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Ahydrogen-like atom (orhydrogenic atom) is anyatom orion with a singleelectron.[1][2][3] Examples of hydrogen-like atoms areH,He+,Li2+,Be3+ and so on, as well as any of theirisotopes. These ions areisoelectronic with hydrogen and are sometimes calledhydrogen-like ions.[4] The non-relativisticSchrödinger equation and relativisticDirac equation for the hydrogen atom and hydrogen-like atoms can be solved analytically, owing to the simplicity of the two-particle physical system. The one-electronwave function solutions are referred to ashydrogen-like atomic orbitals. Hydrogen-like atoms are of importance because their corresponding orbitals bear similarity to the hydrogen atomic orbitals.
The definition of hydrogen-like atoms can be extended to also include any system with only onevalence electron (but morecore electrons). Examples such atoms include, but are not limited to, allalkali metals such asRb andCs and singly ionizedalkaline earth metals such asCa+ andSr+. In such a case, the hydrogen-like atom includes a positively charged core consisting of theatomic nucleus and anycore electrons, as well as a single valence electron. Because helium is common in the universe, the spectroscopy of singly ionized helium is important inEUV astronomy, for example, of DOwhite dwarf stars.
Other systems may also be referred to as "hydrogen-like atoms", such asmuonium (an electron orbiting anantimuon),positronium (an electron and apositron), certainexotic atoms (formed with other particles), orRydberg atoms (in which one electron is in such a high energy state that it sees the rest of the atom effectively as apoint charge).
Highly excited states of neutral atoms are well described in terms of one electron around a nucleus of a single positive charge resembling a hydrogen atom. These states are called Rydberg atoms. They have important applications inastrophysics, including in the dynamics of the primordial gas of theBig Bang.[5]
In the solution to the Schrödinger equation, which is non-relativistic, hydrogen-like atomic orbitals areeigenfunctions of the one-electronangular momentum operatorL (more precisely, its square,L2) and its z-componentLz. A hydrogen-like atomic orbital is uniquely identified by the values of theprincipal quantum numbern, theangular momentum quantum numberℓ, and themagnetic quantum numberm. The energy eigenvalues do not depend onℓ orm, but solely onn. To these must be added the two-valuedspin quantum numberms = ±1/2, setting the stage for theAufbau principle. This principle restricts the allowed values of the four quantum numbers inelectron configurations of more-electron atoms. In hydrogen-like atoms all degenerate orbitals of fixedn andℓ,m ands varying between certain values (see below) form anatomic shell.
The Schrödinger equation of atoms or ions with more than one electron has not been solved analytically, because of the computational difficulty imposed by the Coulomb interaction between the electrons. Numerical methods must be applied in order to obtain (approximate) wavefunctions or other properties from quantum mechanical calculations. Due to the spherical symmetry (of theHamiltonian), the total angular momentumJ of an atom is a conserved quantity. Many numerical procedures start from products of atomic orbitals that are eigenfunctions of the one-electron operatorsL andLz. The radial parts of these atomic orbitals are sometimes numerical tables or are sometimesSlater orbitals. Byangular momentum coupling many-electron eigenfunctions ofJ2 (and possiblyS2) are constructed.
In quantum chemical calculations hydrogen-like atomic orbitals cannot serve as an expansion basis, because they are not complete. The non-square-integrable continuum (E > 0) states must be included to obtain a complete set, i.e., to span all of one-electron Hilbert space. This was observed as early as 1928 by E. A. Hylleraas,[6] and later by Harrison Shull andPer-Olov Löwdin.[7]
In the simplest model, the atomic orbitals of hydrogen-like atoms/ions are solutions to theSchrödinger equation in a spherically symmetric potential. In this case, thepotential term is the potential given byCoulomb's law:where
After writing the wave function as a product of functions:(inspherical coordinates), whereYℓm arespherical harmonics, we arrive at the following Schrödinger equation:whereμ is, approximately, themass of theelectron (more accurately, it is thereduced mass of the system consisting of the electron and the nucleus), andħ is the reducedPlanck constant.
Different values ofℓ give solutions with differentangular momentum, whereℓ (a non-negative integer) is thequantum number of the orbitalangular momentum. Themagnetic quantum numberm (satisfying−ℓ ≤m ≤ℓ) is the (quantized) projection of the orbital angular momentum on the z-axis. SeeParticle in a spherically symmetric potential § Hydrogen-like atoms for the steps leading to the solution of this equation.
In addition toℓ andm, a third integern > 0, emerges from the boundary conditions placed onR. The functionsR andY that solve the equations above depend on the values of these integers, calledquantum numbers. It is customary to subscript the wave functions with the values of the quantum numbers they depend on. The final expression for the normalized wave function is:where:
parity due to angular wave function is(−1)ℓ.
The quantum numbersn,ℓ andm are integers and can have the following values:
For a group-theoretical interpretation of these quantum numbers, seeRunge–Lenz § Quantum mechanics of the hydrogen atom. Among other things, this article gives group-theoretical reasons whyℓ <n and−ℓ ≤m ≤ℓ.
Each atomic orbital is associated with anangular momentumL. It is avector operator, and the eigenvalues of its squareL2 ≡Lx2 +Ly2 +Lz2 are given by:
The projection of this vector onto an arbitrary direction isquantized. If the arbitrary direction is labelled 'z', the quantization is given by:wherem is restricted as described above. Note thatL2 andLz commute and have a common eigenstate, which is in accordance with Heisenberg'suncertainty principle. SinceLx andLy do not commute withLz, it is not possible to find a state that is an eigenstate of all three components simultaneously. Hence the values of the x- and y-components are not sharp, but are given by a probability function of finite width. The fact that the x- and y-components are not well-determined, implies that the direction of the angular momentum vector is not well determined either, although its component along the z-axis is sharp.
These relations do not give the total angular momentum of the electron. For that, electronspin must be included.
This quantization of angular momentum closely parallels that proposed byNiels Bohr (seeBohr model) in 1913, with no knowledge of wavefunctions.
In a real atom, thespin of a moving electron can interact with theelectric field of the nucleus through relativistic effects, a phenomenon known asspin–orbit interaction. When one takes this coupling into account, thespin and theorbital angular momentum are no longerconserved, which can be pictured by theelectronprecessing. Therefore, one has to replace the quantum numbersℓ,m and the projection of thespinms by quantum numbers that represent the total angular momentum (includingspin),j andmj, as well as thequantum number ofparity.
See thenext section on the Dirac equation for a solution that includes the coupling.
In 1928 in EnglandPaul Dirac foundan equation that was fully compatible withspecial relativity. The equation was solved for hydrogen-like atoms the same year (assuming a simple Coulomb potential around a point charge) by the GermanWalter Gordon. Instead of a single (possibly complex) function as in the Schrödinger equation, one must find four complex functions that make up abispinor. The first and second functions (or components of the spinor) correspond (in the usual basis) to spin "up" and spin "down" states, as do the third and fourth components.
The terms "spin up" and "spin down" are relative to a chosen direction, conventionally the z-direction. An electron may be in a superposition of spin up and spin down, which corresponds to the spin axis pointing in some other direction. The spin state may depend on location.
An electron in the vicinity of a nucleus necessarily has non-zero amplitudes for the third and fourth components. Far from the nucleus these may be small, but near the nucleus they become large.
Theeigenfunctions of theHamiltonian, which means functions with a definite energy (and which therefore do not evolve except for a phase shift), have energies characterized not by the quantum numbern only (as for the Schrödinger equation), but byn and a quantum numberj, thetotal angular momentum quantum number. The quantum numberj determines the sum of the squares of the three angular momenta to bej(j + 1) (timesħ2, seePlanck constant). These angular momenta include both orbital angular momentum (having to do with the angular dependence ofψ) and spin angular momentum (having to do with the spin state). The splitting of the energies of states of the sameprincipal quantum numbern due to differences inj is calledfine structure. The total angular momentum quantum numberj ranges from1/2 ton − 1/2.
The orbitals for a given state can be written using two radial functions and two angle functions. The radial functions depend on both the principal quantum numbern and an integerk, defined as:
whereℓ is theazimuthal quantum number that ranges from0 ton − 1. The angle functions depend onk and on a quantum numberm, which ranges from−j toj by steps of1. The states are labeled using the letters S, P, D, F etc. to stand for states withℓ equal to0,1,2,3, etc. (seeAzimuthal quantum number), with a subscript givingj. For instance, the states forn = 4 are given in the following table (these would be prefaced byn, for example 4S1/2):
| m = −7/2 | m = −5/2 | m = −3/2 | m = −1/2 | m = 1/2 | m = 3/2 | m = 5/2 | m = 7/2 | |
|---|---|---|---|---|---|---|---|---|
| k = 3,ℓ = 3 | F5/2 | F5/2 | F5/2 | F5/2 | F5/2 | F5/2 | ||
| k = 2,ℓ = 2 | D3/2 | D3/2 | D3/2 | D3/2 | ||||
| k = 1,ℓ = 1 | P1/2 | P1/2 | ||||||
| k = 0 | ||||||||
| k = −1,ℓ = 0 | S1/2 | S1/2 | ||||||
| k = −2, ℓ = 1 | P3/2 | P3/2 | P3/2 | P3/2 | ||||
| k = −3, ℓ = 2 | D5/2 | D5/2 | D5/2 | D5/2 | D5/2 | D5/2 | ||
| k = −4, ℓ = 3 | F7/2 | F7/2 | F7/2 | F7/2 | F7/2 | F7/2 | F7/2 | F7/2 |
These can be additionally labeled with a subscript givingm. There are2n2 states with principal quantum numbern,4j + 2 of them with any allowedj except the highest (j =n − 1/2) for which there are only2j + 1. Since the orbitals having given values ofn andj have the same energy according to the Dirac equation, they form abasis for the space of functions having that energy.
The energy, as a function ofn and|k| (equal toj + 1/2), is:
(The energy of course depends on the zero-point used.) Note that ifZ were able to be more than137 (higher than any known element) then we would have a negative value inside the square root for the S1/2 and P1/2 orbitals, which means they would not exist. The Schrödinger solution corresponds to replacing the inner bracket in the second expression by1. The accuracy of the energy difference between the lowest two hydrogen states calculated from the Schrödinger solution is about 9 ppm (90 μeV too low, out of around10 eV), whereas the accuracy of the Dirac equation for the same energy difference is about 3 ppm (too high). The Schrödinger solution always puts the states at slightly higher energies than the more accurate Dirac equation. The Dirac equation gives some levels of hydrogen quite accurately (for instance the 4P1/2 state is given an energy only about2×10−10 eV too high), others less so (for instance, the 2S1/2 level is about4×10−6 eV too low).[8] The modifications of the energy due to using the Dirac equation rather than the Schrödinger solution is of the order ofα2, and for this reasonα is called thefine-structure constant.
In the general case, the solution to the Dirac equation for quantum numbersn,k, andm, is:where the Ωs are columns of the twospherical harmonics functions shown to the right. signifies a spherical harmonic function:
in whichPb
a is anassociated Legendre polynomial. (Note that the definition ofΩ may involve a spherical harmonic that doesn't exist, likeY0,1, but the coefficient on it will be zero.)
Here is the behavior of some of these angular functions. The normalization factor is left out to simplify the expressions.
From these we see that in the S1/2 orbital (k = −1), the top two components ofΨ have zero orbital angular momentum like Schrödinger S orbitals, but the bottom two components are orbitals like the Schrödinger P orbitals. In the P1/2 solution (k = 1), the situation is reversed. In both cases, the spin of each component compensates for its orbital angular momentum around the z-axis to give the right value for the total angular momentum around the z-axis.
The twoΩ spinors obey the relationship:
To write the functionsgn,k(r) andfn,k(r) let us define a scaled radiusρ:
with
whereE is the energy (Enj) given above. We also defineγ as:
gn,k(r) andfn,k(r) are based on twogeneralized Laguerre polynomials of ordern − |k| − 1 andn − |k|:
whereA is a normalization constant involving thegamma function:
f is small compared tog (except at very smallr) because whenk is positive the first terms dominate, andα is big compared toγ −k, whereas whenk is negative the second terms dominate andα is small compared toγ −k. Note that the dominant term is quite similar to corresponding the Schrödinger solution – the upper index on the Laguerre polynomial is slightly less (2γ + 1 or2γ − 1 rather than2ℓ + 1, which is the nearest integer), as is the power ofρ (γ orγ − 1 instead ofℓ, the nearest integer). The exponential decay is slightly faster than in the Schrödinger solution.
The normalization factor makes the integral over all space of the square of the absolute value equal to1.
Whenk = −n (which corresponds to the highestj possible for a givenn, such as 1S1/2, 2P3/2, 3D5/2, ...), thengn,k(r) andfn,k(r) are:
WithA now reduced to:
Notice that because of the factorZα,f(r) is small compared tog(r). Also notice that in this case, the energy is given by
and the radial decay constantC by
Here is the 1S1/2 orbital, spin up, without normalization:
Note thatγ is a little less than1, so the top function is similar to an exponentially decreasing function ofr except that at very smallr it theoretically goes to infinity. But the value of therγ−1 surpasses 10 only at a value ofr smaller than101/(γ−1), which is very small (much less than the radius of a proton) unlessZ is very large.
The 1S1/2 orbital, spin down, without normalization, comes out as:
We can mix these in order to obtain orbitals with the spin oriented in some other direction, such as:
which corresponds to the spin and angular momentum axis pointing in the x-direction. Addingi times the "down" spin to the "up" spin gives an orbital oriented in the y-direction.
To give another example, the 2P1/2 orbital, spin up, is proportional to:
(Remember thatρ = 2rC.C is about half what it is for the 1S orbital, butγ is still the same.)
Notice that whenρ is small compared toα (orr is small compared toħc/(μc2)) the "S" type orbital dominates (the third component of the bispinor).
For the 2S1/2 spin up orbital, we have:
Now the first component is S-like and there is a radius nearρ = 2 where it goes to zero, whereas the bottom two-component part is P-like.
In addition to bound states, in which the energy is less than that of an electron infinitely separated from the nucleus, there are solutions to the Dirac equation at higher energy, corresponding to an unbound electron interacting with the nucleus. These solutions are not normalizable, but solutions can be found which tend toward zero asr goes to infinity (which is not possible when|E| <μc2 except at the above-mentioned bound-state values ofE). There are similar solutions withE < −μc2. These negative-energy solutions are just like positive-energy solutions having the opposite energy but for a case in which the nucleus repels the electron instead of attracting it, except that the solutions for the top two components switch places with those for the bottom two.
Negative-energy solutions to Dirac's equation exist even in the absence of a Coulomb force exerted by a nucleus. Dirac hypothesized that we can consider almost all of these states to be already filled. If one of these negative-energy states is not filled, this manifests itself as though there is an electron which isrepelled by a positively-charged nucleus. This prompted Dirac to hypothesize the existence of positively-charged electrons, and his prediction was confirmed with the discovery of thepositron.
The Dirac equation with a simple Coulomb potential generated by a point-like non-magnetic nucleus was not the last word, and its predictions differ from experimental results as mentioned earlier. More accurate results include theLamb shift (radiative corrections arising fromquantum electrodynamics[9]) andhyperfine structure.
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