measure for measure Simplify your life Within the Hartree atomic unit systems, the Schrödinger equation becomes parameter free. But there’s more to it than making a student’s life easier, as Gordon Drake and Eite Tiesinga recount.

ntroductory courses on quantum of the experiment in terms of the Hartree mechanics typically include the energy requires knowledge of many small Ithree-dimensional Schrödinger equation theoretical corrections to the lowest-order for the idealized hydrogen atom result from the Schrödinger equation. Most relevant here is that the root-mean-square ℏ2 e2 1 =2ψ r ψ r Eψ r charge radius of the proton leads to a small, � 2me ð Þ�4πε0 r ðÞ¼ ðÞ hard-to-pin-down energy correction. In fact, until recently, the entire field was to determine the allowed wavefunctions clouded by the ‘proton radius puzzle’: the ψ(r), where r is the separation between extremely accurate value obtained from the electron and proton, and the energies muonic hydrogen3 differed by several or eigenvalues E are given in the metric SI standard deviations from a less accurate (Système International) system of units. value from electronic transitions and Here ħ = h/(2π) is the reduced Planck’s electron–proton scattering. This puzzle constant, e is the elementary charge, me gave rise to an attempt to account for the is the electron mass and ε0 is the vacuum discrepancy that lasted over ten years and permittivity. For students, this equation has now been resolved in favour of the can be intimidating, and defining the muonic-hydrogen value4. welter of fundamental constants as In principle, this resolution opens well as giving their numerical values the way to a much more accurate is distracting, particularly, as some of determination of the Hartree energy. them are known to about ten significant Credit: Science & Society Picture Library / SSPL / However, as recently pointed out by digits while others, such as ħ, are exactly Getty Images Yerokhin et al.5, this will require the defined. The main goal of the exercise is theoretical evaluation of as-yet-unknown to teach how boundary conditions near two- and three-loop quantum the origin and at infinity determine the In a series of papers beginning in 19281, electrodynamic effects, because these are eigenvalues. Although the exact values of Hartree simplified the complex system now the main source of uncertainty, and the constants are not needed until the end of equations by introducing what have such evaluations are truly challenging. of the calculation, it is hard to even make now become known as ‘atomic units’. This Results from upcoming searches for new an order-of-magnitude estimation. system of units boldly cuts through the physics beyond the standard model and In the earliest days of quantum numerical complexities of the fundamental variations of the fundamental constants mechanics, these same problems must have constants by defining a length scale over cosmological time periods might plagued Douglas Hartree (pictured) as he a0 (the Bohr radius) and energy scale make these evaluations even more 2 2 struggled with the complexities of solving Eh ℏ =mea0 (the Hartree energy). With important. It is in the tiny deviations ¼ 2 2 the Schrödinger equation for many-electron someI algebra, we get a0 = 4πε0ħ /mee from conventional theory that the first atoms and molecules. This brought Hartree = 5.29177210903(80) × 10−11 m (ref. 2), hints of new physics may appear. ❐ face to face with the many-body problem in where the number in parenthesis is its quantum mechanics. Just as in the classical one-standard-deviation uncertainty. Gordon W. F. Drake1 ✉ and Eite Tiesinga2 ✉ equivalent, the equations of motion for an If we agree to measure energy in units 1University of Windsor, Windsor, Ontario, Canada. atom with two or more electrons cannot be of the Hartree energy and distance in units 2Joint Quantum Institute, National Institute of solved exactly, and so approximations must of the Bohr radius, then the Schrödinger Standards and Technology and the University of be used. Hartree’s idea was to assume that equation for hydrogen assumes a simple Maryland, Gaithersburg, MD, USA. each electron moves in the average field parameter-free form and we can just ✉e-mail: [email protected]; [email protected] produced by all the others. The resulting perform the calculations in atomic units, equations are now known as the Hartree and multiply the resulting energies by Published online: 2 December 2020 equations, and, if the indistinguishability the currently most accurate value of Eh, https://doi.org/10.1038/s41567-020-01095-x of the electrons is taken into account, which is 27.211386245988(53) eV (ref. 2). the Hartree–Fock equations after the The technique used to determine this References Russian physicist Vladimir Aleksandrovich value takes us to the forefront of modern 1. Hartree, D. R. Math. Proc. Camb. Phil. Soc. 24, 89110 (1928). 2. Tiesinga, E., Mohr, P. J., Newell, D. B. & Taylor, B. N. Values of Fock. Today, these equations and their physics. The numerical value of the Hartree fundamental physical constants. NIST https://physics.nist.gov/ extensions are widely used throughout the energy, one of the most precisely known cuu/Constants/index.html (2019). atomic physics and quantum chemistry physical constants, is set by high-precision 3. Pohl, R. et al. Nature 466, 213–216 (2010). 4. Bezginov, N. et al. Science 365, 1007–1012 (2019). communities to predict properties of measurements of the 1s–2s transition energy 5. Yerokhin, V. A., Pachucki, K. & Patkóš, V. Ann. Phys. 531, atoms and molecules. in atomic hydrogen. The interpretation 1800324 (2019).

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