Appendix C: Physical Constants and Unit Conversions

This appendix collects the physical constants, unit systems, and conversion factors most frequently needed in quantum mechanics. All values are from the 2018 CODATA adjustment unless otherwise noted. We quote more digits than you will typically need so that intermediate calculations do not accumulate rounding errors.

C.1 Fundamental Constants

C.2 Atomic and Quantum Constants

C.3 Energy Conversion Factors

Quantum mechanics problems express energies in a bewildering variety of units. The table below provides the conversion factors between the most common ones.

Practical rules of thumb:

• 1 eV $\approx$ 11,600 K (thermal energy at "1 eV temperature")
• 1 eV $\approx$ 8066 cm$^{-1}$ (spectroscopic wavenumber)
• Room temperature ($T = 300$ K) $\approx$ 1/40 eV $\approx$ 25.9 meV
• Visible light: 1.65 eV (red) to 3.1 eV (violet)
• Hydrogen ground state: $-13.6$ eV $= -1$ Rydberg $= -0.5$ Hartree

C.4 Natural Units ($\hbar = c = 1$)

In particle physics and relativistic quantum mechanics (Chapters 36-38), it is conventional to set $\hbar = 1$ and $c = 1$. This reduces all physical quantities to powers of a single dimension, usually chosen as energy (eV or GeV).

Conversion to SI:

Useful numerical values in natural units:

$$ \hbar c = 197.327 \text{ MeV fm} = 0.197327 \text{ GeV fm} $$

$$ (\hbar c)^2 = 0.389379 \text{ GeV}^2 \text{ mb} \quad (\text{where 1 mb} = 10^{-31} \text{ m}^2) $$

$$ m_e c^2 = 0.51100 \text{ MeV}, \quad m_p c^2 = 938.272 \text{ MeV} $$

C.5 Atomic Units

Atomic units are the natural unit system for non-relativistic quantum mechanics and are used extensively in computational chemistry and in this textbook's perturbation-theory chapters. In atomic units:

$$ \hbar = m_e = e = 4\pi\varepsilon_0 = 1 $$

The hydrogen atom in atomic units:

The Hamiltonian is simply:

$$ \hat{H} = -\frac{1}{2}\nabla^2 - \frac{1}{r} $$

and the ground-state energy is $E_1 = -1/2$ (in Hartrees). This is why atomic units are so popular: factors of $\hbar$, $m_e$, $e$, and $4\pi\varepsilon_0$ simply disappear.

C.6 SI vs. Gaussian (CGS) Unit Conventions

Some classic quantum mechanics textbooks (notably Jackson for electromagnetism, and Sakurai in places) use Gaussian units. The key differences affecting quantum mechanics are in Coulomb's law and the vector potential.

Translation rule: To convert a Gaussian formula to SI, replace every occurrence of $e^2$ with $e^2/(4\pi\varepsilon_0)$. This textbook uses SI throughout except where explicitly noted, but you will encounter Gaussian units in Griffiths, Sakurai, and Weinberg.

C.7 Useful Numerical Combinations

These combinations appear so frequently in problems that it is worth memorizing them or keeping this page bookmarked.

C.8 The constants.py Module

All constants in this appendix are available programmatically:

from qm_toolkit.constants import (
    hbar, h, c, e_charge, m_e, m_p, m_n,
    k_B, N_A, epsilon_0, mu_0,
    alpha, a_0, R_inf, E_Rydberg, E_Hartree,
    mu_B, mu_N, g_e, g_p,
    lambda_C,
    eV_to_J, J_to_eV, cm_inv_to_eV, K_to_eV,
)

# Example
print(f"Bohr radius = {a_0:.6e} m")
print(f"Ground state H = {-E_Rydberg:.4f} eV")

The module also provides a convert(value, from_unit, to_unit) function:

from qm_toolkit.constants import convert

energy_eV = convert(500, "nm", "eV")     # photon wavelength to energy
print(f"500 nm photon = {energy_eV:.4f} eV")  # 2.4797 eV

When in doubt about which value to use in a homework problem, use the values in this appendix. They match the constants.py module, so your analytical and numerical answers will agree. For higher precision or for constants not listed here, consult the NIST CODATA database at physics.nist.gov/constants.