In the Schrödinger equation for this system of one negative and one positive particle, the atomic orbitals are the eigenstates of the Hamiltonian operator for the energy.

In this type of calculation, there is an approximate Hamiltonian and an approximate expression for the total electron density.

Once the electronic and nuclear variables are separated ( within the Born – Oppenheimer representation ), in the time-dependent approach, the wave packet corresponding to the nuclear degrees of freedom is propagated via the time evolution operator ( physics ) associated to the time-dependent Schrödinger equation ( for the full molecular Hamiltonian ).

High-dimensional spaces occur in mathematics and the sciences for many reasons, frequently as configuration spaces such as in Lagrangian or Hamiltonian mechanics ; these are abstract spaces, independent of the physical space we live in.

The number of times the interaction Hamiltonian acts is the order of the perturbation expansion, and the time-dependent perturbation theory for fields is known as the Dyson series.

For this reason cross terms for kinetic energy may appear in the Hamiltonian ; a mix of the gradients for two particles:

where the sum is taken over all particles and their corresponding potentials ; the result is that the Hamiltonian of the system is the sum of the separate Hamiltonians for each particle.

This follows from rotational invariance of the Hamiltonian, which in turn is guaranteed for a spherically symmetric potential.

This was rigorously proved and extended by Vladimir Arnold ( in 1963 for analytic Hamiltonian systems ) and Jürgen Moser ( in 1962 for smooth twist maps ), and the general result is known as the KAM theorem.

In his PhD thesis project, Paul Dirac discovered that the equation for the operators in the Heisenberg representation, as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, when one expresses them through Poisson brackets, a procedure now known as canonical quantization.

for all times s, t. The existence of a self-adjoint Hamiltonian H such that

For this reason, the Hamiltonian for the observables is called " free Hamiltonian " and the Hamiltonian for the states is called " interaction Hamiltonian ".

The Heisenberg picture is the closest to classical Hamiltonian mechanics ( for example, the commutators appearing in the above equations directly translate into the classical Poisson brackets ); but this is already rather " high-browed ", and the Schrödinger picture is considered easiest to visualize and understand by most people, to judge from pedagogical accounts of quantum mechanics.

Time would be replaced by a suitable coordinate parameterizing the unitary group ( for instance, a rotation angle, or a translation distance ) and the Hamiltonian would be replaced by the conserved quantity associated to the symmetry ( for instance, angular or linear momentum ).

Suppose we replace the real Hamiltonian of the model by a trial Hamiltonian, which has different interactions and may depend on extra parameters that are not present in the original model.

Quantum algorithms may also be stated in other models of quantum computation, such as the Hamiltonian oracle model.

No better quantum algorithm for this case was known until one was found for the unconventional Hamiltonian oracle model.

This model doesn't admit a Hamiltonian formulation ( the Legendre transform is multi-valued because the momentum function isn't convex ) because it is acausal.

Note that the Ising model exhibits the following discrete symmetry: If every spin in the model is flipped, such that, where is the value of the spin, the Hamiltonian ( and consequently the free energy ) remains unchanged.

What is now known as the standard Potts model was suggested by Potts in the course of the solution above, and uses a simpler Hamiltonian, given by:

The Hamiltonian of the n-vector model is given by:

In this model the atomic Hamiltonian is a sum of kinetic energies of the electrons and the spherically symmetric electron-nucleus interactions.

The Hubbard model, named after John Hubbard, is the simplest model of interacting particles in a lattice, with only two terms in the Hamiltonian ( see example below ): a kinetic term allowing for tunneling (' hopping ') of particles between sites of the lattice and a potential term consisting of an on-site interaction.

Expressed in terms of the Hubbard model, on the other hand, the Hamiltonian is now made up of two components.

This scheme maps the Hubbard Hamiltonian onto a single site impurity model, which allows one to compute the local Green's function of the Hubbard model for a given and a given temperature.

The model is defined through the following Hamiltonian:

Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.

Specifically, in quantum mechanics, the state of an atom, i. e. an eigenstate of the atomic Hamiltonian, is approximated by an expansion ( see configuration interaction expansion and basis set ) into linear combinations of anti-symmetrized products ( Slater determinants ) of one-electron functions.

Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations.

In particular, quantum phase transitions refer to transitions where the temperature is set to zero, and the phases of the system refer to distinct ground states of the Hamiltonian.

Semi-empirical methods follow what are often called empirical methods, where the two-electron part of the Hamiltonian is not explicitly included.

In the Hamiltonian formalism, given a coordinate it is possible to derive the appropriate ( generalized ) momentum such that the associated volume is preserved by the flow.

Although the resulting energy eigenfunctions ( the orbitals ) are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely generally from this isotropy of the underlying potential: the eigenstates of the Hamiltonian ( that is, the energy eigenstates ) can be chosen as simultaneous eigenstates of the angular momentum operator.

In quantum mechanics, the Hamiltonian is the operator corresponding to the total energy of the system.

The Hamiltonian is the sum of the kinetic energies of all the particles, plus the potential energy of the particles associated with the system.

For different situations and / or number of particles, the Hamiltonian is different since it includes the sum of kinetic energies of the particles, and the potential energy function corresponding to the situation.

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.

The general form of the Hamiltonian in this case is:

( It takes the same form as the Hamilton – Jacobi equation, which is one of the reasons H is also called the Hamiltonian ).

If the Hamiltonian is time-independent,