Relativistic hydrogen problem

The relativistic hydrogen problem is the generalization of the hydrogen problem of non-relativistic quantum mechanics to relativistic quantum mechanics. Instead of solving the Schrödinger equation for the Coulomb potential of a point charge, the relativistic generalization requires the solution of the Dirac equation . The solution to the relativistic hydrogen problem fully explains the fine structure of the hydrogen spectrum .

From a physical point of view, the relativistic hydrogen problem is a two-body problem in a central potential , the electric Coulomb potential , because it deals with the interaction of two electrically charged particles, assumed to be point-like, electron and atomic nucleus . It generalizes the treatment of the classical hydrogen problem to the special theory of relativity and that of the celestial mechanical Kepler problem both to the theory of relativity and to quantum mechanics. Like these two, it is one of the few precisely solvable problems in physics.

history

In 1885 Johann Jakob Balmer found the empirical Balmer formula for calculating the wavelengths of the spectral lines of the hydrogen atom in the visible range of the electromagnetic spectrum, which was generalized by Johannes Rydberg in 1888 to the Rydberg formula . In this formula, integer quantities appear as parameters , the meaning of which found its first plausible explanation in the semi-classical Bohr model of the atom from 1913, since electrons can only move on discrete orbits. After all, quantum mechanics with the Schrödinger equation provided the model of the hydrogen atom, which is still accepted today, according to which the electron resides in orbitals around the nucleus and the energies of the electrons can only assume discrete values: when the electron passes from an orbital of higher energy into a lower light is emitted with exactly one frequency and wavelength. ${\displaystyle n_{i}}$

More precise measurements of the spectrum showed in 1887 that the lines of the hydrogen spectrum are not individual lines, but consist of a bundle of closely spaced spectral lines. This division is called fine structure. It was partially explained by Arnold Sommerfeld in the semi-classical atomic model with the aid of elliptical orbits and heuristic assumptions. The full explanation was achieved in 1928 by the formulation of the Dirac equation by Paul Dirac and its solution by Charles Galton Darwin in the same year.

Later, with the Lamb shift, further splits of the hydrogen lines were found, which can only be explained with the help of quantum field theory .

Fine structure

→ Main article : Fine structure (physics)

The corrections of the energy levels in the hydrogen atom and thus also the fine structure are induced by three causes: the consideration of the relativistic energy-momentum relation , the spin of the electron and the Darwin term . The replacement of the classical kinetic energy by the relativistic can be treated in terms of perturbation theory and causes a shift in the energy levels, but not their splitting. The spin is a quantum mechanical quantity that had to be introduced ad hoc in classical quantum mechanics in order, for example, to explain the anomalous Zeeman effect and is coupled to the orbital angular momentum , so that the energies of different angular momentum states are split up. These two effects, inserted “by hand” into the theory, lead to correct results.

The last effect of the fine structure, the Darwin term, comes from the trembling movement of the electron due to the uncertainty relation . If you insert this heuristically, it leads to a wrong result, because it is then too small by a factor of 3/4.

The use of a relativistic equation, the aim of which was to implement the relativistic energy-momentum relation in quantum mechanics, not only leads to the consideration of the relativistic correction, but also to the correct Darwin term and, in the case of the Dirac equation, also to the consideration of the spin -Bahn coupling

Classic hydrogen problem

→ Main article : Hydrogen problem

The hydrogen problem is based on a Coulomb potential of the atomic nucleus and a single electron in the atomic shell . After the transition to the center of gravity system , the Schrödinger equation to be solved for the quantum mechanical wave function is : ${\displaystyle \psi }$

${\displaystyle \left(-{\frac {\hbar ^{2}}{2\mu }}\Delta -\left(E+Z\alpha \hbar c{\frac {1}{r}}\right)\right)\psi ({\vec {r}})=0}$

There are:

• ${\displaystyle \hbar }$the reduced Planck quantum of action
• ${\displaystyle \mu }$the reduced mass of the system
• ${\displaystyle \Delta }$the Laplace operator
• ${\displaystyle Z}$the atomic number
• ${\displaystyle \alpha }$the fine structure constant
• ${\displaystyle c}$the speed of light
• ${\displaystyle E}$the energy of the system
• ${\displaystyle {\vec {r}}}$the distance vector between nucleus and electron and
• ${\displaystyle r}$its amount

The solution of this equation leads to certain forms of the wave function, which can be described by three quantum numbers . is called principal quantum number and is identical to that of the Rydberg formula; 's angular momentum quantum number , since the rail angular momentum only dependent and is called magnetic quantum number , as it is responsible for the splitting of spectral lines in the magnetic field ( Zeeman effect ). ${\displaystyle n,\ell ,m}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \ell }$ ${\displaystyle \ell }$ ${\displaystyle m_{l}}$

The solution functions can be separated into a radial part and an angular part

${\displaystyle \psi _{n\ell m_{l}}(r,\theta ,\phi )={\frac {U_{n\ell }(r)}{r}}Y_{\ell m_{l}}(\theta ,\phi )}$

where are the spherical surface functions and the radial part of the equation ${\displaystyle Y_{\ell m_{l}}}$

${\displaystyle \left(-\hbar ^{2}c^{2}{\frac {\partial ^{2}}{\partial r^{2}}}+\hbar ^{2}c^{2}{\frac {\ell (\ell +1)}{r^{2}}}-Z\alpha \hbar c{\frac {2\mu c^{2}}{r}}\right)U_{n\ell }(r)=2\mu c^{2}E_{n}U_{n\ell }(r)}$

obey. The energy of the system is discrete and only depends on the principal quantum number; it is said that the energy is in and degenerate . The energies that can be accepted are ${\displaystyle \ell }$ ${\displaystyle m}$

${\displaystyle E_{n}=-\mu c^{2}{\frac {(Z\alpha )^{2}}{2n^{2}}}}$.

Another quantum number, the radial quantum number , which counts the number of zeros of the radial wave function , can be introduced as an auxiliary variable . She is about the relationship ${\displaystyle n_{r}}$

${\displaystyle n_{r}=n-\ell -1}$

connected with the principal and the angular momentum quantum number.

Relativistic generalization

There are two different equations to generalize the Schrödinger equation relativistically. The first possibility is the Dirac equation, which describes particles with a spin s = ½, the second possibility is the Klein-Gordon equation , which describes particles with a spin 0. The spin is a quantum mechanical quantity which has no analogue in classical mechanics and which has to be inserted ad hoc by hand in non-relativistic quantum mechanics using the Pauli equation, but which results automatically from the Dirac equation. Since the electron has the spin ½, the correct description for the hydrogen atom is done using the Dirac equation.

Pionic hydrogen

Pionic hydrogen is an exotic atom in which the shell electron has been replaced by a negatively charged pion . Pions belong to the pseudoscalar mesons and therefore have a spin 0, so that the Klein-Gordon equation must be used. In addition, in a complete treatment using quantum field theory, pions also interact with the atomic nucleus due to the strong force , which should be neglected in this treatment. The Klein-Gordon equation does not correctly reproduce the hydrogen spectrum, but it is easier to solve than the Dirac equation. The Klein-Gordon equation is

${\displaystyle \left(-\hbar ^{2}c^{2}\Delta +{\frac {\mu ^{2}c^{2}}{\hbar ^{2}}}-\left(E+Z\alpha \hbar c{\frac {1}{r}}\right)^{2}\right)\phi ({\vec {r}})=0}$

where denotes a scalar wave function. As in the non-relativistic Limes, a separation approach is used to solve the equation ${\displaystyle \phi }$

${\displaystyle \phi ={\frac {U(r)}{r}}Y_{lm_{l}}(\theta ,\phi )}$

used to separate the radial dependence from the angular dependence. The Laplace operator can also be separated into a radial and an angular component,

${\displaystyle \Delta ={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r-{\frac {{\vec {L}}^{2}}{\hbar ^{2}r^{2}}}}$

where is the angular momentum operator . This acts by means of the spherical surface functions. The equation for the radial wave function is therefore: ${\displaystyle {\vec {L}}}$ ${\displaystyle {\vec {L}}^{2}Y_{lm_{l}}(\theta ,\phi )=\hbar ^{2}l(l+1)Y_{lm_{l}}(\theta ,\phi )}$

${\displaystyle \left(-\hbar ^{2}c^{2}{\frac {\partial ^{2}}{\partial r^{2}}}+\hbar ^{2}c^{2}{\frac {l(l+1)-(Z\alpha )^{2}}{r^{2}}}-Z\alpha \hbar c{\frac {2E}{r}}\right)U(r)=(E^{2}-\mu ^{2}c^{4})U(r)}$

In this form, the Klein-Gordon equation has the same structure as the Schrödinger equation from the non-relativistic hydrogen problem with the following substitutions:

order	Non-relativistic Schrödinger equation	Relativistic Klein-Gordon equation
${\displaystyle r^{-0}}$	${\displaystyle 2\mu c^{2}E}$	${\displaystyle E^{2}-\mu ^{2}c^{4}}$
${\displaystyle r^{-1}}$	${\displaystyle 2\mu c^{2}}$	${\displaystyle 2E}$
${\displaystyle r^{-2}}$	${\displaystyle \ell (\ell +1)}$	${\displaystyle l(l+1)-(Z\alpha )^{2}}$

In order to determine the energy level in pionic hydrogen, it is sufficient to carry out these replacements from analogy in the non-relativistic case. So it applies

Non-relativistic Schrödinger equation	Relativistic Klein-Gordon equation
${\displaystyle 2\mu c^{2}E=-\mu ^{2}c^{4}{\frac {(Z\alpha )^{2}}{(n_{r}+\ell +1)^{2}}}}$	${\displaystyle E^{2}-\mu ^{2}c^{4}=-E^{2}{\frac {(Z\alpha )^{2}}{\left(n_{r}+{\frac {1}{2}}+{\sqrt {\left(l+{\frac {1}{2}}\right)^{2}-(Z\alpha )^{2}}}\right)^{2}}}}$
${\displaystyle E=-\mu c^{2}{\frac {(Z\alpha )^{2}}{2n^{2}}}}$	${\displaystyle E=\mu c^{2}\left(1+{\frac {(Z\alpha )^{2}}{{\tilde {n}}^{2}}}\right)^{-1/2}}$

with . It is obvious that, in contrast to the main quantum number, it does not have to be an integer and instead has taken on the role of a “good” quantum number. ${\displaystyle \textstyle {\tilde {n}}=n_{r}+{\tfrac {1}{2}}+{\sqrt {(l+{\tfrac {1}{2}})^{2}-(Z\alpha )^{2}}}}$ ${\displaystyle {\tilde {n}}}$ ${\displaystyle n}$ ${\displaystyle n_{r}}$

Weak potential

The quantity is in the weak potential, i.e. for atoms with a low atomic number, and a Taylor expansion can therefore be carried out in it. It then applies to the energy: ${\displaystyle (Z\alpha )^{2}\ll 1}$

${\displaystyle E\approx \mu c^{2}\left[1-{\frac {(Z\alpha )^{2}}{2n^{2}}}-{\frac {(Z\alpha )^{4}}{2n^{4}}}\left({\frac {n}{\ell +{\tfrac {1}{2}}}}-{\frac {3}{4}}\right)\right]}$

The first term is the rest energy , the second the result of the non-relativistic calculation and the higher order terms reflect the fine structure in which the degeneracy in is canceled. Nevertheless, this result does not agree with the measurements from the hydrogen spectrum, since the electron spin has been neglected. ${\displaystyle \ell }$

Strong potential

For atomic nuclei with a high atomic number, this can happen . This happens with the s-orbitals with an atomic number of ( thulium ). Then the term in the root for the calculation of the principal quantum number becomes negative and the energy imaginary , which is a physically nonsensical result. The description by the relativistic quantum mechanics breaks down for strong potentials; instead, quantum field theory must be used for correct consideration. ${\displaystyle (Z\alpha )^{2}>(l+{\tfrac {1}{2}})^{2}}$ ${\displaystyle l=0}$ ${\displaystyle Z\geq 69}$

Standard hydrogen

The Dirac equation for the relativistic hydrogen problem is

${\displaystyle \left(-\mathrm {i} \hbar c\gamma ^{0}{\vec {\gamma }}\cdot {\vec {\nabla }}+\gamma ^{0}\mu c^{2}-I_{4}\left(E-Z\alpha \hbar c{\frac {1}{r}}\right)\right)\psi ({\vec {r}})=0}$

In addition to the sizes already introduced:

• ${\displaystyle \mathrm {i} }$the imaginary unit
• ${\displaystyle {\vec {\nabla }}}$the Nabla operator
• ${\displaystyle {\vec {\gamma }}}$and the four Dirac matrices ${\displaystyle \gamma ^{0}}$
• ${\displaystyle I_{4}}$the four-dimensional identity matrix

In contrast to the Schrödinger and Klein-Gordon equations, the Dirac equation is a first order differential equation. In addition, since the Dirac matrices are four-dimensional, the wave function also has four components, so that the equation to be solved actually represents four coupled differential equations. To decouple these equations one finally arrives at a differential equation of the second order after a series of term transformations using the Dirac algebra

${\displaystyle \left(-\hbar ^{2}c^{2}\Delta +{\frac {\mu ^{2}c^{2}}{\hbar ^{2}}}-\left(E+Z\alpha \hbar c{\frac {1}{r}}\right)^{2}+\mathrm {i} Z\hbar c{\frac {1}{r^{3}}}{\begin{pmatrix}{\vec {\sigma }}\cdot {\vec {r}}&0\\0&-{\vec {\sigma }}\cdot {\vec {r}}\end{pmatrix}}\right)\psi ({\vec {r}})=0}$

which, compared to the Klein-Gordon equation for pionic hydrogen, contains an additional term in which the three Pauli matrices occur, which correctly describe the spin of the electron. In this form, the Dirac equation is decoupled into two independent two-component problems, the structure of which is identical except for the sign in the last term. It is therefore only necessary to solve the upper of the two equations, since the solution of the lower one results from this with the replacement . ${\displaystyle {\vec {\sigma }}}$ ${\displaystyle {\vec {r}}\to -{\vec {r}}}$

This spin term complicates the solution of the equation, since the angular momentum operator does not commute with it and the orbital angular momentum is therefore not a conserved quantity. Instead, the (orbit) angular momentum operator must be replaced by the total angular momentum operator

${\displaystyle {\vec {J}}={\vec {L}}+{\frac {1}{2}}{\vec {\Sigma }}={\vec {L}}+{\frac {1}{2}}{\begin{pmatrix}{\vec {\sigma }}&0\\0&{\vec {\sigma }}\end{pmatrix}}}$

be replaced. The eigenvalues ​​of the total angular momentum operator and its eigenfunctions are no longer the spherical surface functions, but rather the two-component functions due to the rules for angular momentum addition using the Clebsch-Gordan coefficients ${\displaystyle j=l\pm s=l\pm {\tfrac {1}{2}}}$

${\displaystyle \Omega _{jlm}={\frac {1}{\sqrt {2l+1}}}{\begin{pmatrix}{\sqrt {l\pm m+1/2}}Y_{lm-1/2}\\\pm {\sqrt {l\mp m+1/2}}Y_{lm+1/2}\end{pmatrix}}}$

with the magnetic quantum number taking into account the spin as the sum of the magnetic orbital angular momentum quantum number and the magnetic spin quantum number . The signs are selected according to. Since there are two possible choices for each choice , the states must be further distinguished. This is done with the help of their parity , i.e. their behavior under spatial reflection . Since the spherical surface functions have definite parity , it also has this property, so parity is also a good quantum number that is directly related to the orbital angular momentum. ${\displaystyle m=m_{s}+m_{l}}$ ${\displaystyle m_{l}}$ ${\displaystyle m_{s}=\pm {\tfrac {1}{2}}}$ ${\displaystyle j=l\pm {\tfrac {1}{2}}}$ ${\displaystyle j}$ ${\displaystyle l}$ ${\displaystyle P=(-1)^{l}}$ ${\displaystyle \Omega _{jlm}}$

For this reason, the wave function can be represented in Bra-Ket notation as a state vector by its quantum numbers : ${\displaystyle j,m,l}$

${\displaystyle |\psi \rangle =|j\,l\,m\rangle =|j\,j\mp 1/2\,m\rangle \equiv |j\,\pm \,m\rangle }$

This state vector is through local operator connected to the wave function: . ${\displaystyle |x\rangle }$ ${\displaystyle \psi ({\vec {r}})=\langle x|\psi \rangle }$

It therefore remains to set up the equations for the radial component. In the basis of these quantum numbers, this results after a series of term transformations to

${\displaystyle \left(-\hbar ^{2}c^{2}{\frac {\partial ^{2}}{\partial r^{2}}}+\hbar ^{2}c^{2}{\frac {\lambda (\lambda +1)}{r^{2}}}-Z\alpha \hbar c{\frac {2E}{r}}\right)U(r)=(E^{2}-\mu ^{2}c^{4})U(r)}$

With

${\displaystyle \lambda ={\sqrt {\left(j+{\tfrac {1}{2}}\right)^{2}-(Z\alpha )^{2}}}-{\frac {1}{2}}\pm {\frac {1}{2}}}$.

Finding the energy eigenvalues ​​results again from analogy conclusions to the Schrödinger equation with the following substitutions

order	Non-relativistic Schrödinger equation	Relativistic Dirac equation
${\displaystyle r^{-0}}$	${\displaystyle 2\mu c^{2}E}$	${\displaystyle E^{2}-\mu ^{2}c^{4}}$
${\displaystyle r^{-1}}$	${\displaystyle 2\mu c^{2}}$	${\displaystyle 2E}$
${\displaystyle r^{-2}}$	${\displaystyle \ell (\ell +1)}$	${\displaystyle \lambda (\lambda +1)}$

to the energy eigenvalues

Non-relativistic Schrödinger equation	Relativistic Dirac equation
${\displaystyle 2\mu c^{2}E=-\mu ^{2}c^{4}{\frac {(Z\alpha )^{2}}{(n_{r}+\ell +1)^{2}}}}$	${\displaystyle E^{2}-\mu ^{2}c^{4}=-E^{2}{\frac {(Z\alpha )^{2}}{n_{r}+\lambda +1}}}$
${\displaystyle E=-\mu c^{2}{\frac {(Z\alpha )^{2}}{2n^{2}}}}$	${\displaystyle E=\mu c^{2}\left(1+{\frac {(Z\alpha )^{2}}{n'^{2}}}\right)^{-1/2}}$

with the effective principal quantum number . Since the radial quantum number can count all possible natural numbers including zero, the last two terms can be absorbed into it. To extract the non-relativistic limit, a zero in the form of ${\displaystyle \textstyle n'=n_{r}+{\sqrt {\left(j+{\tfrac {1}{2}}\right)^{2}-(Z\alpha )^{2}}}+{\frac {1}{2}}\pm {\frac {1}{2}}}$ ${\displaystyle n_{r}}$ ${\displaystyle \textstyle {\frac {1}{2}}\pm {\frac {1}{2}}}$

${\displaystyle n'=n+{\sqrt {\left(j+{\tfrac {1}{2}}\right)^{2}-(Z\alpha )^{2}}}-j-{\frac {1}{2}}}$

inserted, where is. ${\displaystyle \textstyle n=n_{r}+j+{\tfrac {1}{2}}}$

Splitting of the energy levels of the hydrogen atom in spectroscopic notation with the names . You can see the in , but not in, reversed degeneration in the fine structure (blue) ${\displaystyle nl_{j}}$ ${\displaystyle l=0,1,2=S,P,D}$ ${\displaystyle j}$ ${\displaystyle l}$

In the limit of weak potentials or small atomic numbers, this results in:

${\displaystyle E\approx \mu c^{2}\left[1-{\frac {(Z\alpha )^{2}}{2n^{2}}}-{\frac {(Z\alpha )^{4}}{2n^{4}}}\left({\frac {n}{j+{\frac {1}{2}}}}-{\frac {3}{4}}\right)\right]}$

In terms of form, this is the same result as when considering pionic hydrogen, only the orbital angular momentum quantum number is replaced by the total angular momentum quantum number . This has significant effects on the spectrum, since the orbital angular momentum quantum number is an integer ( ), while the total angular momentum quantum number is half- integer ( ). In the case of the real hydrogen atom, the degeneracy of the energy eigenvalues ​​is not in but in , so that configurations with different orbital angular momentum can occupy the same energy level depending on the spin setting. However, these identical energy levels have different parity because they differ by exactly one. In particular, with the relativistic hydrogen problem, with correct treatment, the case occurs that the energy becomes imaginary, which in the case of the orbitals with, however, only happens at an atomic number of . All completely ionized natural elements can therefore be easily described with the relativistic quantum theory. ${\displaystyle l}$ ${\displaystyle j}$ ${\displaystyle l=0,1,\dots ,n-1}$ ${\displaystyle j={\tfrac {1}{2}},{\tfrac {3}{2}},\dots ,n-{\tfrac {1}{2}}}$ ${\displaystyle l}$ ${\displaystyle j}$ ${\displaystyle l}$ ${\displaystyle Z\alpha >j+{\tfrac {1}{2}}}$ ${\displaystyle l=0}$ ${\displaystyle Z\geq 138}$

See also

• Helium atom , treats the analogue of the non-relativistic hydrogen problem as a multi-electron system
• Moseley's law , modification of the Rydberg formula for multi-electron systems
• Hyperfine structure , also takes nuclear spin into account

Individual evidence

1. ↑ Arnold Sommerfeld: On the fine structure of the hydrogen lines. History and current state of the theory . In: Natural Sciences . tape 28 , no. 27 , 1940, p. 417 - 423 , doi : 10.1007 / BF01490583 .
2. ^ Charles Darwin: The wave equation of the electron . In: Proceedings of the Royal Society of London . tape 118 , no. 780 , 1928, pp. 654 - 680 , doi : 10.1098 / rspa.1928.0076 .

literature

• Thorsten Fließbach: Quantum Mechanics . 4th edition. Spectrum, Munich 2005, ISBN 3-8274-1589-6 .
• Armin Wachter: Relativistic Quantum Mechanics . Springer, Berlin Heidelberg New York 2005, ISBN 3-540-22922-1 .
• James Bjorken and Sidney Drell: Relativistic Quantum Mechanics . McGraw-Hill, New York St. Louis San Francisco Toronto London Sydney 1965, ISBN 978-0-07-005493-6 (English).