In quantum mechanics, the Hamiltonian operator Ĥ is aObservable measurement(observable), which corresponds to the total energy of the system.Like all other operators, the spectrum of the Hamiltonian operator ismeasurement system Total energy of all possible resultsaggregate。Like other self adjoint operators, the spectrum of Hamiltonian operators can be decomposed into pure points, absolutely continuoussingularity(singular) Three parts.
The pure point spectrum corresponds to the eigenvector, which in turn corresponds to theBound state(bound states)；The absolute continuous spectrum corresponds to free states;The singularity spectrum is interestingly composed of physically impossible results.For example, consideration is limitedpotential wellWhich allows for a discreteNegative energyAnd free states with continuous positive energy.
The general Hamiltonian operator has the following form:
The Schrodinger equation can be written as:
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The Hamiltonian operator producesquantum stateTime evolution of.If at timetSystem status of, where ℏ isReduced Planck constant。This equation isSchrodinger equation。(It is the same asHamilton Jacobi equationIt has the same form, and because of this,ĤCoronalHamiltonName.)If a given system is at an initial time（t=0), we can get the system state at any time in the future by integrating.In particular, if Ĥ is independent of time, the form of steady state solution remains unchanged.
The Hamiltonian operator in has the following form:
One dimensional case:
Three dimensional situation:
The stationary Schrodinger equation can be transformed into a partial differential equation
Or become
For different potential functionsV, solve the partial differential equationStationary wave function。[1]
Hamiltonian operator
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First“
”This thing has“Dual character”, which is both a vector and adifferential operator （DerivationTherefore, the Hamiltonian operator has the properties of both vector and differential.By definition:
The above formula represents D'sdivergence(also called divD), Dx, Dy and Dz are respectively D in x, y and zAxisComponent on.▽ × H represents H'scurl(It can also be recorded as rotH or curlH).[2]
