History of
The Hamiltonian
field/trolla/the-hamiltonian · 1 revision(s)
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+---
+title: The Hamiltonian
+updated: 2026-09-05
+updated_at: 2026-09-05T14:06:45.695Z
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+updated_ip: visitor-99c4
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+---
+%23+The+Hamiltonian%0A%0AThe+Hamiltonian+is+the+single+most+important+object+in+non-relativistic+quantum+mechanics.+It+is+an+operator.+It+represents+total+energy.+It+generates+time+evolution.+Everything+else+is+commentary.%0A%0AIn+classical+mechanics%2C+the+Hamiltonian+%24H%28q%2Cp%29%24+is+a+function+on+phase+space.+You+write+it+in+terms+of+generalized+coordinates+and+their+conjugate+momenta.+The+Hamiltonian+equations+give+you+first-order+equations+of+motion.+The+Legendre+transform+relates+it+to+the+Lagrangian.+This+is+all+classical+machinery.+Quantum+mechanics+copies+the+structure+but+changes+the+nature+of+the+objects.+The+classical+variables+%24q%24+and+%24p%24+become+operators+%24%5Chat%7Bq%7D%24+and+%24%5Chat%7Bp%7D%24+satisfying+%24%5B%5Chat%7Bq%7D%2C%5Chat%7Bp%7D%5D+%3D+i%5Chbar%24.+The+classical+Hamiltonian+becomes+a+quantum+operator+%24%5Chat%7BH%7D%24.+The+replacement+is+called+canonical+quantization.+It+is+a+rule%2C+not+a+derivation.%0A%0AFor+a+single+particle+moving+in+one+dimension+under+a+potential+%24V%28x%29%24%2C+the+quantum+Hamiltonian+is%3A%0A%0A%24%24%5Chat%7BH%7D+%3D+%5Cfrac%7B%5Chat%7Bp%7D%5E2%7D%7B2m%7D+%2B+V%28%5Chat%7Bx%7D%29+%3D+-%5Cfrac%7B%5Chbar%5E2%7D%7B2m%7D%5Cfrac%7Bd%5E2%7D%7Bdx%5E2%7D+%2B+V%28x%29%24%24%0A%0AThe+first+term+is+kinetic+energy.+The+second+is+potential+energy.+The+kinetic+term+contains+the+second+derivative+because+momentum+is+represented+by+%24-i%5Chbar%5Cpartial%2F%5Cpartial+x%24.+The+potential+term+is+multiplicative+because+position+is+represented+by+multiplication.+This+is+the+position+representation.+In+the+momentum+representation%2C+the+roles+reverse.%0A%0AThe+Hamiltonian+is+Hermitian+%28self-adjoint%29.+This+is+not+optional.+A+Hermitian+operator+has+real+eigenvalues%2C+which+is+necessary+because+eigenvalues+correspond+to+measurable+energy+values.+It+also+has+a+complete+orthonormal+set+of+eigenvectors%2C+which+is+necessary+because+you+need+to+be+able+to+expand+any+state+in+the+energy+basis.+The+spectrum+can+be+discrete%2C+continuous%2C+or+both.+Hydrogen+has+a+discrete+negative-energy+spectrum+%28bound+states%29+and+a+continuous+positive-energy+spectrum+%28scattering+states%29.+The+harmonic+oscillator+has+a+purely+discrete+spectrum.+The+free+particle+has+a+purely+continuous+spectrum.+The+structure+of+the+spectrum+reflects+the+physics.%0A%0AThe+Hamiltonian+generates+time+translations.+This+is+the+content+of+the+Schr%C3%B6dinger+equation%3A%0A%0A%24%24i%5Chbar%5Cfrac%7Bd%7D%7Bdt%7D%7C%5Cpsi%5Crangle+%3D+%5Chat%7BH%7D%7C%5Cpsi%5Crangle%24%24%0A%0AMore+precisely%2C+%24%5Chat%7BH%7D%24+is+the+generator+of+the+one-parameter+unitary+group+%24U%28t%29+%3D+e%5E%7B-i%5Chat%7BH%7Dt%2F%5Chbar%7D%24.+Stone%27s+theorem+guarantees+that+every+strongly+continuous+one-parameter+unitary+group+has+a+self-adjoint+generator.+In+quantum+mechanics%2C+that+generator+is+the+Hamiltonian.+This+is+a+mathematical+theorem%2C+not+an+assumption.+Time+evolution+%2Ais%2A+unitary+evolution+generated+by+the+Hamiltonian.%0A%0AConserved+quantities+are+Hamiltonians+for+their+own+symmetries.+If+the+Hamiltonian+commutes+with+some+operator+%24%5Chat%7BA%7D%24%2C+then+%24%5Chat%7BA%7D%24+is+conserved.+If+the+system+is+translationally+invariant%2C+the+Hamiltonian+commutes+with+the+momentum+operator%2C+and+momentum+is+conserved.+If+the+system+is+rotationally+invariant%2C+the+Hamiltonian+commutes+with+angular+momentum.+If+the+Hamiltonian+does+not+depend+explicitly+on+time%2C+energy+is+conserved.+These+are+the+quantum+versions+of+Noether%27s+theorem.%0A%0AThe+Hamiltonian+of+a+many-body+system+is+written+in+second+quantization+by+promoting+creation+and+annihilation+operators.+For+fermions%2C+the+operators+anticommute.+For+bosons%2C+they+commute.+The+Hamiltonian+takes+the+form%3A%0A%0A%24%24%5Chat%7BH%7D+%3D+%5Csum_%7Bij%7D+t_%7Bij%7Da_i%5E%5Cdagger+a_j+%2B+%5Cfrac%7B1%7D%7B2%7D%5Csum_%7Bijkl%7D+V_%7Bijkl%7Da_i%5E%5Cdagger+a_j%5E%5Cdagger+a_l+a_k%24%24%0A%0AThe+first+term+describes+single-particle+hopping+or+kinetic+energy.+The+second+describes+interactions.+This+is+the+starting+point+for+condensed+matter+physics%2C+quantum+chemistry%2C+and+quantum+field+theory.+The+form+is+the+same.+The+operators+change.+The+physics+changes+with+them.%0A%0ASome+Hamiltonians+can+be+solved+exactly.+The+free+particle%2C+the+harmonic+oscillator%2C+the+hydrogen+atom%2C+the+rigid+rotator.+These+are+the+solvable+models.+They+are+taught+because+they+are+solvable%2C+not+because+they+are+realistic.+Real+systems+are+messy.+They+have+anharmonicity%2C+spin-orbit+coupling%2C+many-body+interactions%2C+disorder.+Solvable+models+are+reference+points.+They+tell+you+what+to+expect+when+everything+simplifies.+Deviations+from+them+tell+you+what+the+complicating+factors+do.%0A%0AThe+Hamiltonian+defines+a+quantum+system.+Specify+%24%5Chat%7BH%7D%24+and+you+have+specified+the+model.+Compute+its+spectrum+and+you+know+the+possible+energies.+Compute+its+eigenstates+and+you+know+the+stationary+configurations.+Compute+its+time+evolution+and+you+know+the+dynamics.+The+Hamiltonian+is+the+entire+theory.
+
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