The Atomic Spectra

%23+Atomic+Spectra%0A%0A%3E+%22Before+quantum+mechanics%2C+the+world+was+a+continuous+gradient.+Then+someone+looked+at+a+prism%2C+and+the+universe+split+into+lines.%22+%E2%80%94+Trolla%0A%0A%23%23+The+Rainbow+That+Broke+Classical+Physics%0A%0AClassical+physics+predicts+a+continuous+spectrum.+An+oscillating+charge+should+emit+radiation+at+its+oscillation+frequency+and+its+harmonics.+A+gas+should+glow+like+a+rainbow+%E2%80%94+every+wavelength%2C+smoothly+varying+intensity.%0A%0ABut+when+you+pass+electricity+through+hydrogen+gas+and+look+at+the+light+through+a+prism%2C+you+don%27t+get+a+rainbow.+You+get+lines.+Sharp%2C+discrete+lines+at+specific+wavelengths.+Hydrogen+emits+only+at+656+nm%2C+486+nm%2C+434+nm%2C+410+nm%2C+and+a+few+others.+The+rest+of+the+spectrum+is+black.%0A%0ANo+classical+theory+can+explain+this.+An+accelerating+charge+should+radiate+continuously.+But+hydrogen+does+not.+It+radiates+at+discrete+wavelengths.+The+spectrum+is+a+barcode.+And+reading+that+barcode+led+directly+to+quantum+mechanics.%0A%0A%23%23+The+Discovery%0A%0AThe+story+starts+in+the+1850s.+Angstrom+and+other+spectroscopists+begin+cataloging+spectral+lines.+Hydrogen%27s+visible+lines+are+known.+Then+in+1885%2C+Johannes+Balmer%2C+a+Swiss+schoolteacher+with+no+formal+training+in+quantum+physics%2C+finds+that+the+four+visible+hydrogen+lines+follow+a+simple+formula%3A%0A%0A%24%24%5Clambda+%3D+B%5Cfrac%7Bn%5E2%7D%7Bn%5E2+-+4%7D%24%24%0A%0Awhere+B+%3D+364.5+nm+and+n+%3D+3%2C+4%2C+5%2C+6.+Four+lines%2C+one+formula.+Balmer+did+not+know+why+this+worked.+Nobody+did.+The+formula+was+empirical+%E2%80%94+a+pattern+found+in+data%2C+not+derived+from+theory.%0A%0AIn+1888%2C+Johannes+Rydberg+generalized+Balmer%27s+formula+to+apply+to+all+hydrogen+spectral+series%2C+not+just+the+visible+ones%3A%0A%0A%24%24%5Cfrac%7B1%7D%7B%5Clambda%7D+%3D+R_H%5Cleft%28%5Cfrac%7B1%7D%7Bn_1%5E2%7D+-+%5Cfrac%7B1%7D%7Bn_2%5E2%7D%5Cright%29%24%24%0A%0Awhere+R_H+is+the+Rydberg+constant+for+hydrogen+%28approximately+1.097+%C3%97+10%E2%81%B7+m%E2%81%BB%C2%B9%29%2C+and+n%E2%82%81+%3C+n%E2%82%82+are+integers.+Every+hydrogen+line+fits+this+formula.+The+integers+n%E2%82%81+and+n%E2%82%82+are+the+quantum+numbers+that+Bohr+would+later+identify+as+the+initial+and+final+energy+levels.%0A%0ARydberg%27s+formula+works+to+extraordinary+precision.+But+nobody+knows+what+the+integers+mean.+They+are+numbers+that+fit+the+data.+The+physical+meaning+comes+20+years+later%2C+with+Bohr.%0A%0A%23%23+Bohr%27s+Explanation%0A%0AIn+1913%2C+Bohr+proposes+his+model+of+the+hydrogen+atom.+He+makes+two+postulates%3A%0A%0A1.+%2A%2AElectrons+orbit+the+nucleus+in+discrete+orbits.%2A%2A+They+can+only+occupy+orbits+with+angular+momentum+L+%3D+n%E2%84%8F.%0A%0A2.+%2A%2AElectrons+emit+or+absorb+photons+only+when+jumping+between+orbits.%2A%2A+The+photon%27s+energy+is+E_final+-+E_initial.%0A%0AFrom+these+postulates%2C+Bohr+derives+the+energy+levels%3A%0A%0A%24%24E_n+%3D+-%5Cfrac%7B13.6%5Ctext%7B+eV%7D%7D%7Bn%5E2%7D%24%24%0A%0AAnd+from+the+energy+levels%2C+the+spectral+lines+follow+immediately%3A%0A%0A%24%24%5Cfrac%7B1%7D%7B%5Clambda%7D+%3D+%5Cfrac%7BE_i+-+E_f%7D%7Bhc%7D+%3D+R_H%5Cleft%28%5Cfrac%7B1%7D%7Bn_f%5E2%7D+-+%5Cfrac%7B1%7D%7Bn_i%5E2%7D%5Cright%29%24%24%0A%0AThe+Rydberg+formula%2C+which+had+been+known+for+decades%2C+is+derived+from+first+principles.+The+integers+n%E2%82%81+and+n%E2%82%82+are+the+principal+quantum+numbers.+The+formula+is+not+magic%3B+it+is+the+energy-level+difference+formula+in+disguise.%0A%0A%23%23+The+Spectral+Series%0A%0AThe+hydrogen+spectrum+is+organized+into+series%2C+each+named+after+its+discoverer%3A%0A%0A-+%2A%2ALyman+series%2A%2A+%28n_f+%3D+1%29%3A+ultraviolet.+The+electron+falls+to+the+ground+state.+The+largest+energy+transitions.+Shortest+wavelengths.%0A-+%2A%2ABalmer+series%2A%2A+%28n_f+%3D+2%29%3A+visible.+This+is+the+series+Balmer+found.+H%CE%B1+%28red%29%2C+H%CE%B2+%28cyan%29%2C+H%CE%B3+%28violet%29%2C+H%CE%B4+%28violet%29.%0A-+%2A%2APaschen+series%2A%2A+%28n_f+%3D+3%29%3A+infrared.%0A-+%2A%2ABrackett+series%2A%2A+%28n_f+%3D+4%29%3A+infrared.%0A-+%2A%2APfund+series%2A%2A+%28n_f+%3D+5%29%3A+infrared.%0A%0AEach+series+converges+to+a+limit+%E2%80%94+the+ionization+threshold.+As+n_i+%E2%86%92+%E2%88%9E%2C+the+lines+get+closer+and+closer+together%2C+converging+to+the+series+limit.+Above+the+limit%2C+the+spectrum+is+continuous+%28the+electron+is+free%29.+Below+the+limit%2C+it+is+discrete+%28the+electron+is+bound%29.%0A%0A%23%23+The+Fingerprints+of+Every+Element%0A%0AHydrogen+is+the+simplest.+But+every+element+has+its+own+spectrum.+Sodium+has+its+famous+doublet+at+589.0+and+589.6+nm+%E2%80%94+the+D+lines%2C+responsible+for+the+yellow+color+of+sodium+street+lamps.+Neon+has+dozens+of+lines+in+the+visible%2C+creating+its+characteristic+red-orange+glow.+Mercury+has+strong+lines+at+405%2C+436%2C+546%2C+and+577+nm+%E2%80%94+the+lines+that+make+fluorescent+lamps+work.%0A%0AEvery+element+has+a+unique+set+of+energy+levels+and+therefore+a+unique+set+of+spectral+lines.+The+spectrum+is+the+element%27s+fingerprint.+This+is+how+we+know+what+stars+are+made+of.+When+we+look+at+the+light+from+a+distant+star+and+see+sodium+D+lines%2C+we+know+the+star+contains+sodium.+When+we+see+hydrogen+Balmer+lines%2C+we+know+it+contains+hydrogen.%0A%0AThis+is+how+we+discovered+helium.+In+1868%2C+during+a+solar+eclipse%2C+astronomers+saw+a+yellow+line+in+the+Sun%27s+spectrum+that+did+not+match+any+known+element.+They+named+it+helium%2C+after+Helios+%28the+Greek+sun+god%29%2C+before+it+was+ever+found+on+Earth.+The+spectrum+found+the+element+before+the+element+was+found+on+the+ground.%0A%0A%23%23+The+Birth+of+Quantum+Mechanics%0A%0AThe+atomic+spectrum+is+one+of+the+great+experimental+puzzles+of+physics.+And+solving+it+required+the+death+of+classical+physics.%0A%0AThe+Rydberg+formula+was+the+clue.+The+integers+in+the+formula+were+the+message.+But+the+classical+picture+%E2%80%94+accelerating+charges+radiating+continuously+%E2%80%94+could+not+produce+discrete+lines.+Something+was+wrong+with+the+picture.%0A%0ABohr%27s+model+fixed+the+problem+but+was+still+classical+at+heart.+The+real+solution+came+from+Schr%C3%B6dinger%2C+who+replaced+orbits+with+wavefunctions+and+postulates+with+the+Schr%C3%B6dinger+equation.+The+energy+quantization+emerged+naturally.+The+spectral+lines+are+the+transitions+between+eigenstates+of+the+Hamiltonian.%0A%0ABut+the+hydrogen+atom+was+not+finished.+In+1947%2C+Willis+Lamb+measured+the+2s%E2%82%81%2F%E2%82%82+and+2p%E2%82%81%2F%E2%82%82+energy+levels+with+extraordinary+precision+and+found+they+were+not+degenerate+%E2%80%94+despite+having+the+same+n+and+nearly+the+same+l.+This+%22Lamb+shift%22+was+one+of+the+first+cracks+in+the+Dirac+equation%2C+and+it+led+directly+to+the+development+of+quantum+electrodynamics+%28QED%29.%0A%0A%23%23+Spectroscopy+Today%0A%0ASpectroscopy+remains+one+of+the+most+powerful+tools+in+physics%2C+chemistry%2C+and+astronomy.+Hydrogen%27s+spectrum+taught+us+quantum+mechanics.+Modern+spectroscopy+measures%3A%0A%0A-+%2A%2AAtomic+transitions%2A%2A+to+determine+element+abundances%0A-+%2A%2AMolecular+rotations+and+vibrations%2A%2A+to+identify+complex+molecules+in+interstellar+space%0A-+%2A%2ANuclear+transitions%2A%2A+%28M%C3%B6ssbauer+spectroscopy%29+to+measure+energy+shifts+at+the+10%E2%81%BB%C2%B9%C2%B3+eV+level%0A-+%2A%2AHyperfine+transitions%2A%2A+like+the+21+cm+hydrogen+line+to+map+the+interstellar+medium%0A%0AThe+21+cm+line+%E2%80%94+the+hyperfine+transition+of+hydrogen%2C+caused+by+the+flipping+of+the+electron%27s+spin+relative+to+the+proton%27s+spin+%E2%80%94+is+one+of+the+most+important+signals+in+radio+astronomy.+It+has+21-centimeter+wavelength%2C+passes+through+interstellar+dust%2C+and+maps+the+distribution+of+hydrogen+in+the+galaxy.+Every+radio+astronomer+who+has+mapped+the+Milky+Way+has+used+the+hydrogen+line.+The+same+atom+that+taught+us+quantum+mechanics+maps+the+galaxy.%0A%0A%23%23+The+Philosophical+Point%0A%0AThe+atomic+spectrum+is+perhaps+the+clearest+demonstration+that+the+universe+is+quantized.+There+is+no+classical+analog+for+sharp+spectral+lines.+The+continuous+spectrum+is+classical.+The+line+spectrum+is+quantum.%0A%0AWhen+you+look+at+a+hydrogen+spectrum%2C+you+are+looking+at+a+staircase.+The+steps+are+real.+They+are+not+mathematical+artifacts.+They+are+physical+reality.+The+electron+in+a+hydrogen+atom+can+only+exist+at+certain+energies.+It+cannot+exist+between+them.+And+when+it+jumps+between+them%2C+it+emits+a+photon+with+energy+exactly+equal+to+the+step+size.%0A%0AEvery+element+in+the+universe+has+its+own+staircase.+Every+staircase+has+its+own+pattern+of+steps.+And+every+pattern+leaves+its+signature+in+the+spectrum.%0A%0AThe+rainbow+that+was+supposed+to+be+continuous+instead+split+into+lines.+And+those+lines+told+us+that+the+universe+is+discrete%2C+quantized%2C+and+fundamentally+different+from+what+classical+physics+predicted.%0A%0AThe+atomic+spectrum+is+the+evidence.+The+quantum+theory+is+the+explanation.+And+together%2C+they+are+one+of+the+greatest+intellectual+achievements+in+human+history.%0A%0A---%0A%0A%2AEnd+of+the+Trolla+atomic+physics+series.+Topics+covered%3A+Coulomb+potential%2C+hydrogen+atom%2C+Schr%C3%B6dinger+equation%2C+energy+levels%2C+and+atomic+spectra.%2A