Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.
The Hamiltonian is named after William Rowan Hamilton, who developed a revolutionary reformulation of Newtonian mechanics, known as Hamiltonian mechanics, which was historically important to the development of quantum physics.
The Hamiltonian takes different forms and can be simplified in some cases by taking into account the concrete characteristics of the system under analysis, such as single or several particles in the system, interaction between particles, kind of potential energy, time varying potential or time independent one.
By analogy with classical mechanics, the Hamiltonian is commonly expressed as the sum of operators corresponding to the kinetic and potential energies of a system in the form
Although this is not the technical definition of the Hamiltonian in classical mechanics, it is the form it most commonly takes.
which allows one to apply the Hamiltonian to systems described by a wave function
One can also make substitutions to certain variables to fit specific cases, such as some involving electromagnetic fields.
This result can be used to calculate the expectation value of the total energy which is given for a normalized wavefunction as:
Similarly, the condition can be generalized to any higher dimensions using divergence theorem.
is the potential energy function, now a function of the spatial configuration of the system and time (a particular set of spatial positions at some instant of time defines a configuration) and
For this reason cross terms for kinetic energy may appear in the Hamiltonian; a mix of the gradients for two particles:
denotes the mass of the collection of particles resulting in this extra kinetic energy.
is not simply a sum of the separate potentials (and certainly not a product, as this is dimensionally incorrect).
This is an idealized situation—in practice the particles are almost always influenced by some potential, and there are many-body interactions.
The exponential operator on the right hand side of the Schrödinger equation is usually defined by the corresponding power series in
One might notice that taking polynomials or power series of unbounded operators that are not defined everywhere may not make mathematical sense.
It is the time evolution operator or propagator of a closed quantum system.
form a one parameter unitary group (more than a semigroup); this gives rise to the physical principle of detailed balance.
However, in the more general formalism of Dirac, the Hamiltonian is typically implemented as an operator on a Hilbert space in the following way: The eigenkets of
The spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted
However, all routine quantum mechanical calculations can be done using the physical formulation.
This applies to the elementary "particle in a box" problem, and step potentials.
For a simple harmonic oscillator in one dimension, the potential varies with position (but not time), according to:
For a rigid rotor—i.e., system of particles which can rotate freely about any axes, not bound in any potential (such as free molecules with negligible vibrational degrees of freedom, say due to double or triple chemical bonds), the Hamiltonian is:
(i.e., those that have no spatial extent independently), in three dimensions, is (in SI units—rather than Gaussian units which are frequently used in electromagnetism):
A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves.
The energy of each of these plane waves is inversely proportional to the square of its wavelength.
In the case of the free particle, the conserved quantity is the angular momentum.
We now perform the following trick: instead of using the real and imaginary parts as the independent variables, we use
By applying Schrödinger's equation and using the orthonormality of the basis states, this further reduces to