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(Figure above includes components from: https:\/\/www.thoughtco.com\/definition-of-balmer-series-604381 and https:\/\/byjus.com\/chemistry\/hydrogen-spectrum\/)<\/p>\n
2. Balmer (in 1885) figured out a formula that predicted the wavelengths of the first 4 lines of the visible spectrum for hydrogen.\u00a0 His formula for these first 4 lines was: lambda = 3645.6 * m^2 \/ (m^2 – 2^2) where lambda is the wavelength in units of 10^-10 meters (now known as Angstroms), and m was an integer series starting with m=3 – he used this same formula to match the wavelengths of 10 additional ultraviolet lines in the series, extending the series up to m=16.\u00a0 The ultraviolet lines came from observations of the spectra of stars.\u00a0 This series of wavelengths is now known as the Balmer series.<\/p>\n
3. Rydberg (in 1890) expanded on Balmer’s formula for use with other atoms (especially alkali atoms) – Rydberg’s version of the formula is inverted and expressed as a measure of reciprocal wavelength: (1\/lambda) = 109721.6 * (1 \/ n^2 – 1\/m^2). This is the form that is used today.\u00a0 The current value of the constant in this formula (now known as the Rydberg constant) is 109737.316 in units of cm^-1 (also known as reciprocal centimeters or wavenumber).\u00a0 Balmer and Rydberg’s work inspired Nils Bohr to develop a theory that would explain these formulas.<\/p>\n
4. Planck (in 1900) introduced a theory to explain the continuous spectral emissions of a hot body (blackbody spectrum).\u00a0 Planck’s theory introduced the concept of quantum states of energy\u00a0 E = h * nu, where E is energy, nu is frequency, and the constant h.\u00a0 h is now known as Planck’s constant.<\/p>\n
5. Bohr (in 1913) introduced a theory of the atom now known as the “old quantum theory” that combined classical ideas with the then new quantum ideas of Planck.\u00a0 Bohr postulated that an atom was akin to a planetary system with electrons orbiting in circular orbits.\u00a0 Bohr’s model introduced the concept of stationary states and a lowest state with quantum number n=1.\u00a0 Higher energy states (circular electron orbits) had increasing quantum numbers n=2,3,4, …. Bohr’s work followed the work of Rutherford (in 1911) who postulated that an atom was made of a positive charge center surrounded by electrons.\u00a0 Bohr was able to derive the Rydberg formula,\u00a0 as well as an expression for the Rydberg constant based on fundamental constants of the mass of the electron, charge of the electron, Planck’s constant, and the permittivity of free space.<\/p>\n
6. Sommerfeld (in 1916) expanded on Bohr’s ideas by introducing elliptical orbits into Bohr’s model.\u00a0 Sommerfeld’s improvement allowed the Bohr model to explain the splitting of the Balmer lines when the hydrogen atom was in an electric field.\u00a0 Electric field splitting of lines is known as the Stark effect.\u00a0 Today the “old quantum theory” is known as the Bohr-Sommerfeld Model.<\/p>\n
7. de Broglie (in 1924) in his PhD thesis developed the idea of matter waves that have a wavelength given by lambda = h \/ p, where h is Planck’s constant and p is the particle momentum.<\/p>\n
8. Schrodinger (in 1926) developed a theory that launched the field of Quantum Mechanics, known as Schrodinger’s equation.\u00a0 It is a non-relativistic theory that is used today to describe atomic structure.<\/p>\n
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A detailed timeline with links to the original papers follows ….<\/p>\n
1814 \u2013 Fraunhofer observes dark lines in solar spectrum \u2013 gives major lines a letter based designation – the C line is H-alpha, the F line is H-beta, the G’ line is H-gamma, and the h line is H-delta.\u00a0 D1\/D2 are the doublet lines of Sodium.<\/p>\n
Fraunhofer 1814 Denkschriften der Ko\u0308niglichen Akademie der Wissenschaften zu Mu\u0308nchen<\/a><\/p>\n 1853 \u2013 Angstrom first to observe visible spectrum of hydrogen \u2013 atmospheric pressure\/electric spark\/Leyden Jar<\/p>\n Angstrom 1871<\/a><\/p>\n 1868 \u2013 Angstrom\u2019s detailed map of the solar spectrum<\/p>\n Angstrom 1868 spectre solaire<\/a><\/p>\n 1870 \u2013 Virial Theorem by Classius \u2013 applies for all stable systems: for 1\/r2<\/sup> force law with particles in stable orbit: <T> = – 0.5 <V><\/p>\n 1870 Virial Thm Classius The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science<\/a><\/p>\n 1885 \u2013 Balmer formula matching wavelengths Angstrom\u2019s hydrogen measurements of 4 spectral lines \u2013 uses formula to analyze spectral observations from stars \u2013 formula matches the wavelengths of ten additional hydrogen spectral lines<\/p>\n Balmer 1885<\/a><\/p>\n MGL translation of Balmer paper<\/a><\/p>\n 1890 \u2013 Rydberg formula for spectral lines in alkali atoms<\/p>\n Rydberg 1890<\/a><\/p>\n 1897 \u2013 Zeeman shows that magnetic field splits spectral lines<\/p>\n Zeeman Philosophical Magazine 1897<\/a><\/p>\n 1897 \u2013 Larmor theory of magnetic field effect on spectra, and Radiation from moving charges<\/p>\n Larmour 1897 On the theory of the magnetic influence on spectra; and on the radiation from moving ions _ Zenodo<\/a><\/p>\n 1900 \u2013 Planck theory of blackbody radiation: E = h * nu<\/p>\n Planck (1900), Distribution Law<\/a><\/p>\n Planck 1910<\/a><\/p>\n Planck 1912<\/a><\/p>\n 1911 \u2013 Rutherford model of atom \u2013 nucleus plus orbiting electrons<\/p>\n 1913 \u2013 Bohr model of atom \u2013 circular orbits \u2013 derives Balmer formula, Rydberg constant, radius of atom, and more<\/p>\n Bohr 1913<\/a><\/p>\n Bohr1920_Article_\u00dcberDieSerienspektraDerElement<\/a><\/p>\n 1914 \u2013 Stark shows Balmer lines split in an electric field<\/p>\n (Figure above from Bethe and Salpeter’s 1957 book on Quantum Mechanics of one and two electron atoms)<\/p>\n andp.19143480703<\/a><\/p>\n andp.19143480702<\/a><\/p>\n 1915 \u2013 Sommerfeld\u2019s improvement to Bohr model \u2013 elliptical orbits \u2013 angular momentum added to the Bohr model<\/p>\n How Sommerfeld<\/a><\/p>\n 1924 \u2013 deBroglie hypothesis \u2013 PhD thesis under Langevin \u2013 matter waves: \u00a0lambda = h \/ p<\/p>\n 1925 – Goudsmit and Uhlenbeck – introduce the concept of electron spin – electrons are magnetic!<\/p>\n Goudsmit recollections<\/a><\/p>\n 1926 \u2013 Schrodinger atomic model<\/p>\n ——————————————————————————-<\/p>\n Now let’s use the Bohr model to derive some important relationships for atomic hydrogen ……<\/p>\n Below is an illustration of the Bohr atom for the lowest 5 orbits (n=1,2,3,4,5) – the circular orbits are shown in red dashed circles – notice that the lowest state, which has radius a0, has a circumference that equals one deBroglie wavelength, that is deBroglie wavelength for n=1 is 2*pi*a0 – the second state has a radius that is 2^2 * a0 = 4 * a0 – its circumference corresponds to two deBroglie wavelengths, that is deBroglie wavelength for n=2 is 2*pi*4*a0, and so on – derivations to follow ….<\/p>\n <\/p>\n When compared with the Schrodinger Equation solution, the Bohr model is off by 1 deBroglie wavelength – that is, it overestimates the number of deBroglie wavelengths by 1 for each level.\u00a0 So according to Schrodinger’s analysis for n = 1, the number of deBroglie wavelengths is 0, n = 2 the number is 1, and so on.\u00a0 We can see this by comparing an overlay of the Schrodinger solution (contour lines indicate probability of electron position – probability is proportional to the square of the wavefunction – wavefunction goes positive and then negative – the square is positive definite – so there are two lobes for each deBroglie wavelength) for n = 10, with the Bohr solution (red dotted circle with wavy black line indicating deBroglie wave) for n = 9 – in these two cases there are 9 deBroglie wavelengths – the units in the graph below is in ao (that is, the Bohr radius) – so you can see that the Schrodinger solution for n = 10 has a radius of about 100 * a0, and the overlay from the Bohr model for n=9 has a radius of precisely 81 * a0.\u00a0 When the quantum number increases, the relative radius error between Schrodinger and Bohr decreases.\u00a0 To recap, the Bohr model does a reasonable job at estimating the size of the atom, but misses the number of deBroglie waves in an orbit by 1.<\/p>\n A short history of the beginning of Quantum Mechanics (1853 – 1926) …… Consider the visible emission spectrum of hydrogen – the 4 most prominent visible lines start the Balmer series of lines (H-alpha, H-beta, H-gamma, H-delta).\u00a0 This was the state of affairs in the mid-19th century following the first observations of the spectrum of … Continue reading 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