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The Dirac Equation in Low Energy Condensed Matter Physics

Authors:
PoS(ICHEP2016)346
The Dirac Equation in Low Energy Condensed
Matter Physics
John Swain
Dept. of Physics, Northeastern University, Boston, MA 02115
E-mail: john.swain@cern.ch
Allan Widom
Dept. of Physics, Northeastern University, Boston, MA 02115
E-mail: allan_widom@yahoo.com
Yogendra Srivastava
Department of Physics and Geology, University of Perugia, Perugia, Italy
Department of Physics, Northeastern University, Boston, MA 02115 USA
E-mail: yogendra.srivastava@gmail.com
Electrons in low energy condensed matter physics are typically treated using the non-relativistic
Schrödinger or Pauli equations, with relativistic effects included, if at all, via corrections of order
v/c. We show that using the full Dirac equation with 4-component spinors leads to a number
of important qualitative effects and, perhaps surprisingly, some striking simplifications over the
conventional lower energy treatments, already for nuclei with Z >2.
38th International Conference on High Energy Physics
3-10 August 2016
Chicago, USA
Speaker.
J. S. thanks the National Science Foundation for support via NSF grant PHY-1205845.
c
Copyright owned by the author(s) under the terms of the Creative Commons
Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/
PoS(ICHEP2016)346
The Dirac Equation in Low Energy Condensed Matter Physics John Swain
1. Introduction
The idea that the Schrödinger equation gives an adequate and indeed correct description of
chemistry, from the periodic table to chemical bonding is rarely ever called into question. Dirac
had famously quipped that the laws of physics as then understood could explain “most of physics
and all of chemistry”, with, at the time, chemistry being understood through the (non-relativistic)
Schrödinger equation despite Dirac having discovered the correct (relativistic) equation which de-
scribes the electron. Richard Feynman, however, almost as famously quipped “There’s a reason
physicists are so successful with what they do, and that is they study the hydrogen atom and the
helium ion and then they stop.
How well does the Schrödinger equation really describe the quantum mechanics of electrons
in chemistry and, by extension, in condensed matter physics? In this paper we briefly review
some history, starting with the original relativistic wave equation which Schrödinger had found,
now called the Klein-Gordon equation, which was replaced by the non-relativistic Schrödinger
equation, whose defects in matching experiment were then fixed, essentially by hand to mimic
what one would get from a more correct treatment via the relativistic Dirac equation.
We then briefly review work done in “relativistic quantum chemistry” to incorporate correc-
tions due to relativistic effects, noting the generally ignored but crucially important difference that
comes about with relativity, which is that the labeling of states is radically different.
We then point out that, even when relativistic effects on energy levels are small, the change
in angular distributions of electron orbitals due to the different labeling of states, and in particular,
the fact that ~
Land ~
Sare not separately conserved, is significant and has been worked around by
ad-hoc means. Despite the additional complications involved in using a 4-component Dirac spinor
in place of a single component Schrödinger wave function, certain important features of the angular
distribution of electrons in atoms are almost obvious, and do not require the ad-hoc additions to the
Schrödinger equation picture which normally must be made.
2. The Quantum Mechanical Treatment of the Electron
The Klein-Gordon equation, named for Oskar Klein and Walter Gordon in 1926 was proposed,
along with many others, as possible wave equation for the electron. The equation apparently ap-
pears in Schrödinger’s notebooks from late 1925, but it got the fine structure of the hydrogen
atom wrong [1] and of course was plagued by negative energies. In 1926 Schrödinger submitted
for publication[2] the nonrelativistic equation which bears his name, which reproduced the Bohr
model energy levels, but without fine structure.
The orbitals, labelled by a radial quantum number n, orbital ones `and m, and one representing
spin sare well-known even to high school students, and the entire structure of the periodic table is
based on the extension of these “hydrogenic” wave functions. The energies depend only on nare
given by the well-known[3] expression
En=mec2α2
2n2(2.1)
with L2=~
L·~
L=`(`+1)¯
h2,mrunning from `to `in integer steps, and the spin s=±1
2¯
h. Here me
is the (reduced, to account for finite proton mass) electron mass, cthe speed of light, and α=e2
¯
hc
1
PoS(ICHEP2016)346
The Dirac Equation in Low Energy Condensed Matter Physics John Swain
the fine structure constant. For atoms beyond hydrogen, with nuclei of charge Z, and only one
electron, one replaces αwith Zα.
With more than one electron (and ignoring the complications of electron-electron interactions)
a series of ad-hoc principles are applied[4]. First of all, a rule is invoked that only two electrons
will be placed in a given wavefunction labelled by n,`,m. This is done via an appeal to the Pauli
exclusion principle which cannot be derived from the Schrödinger equation, and indeed depends
crucially on the electron having spin-1/2 (and, in fact, really requires a quantum field theoretic
treatment[1]). This leads to a picture of “shells” which are filled in an order given by the n+`rule,
also known as the Madelung rule (after Erwin Madelung), or the Janet rule or the Klechkowsky rule
(after Charles Janet or Vsevolod Klechkovsky, in some countries). Electrons in higher shells feel
an effective screened nuclear charge Ze f f due to electrons in lower shells, which is determined via
Slater’s rules Zef f =Zσfor suitable σ. These σwere determined semi-empirically by Slater[5]
in 1930 and revised based on Hartree-Fock calculations by Clementi et al.[6] in the 1960’s.
The essential point to note is that everything in chemistry and condensed matter is, in almost
all cases, based on electron wavefunctions modeled on what is obtained by the non-relativistic
Schrödinger equation for one electron and a charge Z(or, rather, Ze f f )nucleus.
3. The Dirac Equation
The Dirac equation, which everyone would agree is actually the correct equation to describe
electrons, is, unlike the Schrödinger equation, relativistic, which implies several major departures
from the non-relativistic wavefunction description. First of all, it requires a 4-component wave
function, the Dirac spinor, in order to describe electrons. The spin quantum number and its as-
sociation with angular momentum of 1
2¯
harises naturally, as does, via a quantum field theoretic
treatment of the equation, the Pauli exclusion principle[1]. A key point to which we will return
shortly is that the (non-relativistic) quantum numbers `and sno longer make sense, as neither the
orbital angular momentum ~
Lnor the spin angular momentum ~
Sare separately conserved. That is,
for the Dirac Hamiltonian HD, neither [HD,~
L]nor [HD,~
S]vanish. However, the quantity ~
J=~
L+~
S
is conserved, and the value jdetermined by ~
J·~
J=j(j+1)is a good quantum number. Energy
levels now depend not just on a radial quantum number nbut on jas well, and
En j =mec2
1
1+
Zα
nj1
2+q(j+1
2)2Z2α2
2
1
2
(3.1)
and it is not hard to see that for small Zthis gives much the same result as the Schrödinger equation,
with no dependence on j.
While this expression is often well-approximated by the Schrödinger equation values, this
is not always the case, and the corresponding radial wavefunctions can, for large enough Z, be
affected significantly enough to have important quantitative and even qualitative effects. This leads
to the subject of relativistic quantum chemistry which we touch on in the next section before coming
to the central issue of this work which is the effect on the angular part of the wavefunction, which
will turn out to be, though often ignored, very important to treat relativistically, even for low Z.
2
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The Dirac Equation in Low Energy Condensed Matter Physics John Swain
4. Relativistic Effects in Chemistry as Usually Treated
The study of relativistic effects in chemistry is a well-advanced one (see, for example, the
review by Pyykkö [7] or books such as[8,9]). Relatively well-known examples of relativistic
effects in chemistry are the nobility, trivalency[10] and yellow color of gold[11,12], attributed to
the contraction of the 6s1orbital. In fact, with the single outer 6s electron, one might expect gold
to behave more like an alkali metal rather than being the corrosion resistant metal we know it to be.
The intuitive explanation often given for this is that the s-wave electron wave functions are more
localized near the nucleus and thus feel a less screened charge than those in higher orbital angular
momentum states and are more susceptible to relativistic effects.
McKelvey[13], writing for students, makes the simplistic and yet arguably charming argument
that one can associate a speed v= (Zα)cby writing Bohr’s quantization rule as mvr =n¯
hand
inserting the Bohr radius for r. Then for gold, with Z=80, a chemist can argue for a “relativistic
mass” m=me/q1(v
c)2of about 1.2mewhich inserted into the Bohr formula implies a roughly
20% contraction of the 1s electron orbitals, as well as the other s orbitals since, citing Pyykko and
Desclaux [11] “all the others shells, up to the valence shell, contract roughly as much because their
electron speeds near the nucleus are comparable and the contraction of the inner part of the wave
function pulls in the outer tails.”. The low melting point of mercury[14] (also related to the fact that
it is monatomic in the gas phase) is similarly explained by the contraction of the 6s2orbital so that it
is mainly van der Waals forces between atoms. Rather remarkably, it has been recently shown[15]
that for lead-acid batteries, over 1.7 V of the 2.1 V per cell comes from relativistic effects!
The rigorous approaches to relativistic quantum chemistry as are discussed in references such
as [7,8,9] all share one important feature: while they include relativistic corrections to the non-
relativistic Schr¨
dinger equation in the form of additional spin and momentum dependent terms go-
ing to higher order in v/c, they invariably go to 2-component spinors and the usual (non-relativistic,
we hasten to add) quantum numbers for orbital and spin angular momentum.
That this is a potentially dangerous thing to do is perhaps made most clear by the tendency to
express the 4-component spinor in terms of “large” and “small” components, retaining the “large”
ones and neglecting the “small” ones [3]. This gets one to the Schrödinger equation, now with a
2-component Pauli spinor which contains the spin information. However, the Dirac matrices can be
chosen in any representation, and the distinction between “large” and “small” is, in fact, dependent
on the representation. For example, in the chiral representation, there is no longer such a clear
separation into “large” and “small”. Even in the usual representation, the “small” components are
no longer “small” as Zbecomes non-negligible compared to 1/α137.
In fact, as is well-known [3], the correct (Dirac) hydrogenic wavefunctions, labelled by j, con-
tain spherical harmonics Y`mwith `=j±1
2, each of the two `values appearing with a different
spin. This simply indicates, as noted earlier, that `and sare not good quantum numbers and should
not be used to label states. The fact that reasonable energy eigenvalues for the Hamiltonian can
be achieved with wavefunctions with incorrect angular properties is perhaps not entirely surpris-
ing since the potential is spherically symmetric. The angular dependence of the wave functions,
however, is not at all well-described by non-relativistic wavefunctions.
To the best of our knowledge, the simple fact that there are large qualitative differences in
using the full relativistic treatment of angular momentum quite independent of any relativistic
3
PoS(ICHEP2016)346
The Dirac Equation in Low Energy Condensed Matter Physics John Swain
effects on energy levels and the radial part of the wavefunction (such as those described above)
has not been explicitly stated in the literature. It is to this we now turn.
5. Relativistic Quantum Numbers and Angular Wavefunctions
As noted above, the energy levels En j for the relativistic (Dirac) electron depend on a radial
quantum number nand an angular one j. The correct quantum numbers [3] are n,jand the parity
Pof the state. For a high energy physicist this is immediately clear since one is accustomed to
particles being labelled by, mass and so-called (colloquially) “spin-parity” or JP.
We have already reviewed some of the ad-hoc additions needed to make the non-relativistic
Schrödinger equation, which really describes spin-0 particles, work for atomic physics. Now we
describe another ad-hoc procedure, which is of enormous import, which also finds its honest origin
in the use of the Dirac equation.
One of the most important atoms in the periodic table is undoubtedly carbon a key building
block for all life and perhaps the most important element in chemistry, with the entire subfield of
organic chemistry devoted to it. This all arises from its ability to form 4 bonds with other atoms,
exemplified in the tetrahedral structure of methane, or the carbon bonding in diamond.
Carbon has Z=6. The usual description of its orbital structure is 1s22s22p2meaning a filled
n=1, `=0 s-shell with two electrons, spin up and spin down, and for n=2, a similarly filled
spherical 2s orbital and 1 electron in each of 2 of the three roughly dumbbell-shaped p-orbitals,
oriented along three perpendicular axes. However, in reality, this is not what is seen. Rather, one
finds 4 orbitals with tetrahedral symmetry. This conundrum is resolved by a clever but ad-hoc
procedure, invented[16] by Linus Pauling in 1931, called hybridization. The idea is to form a linear
combination of one s and three p orbitals to obtain agreement with experiment. This, however, is
something which must be added by hand to the Schrödinger description to accommodate data.
Consider now the Dirac description. Despite the apparent complications of the Dirac equation,
relativity, and 4-component spinors, one need only look at jto understand this. The lowest possible
value of jis 1/2 and corresponds to the non-relativistic `=0 state. There are 2 j+1 or 2 electrons
which can go in, one spin-up and one spin-down. The next value of jis 3/2 and we have 2 j+1=4
states, all of the same energy. By symmetry they will be tetrahedrally oriented. Thought of non-
relativistically, they are made of “mixed” or “hybridized” `values, but this in fact need not be done
by hand. Rather, one now has a correct prediction instead of a problem with incorrect angular
distributions that must be fixed by hand to get the right result.
This is a simple observation, but note that already for any Z>2 (where in the non-relativistic
picture, ` > 0) we can expect significant discrepancies with the non-relativistic spin-0 description,
supporting quite clearly Feynman’s comment about some of the apparent “success” he describes
being indeed due to stopping at hydrogen and helium.
6. Further Implications
Now that we have made the simple observation that the Dirac equation should be used to
describe the electron with orbital and spin angular momentum now replaced by jand parity, this
has many far-reaching implications. First of all, one should be careful with any work referring
4
PoS(ICHEP2016)346
The Dirac Equation in Low Energy Condensed Matter Physics John Swain
to “s-wave”, “p-wave”, “d-wave”, etc. in chemistry or condensed matter physics which involves
any angular dependence (even if correct angular distributions can be forced by ad-hoc means).
The same sort of consideration applies to nuclear physics and in fact even to hadron spectroscopy,
which notwithstanding the observations about JPmade above, often calculate in terms of Land S.
An additional upside is that typically large
~
L·~
S couplings now arise naturally whenever there is
aJ2involved. In a sense, part of the usual non-relativistic “Clebsching” of orbital and spin angular
momenta is done automatically by the Dirac equation.
It is perhaps surprising that such an elementary observation with such important consequences
has not been made before, but tradition and custom are powerful forces. An additional simple
observation may help to make this all seem obvious (in hindsight): nobody would be surprised to
find the angular dependence of scattering off a proton to be different for a spinless particle and
a spin-1/2 particle. Given the relationship between poles in the scattering amplitude and bound
states, it is then perhaps not surprising that angular distributions of bound-state wavefunctions
would depend on the spin of what is being bound.
References
[1] C. Itzykson and J.-B. Zuber, “Quantum Field Theory”, McGraw-Hill Co. (1985).
[2] E. Schrödinger, Ann. Physik (4) 79 361 (1925); 79 489 (1925); 80 437 (1926); 81 109 (1926).
[3] A. Messiah, “Quantum Mechanics”, Dunod, Paris (1995).
[4] See, for example, Gary L. Miessler, Donald A. Tarr, “Inorganic Chemistry”, Prentice Hall (2003).
[5] J. C. Slater, Phys. Rev. 36 (1) 57 (1930).
[6] Clementi, E.; Raimondi, D. L. (1963) J. Chem. Phys. 38 (11) 2686 (1963); Clementi, E.; Raimondi,
D. L.; Reinhardt, W. P. (1967) Journal of Chemical Physics 47 (4): 1300 (1967).
[7] P. Pyykkö, Chemical Reviews,112 371 (2012).
[8] K. G. Dyall and K. Fægri, Jr., “Relativistic Quantum Chemistry”, Oxford University Press, New York
(2007).
[9] K. Balasubramanian, “Relativistic Effects in Chemistry Part A”, John Wiley & Sons, New York
(1997); “Relativistic Effects in Chemistry Part B”, John Wiley & Sons, New York (1997).
[10] P. J. Schwerdtfeger, Am.Chem.Soc. 111 7261 (1989).
[11] P. Pyykkö and J. P. Desclaux, Acc. Chem. Res. 12 276 (1979)
[12] P. Romaniello and P. L. de Boeij, J. Chem. Phys. 164303 122 (2005); J. Chem. Phys. 174111 127
(2007).
[13] D. R. McKelvey, J. Chem. Educ. 60 112 (1982).
[14] P. Pyykkö, Adv. Quantum Chem. 11 353 (1978).
[15] R. Ahuja et al.,Phys. Rev. Lett. 106 018301 (2011).
[16] L. Pauling, Journal of the American Chemical Society 53 (4): 1367 (1931).
5
... This construction inherently necessitates the use of relativistic Dirac fermions. The need for Dirac fermions in chemistry and condensed matter has previously been reported in Ref. [27]. The requirement of relativistic 4-component Dirac spinors in the description of condensed matter systems is relevant, for example, in heavy nucleon compounds, where Z 1. ...
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  • J C Slater
J. C. Slater, Phys. Rev. 36 (1) 57 (1930).
  • E Clementi
  • D L Raimondi
  • W P Reinhardt
Clementi, E.; Raimondi, D. L.; Reinhardt, W. P. (1967) Journal of Chemical Physics 47 (4): 1300 (1967).
  • P Pyykkö
P. Pyykkö, Chemical Reviews, 112 371 (2012).
  • P J Schwerdtfeger
P. J. Schwerdtfeger, Am.Chem.Soc. 111 7261 (1989).
  • P Pyykkö
  • J P Desclaux
P. Pyykkö and J. P. Desclaux, Acc. Chem. Res. 12 276 (1979)
  • P Romaniello
  • P L De Boeij
P. Romaniello and P. L. de Boeij, J. Chem. Phys. 164303 122 (2005);