In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. 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. Similar to vector notation.

Introduction

The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kinetic and potential energies of all particles associated with the system. 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.

Schrödinger Hamiltonian

One particle

By analogy with classical mechanics, the Hamiltonian is commonly expressed as the sum of operators corresponding to the kinetic and potential is the potential energy is the kinetic energy is the mass of the particle, the dot denotes the dot product is the momentum operator is the del operator. The dot product with itself is the Laplacian. In three dimensions using Cartesian coordinates

Although this is not the technical definition of the Hamiltonian in classical mechanics, it is the form it most commonly takes. Combining these yields the form used in the Schrödinger equation which allows one to apply the Hamiltonian to systems described by a wave function. This is the approach commonly taken in introductory treatments of quantum mechanics, using the formalism of Schrödinger's wave mechanics.

One can also make substitutions to certain variables to fit specific cases, such as some involving electromagnetic fields.

Expectation value

It can be shown that the expectation value of the Hamiltonian which gives the energy expectation value will always be greater than or equal to the minimum potential of the system.

which complete the proof. Similarly, the condition can be generalized to any higher dimensions using the divergence theorem.

Many particles

is the Laplacian for particle

However, complications can arise in the many-body problem denotes the mass of the collection of particles resulting in this extra kinetic energy. Terms of this form are known as mass polarization terms, and appear in the Hamiltonian of many-electron atoms (see below).

is not simply a sum of the separate potentials (and certainly not a product, as this is dimensionally incorrect). The potential energy function can only be written as above: a function of all the spatial positions of each particle.

For non-interacting particles, i.e. particles which do not interact mutually and move independently, the potential of the system is the sum of the separate potential energy for each particle,

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 is an idealized situation—in practice the particles are almost always influenced by some potential, and there are many-body interactions. One illustrative example of a two-body interaction where this form would not apply is for electrostatic potentials due to charged particles, because they interact with each other by Coulomb interaction (electrostatic force), as shown below.

Schrödinger equation

This equation is the Schrödinger equation. It takes the same form as the Hamilton–Jacobi equation

The exponential operator on the right hand side of the Schrödinger equation is usually defined by the corresponding power series. One might notice that taking polynomials or power series of unbounded operators that are not defined everywhere may not make mathematical sense. Rigorously, to take functions of unbounded operators, a functional calculus is required. In the case of the exponential function, the continuous, or just the holomorphic functional calculus suffices. We note again, however, that for common calculations the physicists' formulation is quite sufficient.

By the *-homomorphism is a unitary operator. It is the time evolution operator or propagator form a one parameter unitary group (more than a semigroup); this gives rise to the physical principle of detailed balance.

Dirac formalism

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, provide an orthonormal basis

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.

Expressions for the Hamiltonian

Following are expressions for the Hamiltonian in a number of situations..

Free particle

Constant-potential well

This applies to the elementary "particle in a box" problem, and step potentials.

Simple harmonic oscillator

For a simple harmonic oscillator where the angular frequency, effective spring constant

Rigid rotor

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 are the moment of inertia components (technically the diagonal elements of the moment of inertia tensor), and are the total angular momentum axes respectively.

Electrostatic (Coulomb) potential

The Coulomb potential energy (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

due to all other charges is (see also Electrostatic potential energy stored in a configuration of discrete point charges):

Electric dipole in an electric field

For an electric dipole moment, in a uniform, electrostatic field

Magnetic dipole in a magnetic field

For a spin-12 particle, the corresponding spin magnetic moment is: is the "spin g-factor" (not to be confused with the gyromagnetic ratio is the spin operator vector, whose components are the Pauli matrices

Charged particle in an electromagnetic field

in an electromagnetic field, described by the scalar potential and vector potential, there are two parts to the Hamiltonian to substitute for. field and fulfils the canonical commutation relation is the kinetic momentum

Energy eigenket degeneracy, symmetry, and conservation laws

In many systems, two or more energy eigenstates have the same energy. 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 direction, but if they have the same wavelength, then their energies will be the same. When this happens, the states are said to be degenerate.

It turns out that degeneracy occurs whenever a nontrivial unitary operator commutes

has at least one pair of degenerate energy eigenkets. In the case of the free particle, the unitary operator which produces the symmetry is the rotation operator, which rotates the wavefunctions by some angle while otherwise preserving their shape.

The existence of a symmetry operator implies the existence of a conserved

In obtaining this result, we have used the Schrödinger equation, as well as its dual

Thus, the expected value is conserved for any state of the system. In the case of the free particle, the conserved quantity is the angular momentum.

Hamilton's equations

Hamilton's equations in classical Hamiltonian mechanics

Note that these basis states are assumed to be independent of time. We will assume that the Hamiltonian is also independent of time.

are complex variables. We can treat them as coordinates which specify the state of the system, like the position and momentum coordinates which specify a classical system. Like classical coordinates, they are generally not constant in time, and their time dependence gives rise to the time dependence of the system as a whole.

in terms of the basis states.

actually corresponds to two and its complex conjugate. With this choice of independent variables, we can calculate the partial derivative

By applying the Schrödinger equation

taking the place of the classical Hamiltonian.

See also

• Birman–Schwinger principle

• Bra–ket notation

• Conservation of energy

• Electric field

• Electrostatics

• Hamiltonian mechanics

• Lieb–Thirring inequality

• Linear algebra

• Magnetic field

• Many-body problem

• Operator (physics)

• Potential theory

• Quantum state

• Two-state quantum system

References

Further reading

External links