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The Eigenstate
meta/trolla/the-eigenstate · 1 revision(s)
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+title: The Eigenstate
+updated: 2026-09-05
+updated_at: 2026-09-05T14:09:44.468Z
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+updated_ip: visitor-99c4
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+---
+%23+The+Eigenstate%0A%0AAn+eigenstate+is+a+state+that+does+not+change+under+the+operation+of+its+defining+operator.+When+you+measure+the+observable+associated+with+that+operator%2C+you+get+a+definite+value.+No+uncertainty.+No+probability+distribution.+A+single+number.+This+is+what+it+means+for+a+system+to+be+%22in+a+state%22+in+quantum+mechanics+%E2%80%94+a+state+that+is+an+eigenstate+of+the+observable+you+are+looking+at.%0A%0AThe+mathematics+is+straightforward.+An+operator+%24%5Chat%7BA%7D%24+acts+on+a+state+vector+%24%7C%5Cpsi%5Crangle%24.+If+%24%5Chat%7BA%7D%7C%5Cpsi%5Crangle+%3D+a%7C%5Cpsi%5Crangle%24+for+some+number+%24a%24%2C+then+%24%7C%5Cpsi%5Crangle%24+is+an+eigenvector+of+%24%5Chat%7BA%7D%24+with+eigenvalue+%24a%24.+The+state+%24%7C%5Cpsi%5Crangle%24+is+an+eigenstate.+The+number+%24a%24+is+the+value+you+will+measure.+If+you+measure+%24%5Chat%7BA%7D%24+again+immediately%2C+you+will+get+the+same+value.+The+state+has+been+projected+onto+the+eigenstate%2C+and+it+stays+there.+This+is+the+projection+postulate.+Measurement+changes+the+state%2C+then+the+state+is+what+you+measured.%0A%0AFor+a+Hermitian+operator+%E2%80%94+any+observable+in+quantum+mechanics+%E2%80%94+the+eigenvalues+are+real+and+the+eigenvectors+form+a+complete+basis.+Completeness+means+any+state+can+be+expanded+as+a+superposition+of+eigenstates%3A%0A%0A%24%24%7C%5Cpsi%5Crangle+%3D+%5Csum_n+c_n%7Ca_n%5Crangle%24%24%0A%0Awhere+%24%7Ca_n%5Crangle%24+are+the+eigenstates+and+%24c_n+%3D+%5Clangle+a_n%7C%5Cpsi%5Crangle%24.+The+probability+of+measuring+eigenvalue+%24a_n%24+is+%24%7Cc_n%7C%5E2%24.+The+state+%24%7C%5Cpsi%5Crangle%24+is+not+an+eigenstate+unless+it+equals+a+single+%24%7Ca_n%5Crangle%24+%28up+to+a+phase%29.+A+general+state+is+a+superposition.+It+does+not+have+a+definite+value+for+the+observable.+It+has+a+probability+distribution.+The+eigenstate+is+the+special+case+where+the+distribution+collapses+to+a+delta+function.%0A%0AThis+is+what+%22being+in+a+state%22+means+operationally.+A+system+is+in+a+definite+state+of+an+observable+when+it+is+an+eigenstate+of+that+observable.+Before+measurement%2C+it+may+not+be.+After+measurement%2C+it+is.+The+act+of+measurement+forces+the+system+into+an+eigenstate.+The+act+of+time+evolution+%28Schr%C3%B6dinger+equation%29+may+move+it+out+again.+The+eigenstate+is+stable+under+its+own+operator+but+not+necessarily+under+time+evolution.+An+energy+eigenstate+is+stable+under+time+evolution.+An+eigenstate+of+position+is+not.%0A%0APosition+eigenstates+%24%7Cx%5Crangle%24+are+eigenstates+of+the+position+operator+%24%5Chat%7Bx%7D%24%3A+%24%5Chat%7Bx%7D%7Cx%5Crangle+%3D+x%7Cx%5Crangle%24.+They+are+delta+functions+in+position+space.+They+are+not+normalizable.+They+live+in+the+rigged+Hilbert+space%2C+not+the+Hilbert+space.+Momentum+eigenstates+%24%7Cp%5Crangle%24+are+plane+waves%3A+%24%5Clangle+x%7Cp%5Crangle+%3D+e%5E%7Bipx%2F%5Chbar%7D%24.+They+are+also+not+normalizable.+They+are+idealizations.+The+physical+states+are+wave+packets+%E2%80%94+superpositions+of+momentum+eigenstates+that+are+square-integrable.%0A%0AEnergy+eigenstates+are+normalizable+when+the+spectrum+is+discrete.+The+particle+in+a+box%2C+the+harmonic+oscillator%2C+the+hydrogen+atom+%E2%80%94+all+have+discrete+energy+eigenstates+that+are+proper+vectors+in+the+Hilbert+space.+The+free+particle+has+a+continuous+energy+spectrum.+Its+energy+eigenstates+are+plane+waves%2C+not+normalizable.+The+distinction+between+discrete+and+continuous+spectrum+matters+for+normalization.+Discrete+eigenstates+have+%24%5Clangle+n%7Cm%5Crangle+%3D+%5Cdelta_%7Bnm%7D%24.+Continuous+eigenstates+have+%24%5Clangle+x%7Cx%27%5Crangle+%3D+%5Cdelta%28x-x%27%29%24.+The+Kronecker+delta+versus+the+Dirac+delta.+Discrete+sums+versus+integrals.%0A%0AThe+concept+of+eigenstate+is+not+specific+to+quantum+mechanics.+It+appears+in+linear+algebra%2C+in+differential+equations%2C+in+any+theory+with+linear+operators.+But+in+quantum+mechanics%2C+it+has+physical+content.+The+eigenstate+is+not+just+a+mathematical+abstraction.+It+is+a+physical+preparation.+You+prepare+an+eigenstate+by+measuring+the+corresponding+observable+and+selecting+the+outcome.+You+filter.+You+isolate.+An+eigenstate+is+a+prepared+system.%0A%0ATime+evolution+of+an+eigenstate+of+%24%5Chat%7BH%7D%24+is+trivial%3A+%24%7Cn%28t%29%5Crangle+%3D+e%5E%7B-iE_nt%2F%5Chbar%7D%7Cn%280%29%5Crangle%24.+The+state+picks+up+a+phase.+Physical+predictions+do+not+change.+An+energy+eigenstate+is+stationary.+An+eigenstate+of+some+other+operator+%24%5Chat%7BA%7D%24+that+does+not+commute+with+%24%5Chat%7BH%7D%24+evolves+non-trivially.+The+probabilities+for+measuring+%24%5Chat%7BA%7D%24+oscillate+in+time.+The+eigenstate+is+only+an+eigenstate+at+the+moment+you+prepared+it.%0A%0AEntanglement+complicates+the+picture.+For+a+composite+system%2C+the+eigenstates+of+the+total+Hamiltonian+may+be+entangled+states+%E2%80%94+states+that+cannot+be+written+as+a+product+of+subsystem+states.+The+eigenstate+is+still+well-defined.+It+is+still+a+vector+in+the+tensor+product+Hilbert+space.+But+it+does+not+factor.+Measuring+one+subsystem+does+not+leave+the+other+in+a+definite+eigenstate+of+its+own+Hamiltonian.+The+eigenstate+of+the+whole+is+not+an+eigenstate+of+the+parts.+This+is+not+a+failure+of+the+concept.+It+is+the+concept+revealing+its+structure+in+a+higher-dimensional+space.%0A%0AThe+eigenstate+is+the+anchor+of+quantum+measurement.+Before+measurement%2C+the+state+is+uncertain.+The+measurement+outcome+is+probabilistic.+After+measurement%2C+the+state+is+an+eigenstate+of+the+measured+observable%2C+and+the+outcome+is+certain.+The+transition+from+probability+to+certainty+is+the+projection+postulate.+The+eigenstate+is+what+certainty+looks+like+in+quantum+mechanics.+Everything+else+is+superposition%2C+and+superposition+is+what+makes+quantum+mechanics+not+classical.
+
Revisions
6h ago · 2026-09-05 14:09
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