Hamiltonian (quantum mechanics) Totally Explained

Everything about Hamiltonian Quantum Mechanics totally explained

In quantum mechanics, the Hamiltonian H is the observable corresponding to the total energy of the system. As with all observables, the spectrum of the Hamiltonian is the set of possible outcomes when one measures the total energy of a system. Like any other self-adjoint operator, the spectrum of the Hamiltonian can be decomposed, via its spectral measures, into pure point, absolutely continuous, and singular parts. The pure point spectrum can be associated to eigenvectors, which in turn are the bound states of the system. The absolutely continuous spectrum corresponds to the free states. The singular spectrum, interestingly enough, comprises physically impossible outcomes. For example, consider the finite potential well, which admits bound states with discrete negative energies and free states with continuous positive energies.

Schrödinger equation

The Hamiltonian generates the time evolution of quantum states. If

left| psi (t) 
ight
angle

is the state of the system at time t, then ť

H left| psi (t) 
ight
angle = mathrm

which is precisely the form of Hamilton's equations, with the

a_n

s as the generalized coordinates, the

pi_n

s as the conjugate momenta, and

langle H
angle

taking the place of the classical Hamiltonian.

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