# Energy level

In physics, an energy level is the potential energy for a quantum mechanical state. The term is most commonly used in reference to the electron configuration in atoms or molecules. According to quantum theory, the electron can only be in certain states, so that only certain energy levels are possible. In other words, the energy spectrum is quantized. As with classical potentials, the potential energy is usually set to zero at infinity, leading to a negative potential energy for bound electron states.

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

Roughly speaking, a molecular energy state is comprised of an electronic, vibrational, and rotational component, such that:

[itex]E = E_\mathrm{electronic}+E_\mathrm{vibrational}+E_\mathrm{rotational}\,[itex]

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

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## Interactions determining the energy of a bound electron in a single atom

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, calculatable using the principal quantum number [itex]n[itex].

However, there are many interactions that lead to small changes to this energy level, which can be calculated involving the other orbital electron quantum numbers [itex]l[itex], [itex]m_l[itex], [itex]m_s[itex]. The more accurate description of the electron wavefunction often leads to the splitting of the energy levels and therefore removes energy level degeneracy.

The following list gives an overview over the most important corrections to the energy level.

## Orbital state energy level

The energy level arrives from the electrostatic interaction of the electron with the positive atomic nucleus and from the energy arising from the angular momentum of the electron on its own (kinetic, magnetic).

Typical magnitude [itex]10 ... 10^{3}[itex] eV.

## Electrostatic interaction of 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.

## Zeeman effect

The orbital angular momentum of the electron corresponds to a magnetic momentum, interacting with the outer magnetic field (electromagnetic interaction). Zeeman effect

The interaction energy is: [itex]U = - \mu B[itex] with [itex]\mu = q L / 2m[itex]

## Zeeman effect taking spin into account

This takes both the magnetic dipole moment due to the orbital angular momentum and the magnetic momentum arrising from the electron spin into account.

Due to relativistic effects (Dirac equation), the magnetic moment arriving from the electron spin is [itex]\mu = - \mu_B g s[itex] with [itex]g[itex] the gyro-magnetic factor (about 2). [itex]\mu = \mu_l + g \mu_s[itex] The interaction energy therefore gets [itex]U_B = - \mu B = \mu_B B (m_l + g m_s)[itex].

## Fine structure splitting

Spin-orbit effect, cf. fine structure. Typical magnitude [itex]10^{-3}[itex] eV.

## Hyperfine structure

Spin-nuclear-spin coupling (cf. hyperfine structure). Typical magnitude [itex]10^{-4}[itex] eV.

## Stark effect

Interaction with an external electric field, cf. Stark effect.

## Paschen-Back effect

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