absoprtion spectra (and corresponding bright lines in emission spectra, "hot spectra") - Empirical laws for the refractive index followed: - 1836, Cauchy proposed a relation between the refractive index $n$ of a material and the wave-length of the radiation $$n^2 = a + \frac{b}{\lambda^2} + \frac{c}{\lambda^4} + \dots \ ,$$ with material-dependent parameters $a$, $b$, $c$,...This model proposed $n(\lambda)$ as a continuous function - 1870, Christiansen observed anomalous dispersion, with apparently discountinuous refractive index at wave lengths that are characteristics of the element under investigation - 1872, Sellmeier proposed that anomalous dispersion should be described in the context of light-matter interaction, in terms of resonance modes of the matter and proposed his law for the refractive index $$n^2 - 1 = \sum_{k} \frac{a_k \lambda_k^2}{\lambda^2 - \lambda^2_k}$$ - 1875, Helmholtz modified Sellmeier formula, introducing a (wrong, this was not a dampening factor, as the one shown by Drude-Lorentz model) dampening factor. - [Drude-Lorentz model](quantum-mechanics:history:dispersion:classical:drude-lorentz). Following the idea of Sellmeier, P.Drude (1900) and then H.Lorentz (1905) proposed a theoretical model for the refractive index of a medium, using a second-order damped oscillator, forced by the electric field of an incoming electromagnetic wave, and evaluated the transmitted wave as the sum of the incident wave and the one generated by the oscillating charges of the matter. - Generalization of the Drude-Lorentz model for systems with multiple modes became the general semi-empirical classical model for light dispersion $$n(\omega)-1 = \frac{q^2}{2 \varepsilon_0 m} \sum_k \frac{N_k}{\omega_k^2 - \omega^2 + i \gamma_k \omega_k}$$ or $$n^2(\omega) - 1 \simeq 2 \delta .$$ - Dispersion becomes quantum: - 1905, Einstein explanation of photoelectric effect - 1906-1908, **Ladenburg** and Loria began experiments about spectral lines, anomalous dispersion, and the microstructure of the light-matter interaction. They found a way to measure the number density of the electrons responsible for spectral lines in absorption spectra in $\text{H}$ atom ($\text{H}_{\alpha}$ line only, while they had not enough accuracy for other weaker lines). Only few electrons resonate while the majority appear to be unaffected. They then focused on $\text{Na}$ for better resolution. They came back to $\text{H}$ few years later (1911), to find $\frac{N_{\text{H}_{\alpha}}}{N_{\text{H}}} \sim \frac{1}{50.000}$, $\frac{N_{\text{H}_{\beta}}}{N_{\text{H}}} \sim \frac{1}{200.000}$. - 1913, Bohr model (incompatible with classical mechanics, no $e^-$ fast collapse on the nucleus, due to classical radiation): - circular stable orbits with quantized angular momentum $n h$ - lines are generated by quantum jumps, with $\nu = \frac{\Delta E}{h}$ - results compatible with the empirical Rydberg formula (1888) - Ladenburg: how do both Bohr model and Drude-Lorentz model work so well in atomic spectra? - 1914, H.Kohn: experimental confirmed radiation emitted by incandescent bodies can be treated as thermal radiation. Ladenburg then used thermodynamics to explain thermal radiation. - Still Ladenburg - 1914, procedure to compute the density of dispersion electrons - 1921, quantum counterpart of the classical dispersion, and link with Einstein coefficients (1916) - the **physical atom is replaced by a representation of oscillators**: only frequencies and intensities of spectral lines can be observed. They abandoned the idea of building a "physical model of the atom" following the electron along an orbit, and built a model that's just compatible with the phhysical quantities they can measure - 1924, **Kramers** (with theoretical derivation from Born, 1924, and Kramers and Heisenberg, 1925): - rule for **polarization** with transition frequencies (both to higher and lower energy levels, modeling both emission and absorption) - Kramers formula and the idea of virtual oscillators are used by Heisenberg in his formulation of matrix mechanics: this formula set the origin for the non-commutativity of physical quantities and, thus, Heisenberg uncertainty principle - tested by Ladenburg, 1928 **Radiation and Light-matter interaction.** - 1896, Wien's law - 1900, Planck described the **continuous spectrum** of black-body radiation, introducing of $h$, and the idea that light radiation with frequency $\nu$ have discrete amount of energy, $h \nu$. He first introduced $h$ as a trick for matching experimental results of the spectrum $\rho(\nu, T)$; a model of statistical mechanics of radiation with discrete levels of energy followed, see [Black-body ratiation using Bose-Einstein statistics](statistical-mechanics:notes:distributions:be), even though that was the original way Planck first found a theoretical model - 1905, Einstein explained the photolectric effects using Planck constant $h$. - 1916, Einstein - derived Planck's law from Bohr's atomic model, using a (quantum) statistical approach to the interaction of atoms (and thier electrons) and electromagnetic radiation in a box. He postulated the existence of three processes, introducing proportionality **Einstein coefficients**, for transition probability (of a $e^-$) between states: *absorption* (rate: $B_{kj} N_k \rho(\nu)$) *spontaneous emission* (rate: $-A_{jk} N_j$), and **stimulated emission** (rate: $- B_{jk} N_j \rho(\nu)$). Stimulated emissions was found to be necessary in order for this model to be compatible with Planck's law $\rho \propto \frac{1}{e^{h\nu/k T}-1}$ for the system at thermal equilibrium (and not with Wien's law $\rho(\nu) \propto e^{-h \nu/ k T}$, the law one would get without stimulated emission) - Einstein couldn't calculate his coefficients but he was positive that they could be computed once a theory about the atom had been be established - recognized that light quanta have momentum, $p = \hbar k = h \frac{\nu}{c}$, then found by Compton (1923) in experiments about X-rays scattering - in 1927, Dirac calculated Einstein coefficients (The Quantum Theory of Emission and Absorption of Radiation) **Origin of quantum mechanics.** * **Matrix mechanics.** (Heisenberg, Born, Jordan), three papers in 1925 * **Ondulatory mechanics.** (Schrodinger) * 1923, L.${}$de Broglie: idea of guiding wave for matter particles * 1925, E.Schrodinger: wave equation. If matter behaves as a wave, a wave equation should be the governing equation of the system * 1923-1927, C.Davisson and L.Germer found a diffraction pattern in an experiment about electron scattering * **Equivalence.** (Dirac)