# Energy level

Energy scheme of the atomic orbital model

An energy level is the discrete energy that, as an energy in its own right, belongs to a quantum mechanical state of a system (such as an atom or an atomic nucleus ). Energy levels are eigenvalues ​​of the Hamilton operator , they are therefore independent of time. The system can only “stay” permanently in one of these states, but not in other, intermediate values ​​of energy. The lowest energy level is called the ground state (or in the case of degeneracy, the "ground state"), all other levels are called excited states.

One can clearly imagine that the arrangement and mode of movement of the electrons in the atomic shell  - or of the nucleons in the nucleus - are only stable in a very specific form. Each of these states has a different, specific energy content. However, above a certain limit energy there is also an energy continuum , a range of any possible energy values.

In a conservative field, for example in the Coulomb field in the atomic shell, this limit corresponds to the binding energy of the most easily separable particle (see e.g. ionization ). The continuum of possible energies results from the fact that this separated particle can fly away with any kinetic energy . In other fields, for example for the nucleons of the atomic nucleus, the continuum boundary does not coincide with the binding energy of a particle.

In both cases there can also be energy levels in the continuum, which can be seen as resonances in cross sections. In atoms this occurs when a state with an asymptotically free particle degenerates - that is, it is equal in energy to a state without an asymptotically free particle.

## Transitions between energy levels

### To higher energy

Term scheme of the hydrogen atom with possible excitations.

Energy absorption into the system can only take place by changing to a higher energy level or into the continuum. This happens, for example, through the absorption of a photon or through the inelastic collision of a particle, as in the Franck-Hertz experiment . At transitions between discrete levels, the appropriate amount of energy must be supplied; the process is called excitation . It leads to discrete absorption lines in the spectrum .

### Too lower energy

The reverse transition from a higher to a lower level with the release of a photon can take place in the atom through spontaneous or externally stimulated emission.

#### Spontaneous

The spontaneous process is called the decay of the excited state or spontaneous emission . Like radioactive decay, it is characterized by a half-life . The energy of the emitted photons corresponds to the energy difference between the two energy levels involved. This causes the discrete spectral lines in the emission spectrum of excited atoms and molecules.

#### Stimulates

An emission process that does not take place spontaneously is stimulated emission , which is used in the laser .

### General

Mathematically, a quantum mechanical transition is calculated with the help of the transition dipole moment , which describes the temporal and spatial course of the quantum mechanically superposed mixed state of ground and excited state. Because of the selection rules z. Sometimes not all transitions are allowed.

Excited states and decay with emission of electromagnetic radiation , d. H. There are transitions in the direction of higher and lower energy not only in atoms, but also in atomic nuclei. In these, the energy differences between the states are significantly higher. This is where the high-energy gamma radiation is created by spontaneous emission .

## Energy levels in the atom

The energy levels of the atoms are described by the main quantum number . The energy of the state with the quantum number in a hydrogen-like atom of the atomic number is approximate ${\ displaystyle n}$${\ displaystyle n}$ ${\ displaystyle Z}$

${\ displaystyle E_ {n} = - {\ frac {Z ^ {2}} {n ^ {2}}} E _ {\ mathrm {R}}}$

with the Rydberg energy . ${\ displaystyle E _ {\ mathrm {R}} = 13, \! 6 \ \ mathrm {eV}}$

In addition, there are fine structure and hyperfine structure corrections and the Lamb shift .