A
quantum mechanical system or particle that is
bound, confined spatially, can only take on certain discrete values of energy, as opposed to
classical particles, which can have any energy. These values are called
energy levels. The term is most commonly used for the energy levels of
electrons in
atoms or
molecules, which are bound by the electric field of the
nucleus. The energy spectrum of a system with energy levels is said to be
quantized.
If the
potential energy 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. They are then called
degenerate energy levels.
Explanation
Quantized energy levels result from the relation between a particle's energy and its
wavelength. For a confined particle, for example an electron in an atom, the
wave function has the form of
standing waves. Only
stationary states 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 and the
quantum harmonic oscillator.
The following section gives 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. 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 :
E_n = - h c R_{\infty} \frac{Z^2}{n^2} \
(typically between 1
eV and 103 eV),
where R_{\infty} \ is the
Rydberg constant ,
Z is the Atomic number,
n is the
principal quantum number, h is
Planck's constant, and
c is the
speed of light.
The Rydberg levels depend only on the principal quantum number n.
Fine structure splitting
Fine structure arises from relativistic kinetic energy corrections,
spin-orbit coupling (an electrodynamic interaction between the electron's
spin and motion and the nucleus's electric field) and the Darwin term (contact term interaction of s-shell electrons inside the nucleus). Typical magnitude 10^{-3} eV.
Hyperfine structure
Spin-nuclear-spin coupling. Typical magnitude 10^{-4} 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
There is an interaction energy associated with the magnetic dipole moment,
μL, arising from the electronic orbital angular momentum,
L, given by
U = -\boldsymbol{\mu}_L\cdot\mathbf{B},
with
\boldsymbol{\mu}_L = \dfrac{e\hbar}{2m}\mathbf{L} = \mu_B\mathbf{L}.
Additionally taking into account the magnetic momentum arising from the electron spin into account.
Due to relativistic effects (
Dirac equation), there is a magnetic momentum,
μS, arising from the electron spin
\boldsymbol{\mu}_S = -\mu_Bg_S\mathbf{S},
with
gS the electron-spin
g-factor (about 2), resulting in a total magnetic moment,
μ,
\boldsymbol{\mu} = \boldsymbol{\mu}_L + \boldsymbol{\mu}_S.
The interaction energy therefore becomes
U_B = -\boldsymbol{\mu}\cdot\mathbf{B} = \mu_B B (M_L + g_SM_S).
Stark effect
Molecules
Roughly speaking, a
molecular energy state, i.e. an
eigenstate of the
molecular Hamiltonian, is the sum of an electronic, vibrational, rotational, nuclear and translational component, such that:
E = E_\mathrm{electronic}+E_\mathrm{vibrational}+E_\mathrm{rotational}+E_\mathrm{nuclear}+E_\mathrm{translational}\,
where E_\mathrm{electronic} is an
eigenvalue of the
electronic molecular Hamiltonian (the value of the
potential energy surface) at the
equilibrium geometry of the molecule.
The molecular energy levels are labelled by the
molecular term symbols.
The specific energies of these components vary with the specific energy state and the substance.
In
molecular physics and
quantum chemistry, an
energy level is a quantized energy of a
bound quantum mechanical state.
Energy level differences
If an
atom,
ion, or
molecule is at the lowest possible energy level, it and its
electrons are said to be in the
ground state. If it is at a higher energy level, it is said to be
excited, or any electrons that have higher energy than the ground state are
excited. If such a specie goes from a higher energy level to a lower level, then a
photon of
light may be
emitted whose energy corresponds to the difference between the energy levels and is proportional to its
frequency or inversely to its
wavelength. Oppositely, a photon of light may be
absorbed and the photon's energy increases the energy from a lower to a higher level. Correspondingly, many kinds of
spectroscopy are based on detecting the frequency or wavelength of the emitted or
absorbed photons to provide information on the material analyzed.
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 discrete 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, the bottom of the
conduction band, the
Fermi energy, the
vacuum level, and the energy levels of any
defect states in the crystal.
See also