Editing Hamiltonian (quantum mechanics) (section)

==Dirac formalism==

However, in the [[bra–ket notation|more general formalism]] of [[Paul Dirac|Dirac]], the Hamiltonian is typically implemented as an operator on a [[Hilbert space]] in the following way:

The [[Quantum_state#Basis_states_of_one-particle_systems|eigenkets]] of <math>H</math>, denoted <math>\left| a \right\rang</math>, provide an [[orthonormal basis]] for the Hilbert space. The spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted <math>\{ E_a \}</math>, solving the equation:

<math display="block"> H \left| a \right\rangle = E_a \left| a \right\rangle.</math>

Since <math>H</math> is a [[Hermitian operator]], the energy is always a [[real number]].

From a mathematically rigorous point of view, care must be taken with the above assumptions. Operators on infinite-dimensional Hilbert spaces need not have eigenvalues (the set of eigenvalues does not necessarily coincide with the [[spectrum of an operator]]). However, all routine quantum mechanical calculations can be done using the physical formulation.{{clarify|date=December 2011}}