1 SI Units

We will often use SI units. SI units are easy for the human world, but inconvenient for many atomic quantities.

1.1 Redeﬁnition Some of the SI units were based on artifacts, especially the kilogram. These artifacts were imperfect, so we wanted a better system. The electron charge deﬁnes the Coulomb, the speed of light deﬁnes meters per second, time is deﬁned in terms of the frequency of a hyperﬁne transition in cesium, Planck’s contant (along with the other factors) deﬁnes the kilogram, Boltzmann’s constant deﬁnes the Kelvin, Avagadro’s number deﬁnes moles, and kcandela deﬁnes candelas and lumens.

2 Atomic Units

Atomic units are a bit more convenient. They are determined by the electron charge e, the reduced Planck constant ~, and the electron mass me. In atomic units, they are each deﬁned to be 1.

2.1 Derived Quantities Length is deﬁned in terms of the Bohr radius. In SI units,

2 ~ ˚ a0 = [4π0] 2 ≈ 0.53A mee

We set 4π0 = 1 in atomic units, and ~ = me = e = 1, so a0 = 1. The Hartree H will have a value of 1 for similar reasons. The atomic deﬁnition of time is ~/H ≈ 2.4 × 10−17 s. Velocity is deﬁned by 1 e2 ≈ 2.2 × 106 m/s 4π0 ~ The fundamental magnetic dipole moment is the Bohr magneton:

~e MHz µB = ≈ h × 1.4 2me G This has a value of 1/2 in atomic units. The unit of electric ﬁeld is 1 e ≈ 5.1 × 109 V/cm 4π0 a0 It has a value of 1 in atomic units.

1 2.2 Alpha Hierarchies Not all numbers in atomic units are 1 (or 1/2). Our velocity unit can be written as

1 e2 e2 1 = = c = cα 4π0 ~ 4π0~c The speed of light in atomic units is thus α−1. This implies that relativity is relevant as a 1/α correction. αa0 is the reduced Compton wavelength, the wavelength of a particle with the rest mass of an electron. ~c a0/α = H ≈ 7.2 nm is the wavelength of light absorbed or emitted in the ground state of an 1 atom. This is about α times the size of the atom. The magnetic ﬁeld in an atom is 1 E v ≈ 2.3 × 105 T c2 atom atom This is α2 in atomic units.

2.3 Other Dimensionless Quantities me ≈ 1 also shows up occasionally. mp 1800

3 Units of Energy

In SI units, energy is measured in Joules. However, the most convenient unit of energy may depend on the instruments. For an instrument that accelerates electrons with an electric ﬁeld (like an electron microscope), eV are much easier to use. Energy can be discussed in terms of frequency, since E = hν. Thus, we can measure energy in Hertz. In a spectrometer, the energy of light depends on the wavelength. Using that 1 = ν , we can λvac c describe energy in cm−1. Sometimes we measure temperature. We can use that E = kBT to measure energy in Kelvin.

3.1 Common Rules 1 mK ∼ 20.8 MHz or 1 µK ∼ 20.8 kHz 1 cm−1 ∼ 30 GHz 1 300 K ∼ eV 40 1 1 eV ∼ 8066 cm−1 = 1.24 µm

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