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A quantum mechanical
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 system or particle that is bound
Bound state

In physics, a bound state is a composite of two or more building blocks that behaves as a single object. In quantum mechanics , a bound state is a state in the Hilbert space that corresponds to two or more particles whose interaction energy is negative, and therefore these particles cannot be separated unless energy is spent....
, confined spatially, can only take on certain discrete values of energy, as opposed to classical
Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
 particles, which can have any energy. These values are called energy levels. The term is most commonly used for the energy levels (electron configuration
Electron configuration

In atomic physics and quantum chemistry, electron configuration is the arrangement of electrons in an atom, molecule, or other physical structure....
) of electron
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
s in atom
Atom

|-! bgcolor=gray | Properties|-||}The atom is a basic unit of matter consisting of a dense, central atomic nucleus surrounded by a electron cloud of electric charge electrons....
s or molecule
Molecule

In chemistry, a molecule is defined as a sufficiently stable, electric charge neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds....
s, which are bound by the electric field of the nucleus
Atomic nucleus

The nucleus of an atom is the very dense region, consisting of nucleons , at the center of an atom. Although the size of the nucleus varies considerably according to the mass of the atom, the size of the entire atom is comparatively constant....
. The energy spectrum of a system with energy levels is said to be quantized
Quantization (physics)

In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field . This is a generalization of the procedure for building quantum mechanics from classical mechanics....
.

If the potential energy
Potential energy

Potential energy can be thought of as energy stored within a physical system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do Mechanical work in the process....
 is set to zero at infinity, the usual convention, then bound electron states have negative potential energy.

Energy levels are said to be degenerate, if the same energy level is obtained by more than one quantum mechanical state
Quantum state

In quantum physics, a quantum State is a mathematical object that fully describes a Quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus....
.






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A quantum mechanical
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 system or particle that is bound
Bound state

In physics, a bound state is a composite of two or more building blocks that behaves as a single object. In quantum mechanics , a bound state is a state in the Hilbert space that corresponds to two or more particles whose interaction energy is negative, and therefore these particles cannot be separated unless energy is spent....
, confined spatially, can only take on certain discrete values of energy, as opposed to classical
Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
 particles, which can have any energy. These values are called energy levels. The term is most commonly used for the energy levels (electron configuration
Electron configuration

In atomic physics and quantum chemistry, electron configuration is the arrangement of electrons in an atom, molecule, or other physical structure....
) of electron
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
s in atom
Atom

|-! bgcolor=gray | Properties|-||}The atom is a basic unit of matter consisting of a dense, central atomic nucleus surrounded by a electron cloud of electric charge electrons....
s or molecule
Molecule

In chemistry, a molecule is defined as a sufficiently stable, electric charge neutral group of at least two atoms in a definite arrangement held together by very strong chemical bonds....
s, which are bound by the electric field of the nucleus
Atomic nucleus

The nucleus of an atom is the very dense region, consisting of nucleons , at the center of an atom. Although the size of the nucleus varies considerably according to the mass of the atom, the size of the entire atom is comparatively constant....
. The energy spectrum of a system with energy levels is said to be quantized
Quantization (physics)

In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field . This is a generalization of the procedure for building quantum mechanics from classical mechanics....
.

If the potential energy
Potential energy

Potential energy can be thought of as energy stored within a physical system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do Mechanical work in the process....
 is set to zero at infinity, the usual convention, then bound electron states have negative potential energy.

Energy levels are said to be degenerate, if the same energy level is obtained by more than one quantum mechanical state
Quantum state

In quantum physics, a quantum State is a mathematical object that fully describes a Quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus....
. They are then called degenerate energy level
Degenerate energy level

In physics two or more different physical states are said to be degenerate if they are all at the same energy level. Physical states differ if and only if they are linearly independent....
s.

Explanation

Quantised energy levels result from the relation between a particle's energy and its wavelength
Wavelength

In physics, wavelength is the distance between repeating units of a propagating wave of a given frequency. It is commonly designated by the Greek language letter lambda ....
. For a confined particle, for example an electron in an atom, the wave function has the form of standing wave
Standing wave

A standing wave, also known as a stationary wave, is a wave that remains in a constant position. This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions....
s. Only stationary state
Stationary state

In quantum mechanics, a stationary state is an eigenstate of a Hamiltonian , or in other words, a state of definite energy. It is called stationary because the corresponding probability density has no time dependence....
s with energies corresponding to integral numbers of wavelengths can exist; for other states the waves interfere destructively, resulting in zero probability density. Elementary examples that show mathematically how energy levels come about are the particle in a box
Particle in a box

In physics, the particle in a box is a problem consisting of a single particle inside of an infinitely deep potential well, from which it cannot escape, and which loses no energy when it collides with the walls of the box....
 and the quantum harmonic oscillator
Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum mechanics analogue of the harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point....
.

The following section give an overview of the most important factors that determine the energy levels of atoms and molecules.

Atoms


Intrinsic energy levels


Orbital state energy level
Assume an electron in a given atomic orbital
Atomic orbital

An atomic orbital is a mathematical function that describes the wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus....
. The energy of its state is mainly determined by the electrostatic interaction of the (negative) electron with the (positive) nucleus. The energy levels of an electron around a nucleus are given by :
,
where is the Rydberg constant
Rydberg constant

The Rydberg Physical constant, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to atomic spectrum in the science of spectroscopy....
 (typically between 1 eV
Electronvolt

In physics, the electron volt is a unit of energy. By definition, it is equal to the amount of kinetic energy gained by a single unbound electron when it accelerates through an Electrostatics potential difference of one volt....
 and 103 eV), Z is the Atomic number, n is the principal quantum number
Principal quantum number

In atomic physics, the principal quantum number symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The quantum number n labels the energy levels of hydrogenic atoms....
, is Planck's constant, and c is the speed of light
Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
.

The Rydberg levels depend only on the principal quantum number .

Fine structure splitting
Fine structure
Fine structure

In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to first order relativistic corrections.The gross structure of line spectra is the line spectra predicted by non-relativistic electrons with no spin....
 arises from relativistic kinetic energy corrections, spin-orbit coupling (an electrodynamic interaction between the electron's spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 and motion and the nucleus's electric field) and the Darwin term (contact term interaction of s-shell electrons inside the nucleus). Typical magnitude eV.

Hyperfine structure
Spin-nuclear-spin coupling. Typical magnitude eV.

Electrostatic interaction of an electron with other electrons
If there is more than one electron around the atom, electron-electron-interactions raise the energy level. These interactions are often neglected if the spatial overlap of the electron wavefunctions is low.

Energy levels due to external fields


Zeeman effect
The interaction energy is: with

Zeeman effect taking spin into account
This takes both the magnetic dipole moment due to the orbital angular momentum and the magnetic momentum arising from the electron spin into account.

Due to relativistic effects (Dirac equation
Dirac equation

In physics, the Dirac equation is a theory of relativity quantum mechanics wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary particle spin-? particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity....
), the magnetic moment arising from the electron spin is with the gyro-magnetic factor (about 2). The interaction energy therefore gets .

Stark effect
Interaction with the external electric field causes the Stark effect.

Molecules

Roughly speaking, a molecular energy state, i.e. an eigenstate of the molecular Hamiltonian
Molecular Hamiltonian

In atomic, molecular, and optical physics as well as in quantum chemistry, molecular Hamiltonian is the name given to the Hamiltonian representing the energy of the electrons and Atomic nucleus in a molecule....
, is the sum of an electronic, vibrational, rotational, nuclear and translational component, such that:

where is an eigenvalue of the electronic molecular Hamiltonian (the value of the potential energy surface
Potential energy surface

A potential energy surface is generally used within the adiabatic approximation or Born?Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems....
) at the equilibrium geometry of the molecule
Molecular geometry

Molecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its Reactivity , Chemical polarity, Phase , color, magnetism, and biological activity....
.

The molecular energy levels are labelled by the molecular term symbol
Molecular term symbol

In molecular physics, the molecular term symbol is a shorthand expression of the group representation and angular momentum that characterize the state of a molecule, i.e....
s.

The specific energies of these components vary with the specific energy state and the substance.

In molecular physics
Molecular physics

Molecular physics is the study of the physical properties of molecules and of the chemical bonds between atoms that bind them. Its most important experimental techniques are the various types of spectroscopy....
 and quantum chemistry
Quantum chemistry

Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry....
, an energy level is a quantized energy of a bound
Bound state

In physics, a bound state is a composite of two or more building blocks that behaves as a single object. In quantum mechanics , a bound state is a state in the Hilbert space that corresponds to two or more particles whose interaction energy is negative, and therefore these particles cannot be separated unless energy is spent....
 quantum mechanical state
Quantum state

In quantum physics, a quantum State is a mathematical object that fully describes a Quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus....
.

Crystalline Materials

Crystalline solids are found to have energy bands, instead of or in addition to energy levels. Electrons can take on any energy within an unfilled band. At first this appears to be an exception to the requirement for energy levels. However as shown in band theory, energy bands are actually made up of many descrete energy levels which are too close together to resolve. Within a band the number of levels is of the order of the number of atoms in the crystal, so although electrons are actually restricted to these energies, they appear to be able to take on a continuum of values. The important energy levels in a crystal are the top of the valence band
Valence band

In solids, the valence band is the highest range of electron energy where electrons are normally present at absolute zero.In semiconductors and Electrical insulations, there is a band gap above the valence band, followed by a conduction band above that....
, the bottom of the conduction band
Conduction band

In the physics field of semiconductors and Electrical insulations, the conduction band is the range of electron energy, higher than that of the valence band, sufficient to make the electrons free to accelerate under the influence of an applied electric field and thus constitute an electric current....
, the Fermi energy
Fermi energy

The Fermi energy is a concept in quantum mechanics usually referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature....
, the vacuum level, and the energy levels of any defect states in the crystal.

See also

  • Perturbation theory (quantum mechanics)
    Perturbation theory (quantum mechanics)

    In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation theory for describing a complicated quantum system in terms of a simpler one....
  • Computational chemistry
    Computational chemistry

    Computational chemistry is a branch of chemistry that uses computers to assist in solving chemical problems. It uses the results of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids....
  • Spectroscopy
    Spectroscopy

    Spectroscopy was originally the study of the interaction between radiation and matter as a function of wavelength . In fact, historically, spectroscopy referred to the use of visible light dispersed according to its wavelength, e.g....