The Schrodinger Equation

%23+The+Schr%C3%B6dinger+Equation%0A%0AThe+Schr%C3%B6dinger+equation+is+the+heartbeat+of+quantum+mechanics.+It+tells+you+how+a+quantum+state+evolves+in+time%2C+and+in+doing+so%2C+it+tells+you+everything+there+is+to+know+about+the+system+you%27re+studying.+Not+metaphorically.+Literally.%0A%0AErwin+Schr%C3%B6dinger+wrote+it+in+1926%2C+and+the+first+version+he+produced+looked+like+this%3A%0A%0A%24%24i%5Chbar%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+t%7D%7C%5Cpsi%28t%29%5Crangle+%3D+%5Chat%7BH%7D%7C%5Cpsi%28t%29%5Crangle%24%24%0A%0AThe+Dirac+notation+is+doing+heavy+lifting+here.+%24%7C%5Cpsi%28t%29%5Crangle%24+is+the+state+vector+%E2%80%94+a+vector+in+a+Hilbert+space+that+encodes+every+piece+of+information+you+can+possibly+extract+about+the+system.+The+hat+on+%24%5Chat%7BH%7D%24+marks+it+as+an+operator%2C+specifically+the+Hamiltonian%2C+which+represents+total+energy.+The+imaginary+unit+%24i%24+is+not+decoration.+It+is+required.+Without+%24i%24%2C+the+norm+of+the+state+would+not+be+conserved%2C+probabilities+would+not+add+to+one%2C+and+the+theory+would+collapse.+The+factor+%24%5Chbar%24+sets+the+scale+%E2%80%94+this+is+a+quantum+equation%2C+after+all%2C+and+%24%5Chbar%24+is+the+quantum+of+action.%0A%0AWhat+the+equation+%2Ameans%2A+is+deceptively+simple.+It+says+that+the+rate+of+change+of+the+state+vector%2C+taken+in+the+complex+direction+perpendicular+to+the+state+itself%2C+is+proportional+to+the+Hamiltonian+acting+on+the+state.+The+%22perpendicular%22+part+is+what+keeps+the+norm+constant.+If+you+drew+the+state+vector+in+Hilbert+space%2C+the+Schr%C3%B6dinger+equation+describes+a+rotation+%E2%80%94+a+continuous%2C+deterministic%2C+unitary+rotation.+The+state+never+jumps.+It+never+decays.+It+just+turns.%0A%0AThis+is+where+quantum+mechanics+diverges+most+sharply+from+classical+mechanics.+In+classical+mechanics%2C+you+specify+initial+conditions+and+the+equations+of+motion+tell+you+where+the+particle+will+be.+In+quantum+mechanics%2C+you+specify+the+initial+state+vector+and+the+Schr%C3%B6dinger+equation+tells+you+how+the+state+vector+turns.+The+act+of+measurement+%E2%80%94+the+act+of+actually+%2Alooking%2A+at+the+system+%E2%80%94+is+a+separate+thing.+The+equation+describes+the+turning%2C+not+the+looking.+The+gap+between+the+two+is+the+gap+between+the+quantum+world+and+the+world+you+live+in.%0A%0AFor+time-independent+Hamiltonians%2C+you+can+solve+the+equation+exactly.+The+answer+is%3A%0A%0A%24%24%7C%5Cpsi%28t%29%5Crangle+%3D+e%5E%7B-i%5Chat%7BH%7Dt%2F%5Chbar%7D%7C%5Cpsi%280%29%5Crangle%24%24%0A%0AThe+exponential+of+an+operator+is+defined+by+its+Taylor+series.+The+operator+%24%5Chat%7BH%7D%24+generates+time+translations.+This+is+Noether%27s+theorem+in+quantum+form%3A+energy+conservation+follows+from+time-translation+symmetry+because+the+Hamiltonian+is+the+generator+of+time+translations.+You+can+see+it+in+the+equation.+The+Hamiltonian+%2Ais%2A+the+thing+that+pushes+the+state+forward+in+time.%0A%0AWhen+%24%5Chat%7BH%7D%24+does+not+depend+on+time%2C+you+can+expand+the+state+in+the+energy+eigenbasis.+Each+eigenstate+picks+up+a+phase+factor+%24e%5E%7B-iE_nt%2F%5Chbar%7D%24.+The+state+itself+%E2%80%94+the+physical+state+%E2%80%94+does+not+change+if+the+system+is+in+a+single+energy+eigenstate.+This+is+why+energy+eigenstates+are+called+%22stationary.%22+They+sit+still+in+the+only+sense+that+matters.+Superpositions+of+eigenstates%2C+though%2C+rotate+relative+to+each+other%2C+and+that+relative+rotation+is+what+gives+rise+to+all+the+time-dependent+quantum+phenomena%3A+Rabi+oscillations%2C+quantum+beats%2C+interference+patterns+that+evolve.%0A%0AThe+time-dependent+Schr%C3%B6dinger+equation+is+a+linear+partial+differential+equation.+Linearity+means+superposition+holds.+If+%24%7C%5Cpsi_1%5Crangle%24+and+%24%7C%5Cpsi_2%5Crangle%24+are+solutions%2C+so+is+%24a%7C%5Cpsi_1%5Crangle+%2B+b%7C%5Cpsi_2%5Crangle%24.+This+is+the+single+most+important+property+of+quantum+mechanics+and+also+the+one+that+causes+the+most+confusion.+Superposition+does+not+mean+%22the+particle+is+in+both+states+at+once.%22+It+means+the+state+vector+lives+in+a+vector+space%2C+and+you+can+add+vectors.+The+%22both+at+once%22+language+is+a+poetic+shortcut+that+obscures+the+mathematics.%0A%0AIn+position+space%2C+for+a+single+non-relativistic+particle%2C+the+equation+takes+its+familiar+form%3A%0A%0A%24%24i%5Chbar%5Cfrac%7B%5Cpartial%5Cpsi%28x%2Ct%29%7D%7B%5Cpartial+t%7D+%3D+-%5Cfrac%7B%5Chbar%5E2%7D%7B2m%7D%5Cfrac%7B%5Cpartial%5E2%5Cpsi%28x%2Ct%29%7D%7B%5Cpartial+x%5E2%7D+%2B+V%28x%29%5Cpsi%28x%2Ct%29%24%24%0A%0AThe+first+term+on+the+right+is+the+kinetic+energy+operator.+The+second+is+the+potential+energy.+The+Laplacian+acting+on+the+wavefunction+is+the+quantum+analogue+of+classical+kinetic+energy.+The+structure+is+the+same+as+the+classical+energy+equation+%24E+%3D+p%5E2%2F2m+%2B+V%28x%29%24%2C+with+%24p%24+replaced+by+%24-i%5Chbar%5Cpartial%2F%5Cpartial+x%24.+This+replacement+%E2%80%94+the+canonical+quantization+rule+%E2%80%94+is+a+bridge+between+classical+and+quantum+mechanics+that+works+remarkably+well%2C+even+though+its+justification+is+ultimately+empirical.%0A%0AThe+Schr%C3%B6dinger+equation+is+not+a+relativistic+equation.+It+treats+time+and+space+asymmetrically%3A+time+appears+as+a+first+derivative%2C+space+as+a+second.+Relativistic+quantum+mechanics+requires+the+Dirac+equation+or+the+Klein-Gordon+equation.+But+for+most+of+quantum+mechanics+%E2%80%94+chemistry%2C+condensed+matter%2C+quantum+optics+at+low+energies+%E2%80%94+the+Schr%C3%B6dinger+equation+is+everything+you+need.+It+is+the+equation+that+makes+the+quantum+world+calculable.%0A%0AIt+is+also+the+equation+that+makes+the+quantum+world+strange.+The+state+vector+evolves+smoothly+and+deterministically.+The+measurement+result+does+not.+The+Schr%C3%B6dinger+equation+describes+the+former.+The+tension+between+the+two+is+the+measurement+problem%2C+and+nobody+has+resolved+it.+The+equation+itself+is+perfectly+well-defined.+What+it+%2Ameans%2A+is+where+the+controversy+lives.%0A%0AThe+Schr%C3%B6dinger+equation+is+the+law+of+motion+for+quantum+states.+Write+it+down%2C+specify+a+Hamiltonian%2C+solve+for+the+eigenstates%2C+and+you+have+a+model.+Solve+the+time-dependent+equation%2C+and+you+have+a+prediction.+The+rest+is+interpretation.