The Klein-Gordon Equation

In 1926 the young quantum theory hit a wall. Schrödinger's equation, barely months old, was already the workhorse of atomic physics — but it was built on the nonrelativistic energy $E=p^2/2m$ , and it singled out time in a way that no relativistic law is allowed to. What happens if you quantize the relativistic energy relation $E^2=p^2c^2+m^2c^4$ instead? The answer is the Klein–Gordon equation, the first relativistic wave equation and the doorway to every field theory that followed. The move is deceptively small: because energy enters the relativistic relation squared, the equation you get is second order in time, matching its second order in space. That single feature makes it manifestly Lorentz covariant — but it also breaks the probability interpretation that made Schrödinger's equation so intuitive. Both signs of the energy survive, and the natural "density" you build from the equation can go negative. Schrödinger found the equation first, saw it predict the wrong hydrogen fine structure, and quietly set it aside. You are ready for this because you already command the nonrelativistic Schrödinger equation, four-vectors, and plane-wave analysis. Here we assemble them into the relativistic scalar field. We will pin the metric and unit conventions used across this whole chapter, derive the equation from the dispersion relation, extract its two energy branches, and prove that it contains ordinary quantum mechanics as a low-energy limit. The diseases it exposes — negative energies, indefinite density — are not failures but signposts, and the next few articles turn each one into physics.

Key takeaways

The Klein–Gordon equation $(\Box+m^2)\phi=0$ comes from promoting $E^2=\mathbf p^2c^2+m^2c^4$ to operators with $E\to i\hbar\partial_t$ , $\mathbf p\to-i\hbar\nabla$ .
It is second order in both time and space — that symmetry is exactly what makes it Lorentz covariant, and it describes spin-0 (scalar) particles like the pion and the Higgs boson.
Plane waves require $p^0=\pm\sqrt{\mathbf p^2c^2+m^2c^4}/c$ : two energy branches appear, unavoidably.
Being second order in time means $\phi$ and $\dot\phi$ are independent initial data — so no positive single-particle probability density exists (unlike Schrödinger's $|\psi|^2$ ).
It correctly reduces to the free Schrödinger equation in the nonrelativistic limit, so it does not throw ordinary quantum mechanics away.
KEY IDEA: quantizing the relativistic $E^2=p^2c^2+m^2c^4$ yields a manifestly covariant, second-order wave equation for spin-0 fields whose two energy branches and indefinite density force it to be read as a field equation, not a one-particle Schrödinger equation.

The core insight

Quantizing $E^2=p^2c^2+m^2c^4$ gives a manifestly Lorentz-covariant, second-order wave equation $(\Box+m^2)\phi=0$ for spin-0 fields — but its second-order-in-time character forbids a positive single-particle probability density, so it must be read as a field equation, not a one-particle Schrödinger equation.

Where this lives today

The Klein–Gordon equation is no historical curiosity: it is the free field equation of every fundamental spin-0 particle we know. The Higgs field obeys it with a self-interaction term added, and its excitation — the Higgs boson found in 2012 — is a Klein–Gordon quantum whose mass fixes the propagator pole. The pions and kaons that bind and shuttle inside nuclei are pseudoscalar Klein–Gordon fields, and the operator $\Box+m^2$ still writes the propagator of every massive scalar exchange in a Feynman diagram. Its reach also runs sideways into analog and condensed-matter physics. Relativistic-like dispersion $\omega=\sqrt{k^2+m^2}$ appears for gapped excitations in many lattice systems, and the Klein paradox — the equation's most notorious prediction — was observed directly in graphene, where charge carriers obey a massless relativistic wave equation and tunnel through barriers with perfect transmission. Wherever a "mass gap" opens in a relativistic spectrum, the Klein–Gordon operator is quietly at work.

The physical picture

Start from the shape of the two energy laws. Nonrelativistically, $E=p^2/2m$ is linear in $E$ , so quantization gives one time derivative and Schrödinger's first-order equation. Relativistically, $E^2=p^2c^2+m^2c^4$ is quadratic in $E$ , so quantization gives two time derivatives. That is the whole story in one line: relativity forces time and space to enter the equation the same way, and the price of that symmetry is a second-order equation.

Figure

Two energy laws, two equations

Think of the operator $\Box+m^2$ (natural units) as a relativistic Laplacian with a mass built in. In momentum space it is simply $m^2-p^2$ , so requiring $(\Box+m^2)\phi=0$ is requiring every Fourier mode to sit on the mass shell $p^2=m^2$ . The mass sets an intrinsic scale; switch it off and you recover the ordinary wave equation $\Box\phi=0$ with everything moving at the speed of light. Because $E$ appears squared, both signs of the energy solve the equation, and you cannot simply cross one out — a complete set of modes needs both branches, and a Lorentz boost mixes them anyway. The static face of the same operator is just as telling: freeze the time dependence and $(\Box+m^2)\phi=0$ becomes $(-\nabla^2+m^2)\phi=0$ , whose point-source solution is the Yukawa potential $e^{-mr}/r$ . The mass term is literally the inverse range of the force a scalar mediates.

Figure

The two energy branches and the mass gap

The core equations

The starting point is the relativistic dispersion relation itself — the on-shell condition that any free relativistic particle must obey. Everything below is the operator version of this one algebraic statement.

E^2=\mathbf p^2c^2+m^2c^4 \qquad (1)

where: $E$ is total relativistic energy; $\mathbf p$ the three-momentum; $m$ the rest mass; $c$ the speed of light. Read it as a constraint: for a given momentum the energy is fixed up to a sign, and that sign ambiguity is the seed of everything that follows. A note on conventions, fixed once for this whole chapter. We use the mostly-minus metric $g_{\mu\nu}=\mathrm{diag}(+1,-1,-1,-1)$ , so $p\cdot x=Et-\mathbf p\cdot\mathbf x$ and $p^2=E^2-\mathbf p^2c^2=m^2c^4/c^2$ on shell. For compactness we work in natural units $\hbar=c=1$ , in which mass, momentum, and energy all carry dimensions of inverse length; a factor-restoring sidebar appears whenever a real number is needed. In this convention the d'Alembertian is $\Box=\partial_\mu\partial^\mu=\partial_t^2-\nabla^2$ . Now promote (1) to an operator equation with the same canonical prescription that produced Schrödinger's equation, applied to a field $\phi$ .

\left(-\hbar^2\partial_t^2+\hbar^2c^2\nabla^2-m^2c^4\right)\phi=0 \qquad (2)

where: $\phi(\mathbf x,t)$ is the (complex) scalar field; $\partial_t^2$ the second time derivative; $\nabla^2$ the spatial Laplacian. This is the Klein–Gordon equation in SI-like form; the two time derivatives sit symmetrically with the two space derivatives.

Derivation AFrom the dispersion relation to the Klein–Gordon equation

We want to turn the algebraic relation $E^2=\mathbf p^2c^2+m^2c^4$ into a differential equation for a field, using nothing more than the canonical substitutions that already work for Schrödinger's equation. The strategy is to substitute the operators, square them, and collect terms; the one thing to watch is that squaring the operators is unambiguous only because $\partial_t$ and $\nabla$ commute with themselves. The payoff is a manifestly covariant equation.

Step 1. Adopt the canonical operator substitutions

E\to i\hbar\partial_t,\qquad \mathbf p\to-i\hbar\nabla,

the identical prescription that turns $E=p^2/2m$ into $i\hbar\partial_t\psi=-(\hbar^2/2m)\nabla^2\psi$ . Why these and not others: they reproduce plane waves $e^{-i(Et-\mathbf p\cdot\mathbf x)/\hbar}$ as eigenfunctions with eigenvalues $E$ and $\mathbf p$ , which is exactly what a momentum-space wave should do.

Step 2. Insert into (1) acting on $\phi$ :

(i\hbar\partial_t)^2\phi=\big[(-i\hbar\nabla)^2c^2+m^2c^4\big]\phi.

Using $(i\hbar\partial_t)^2=-\hbar^2\partial_t^2$ and $(-i\hbar\nabla)^2=-\hbar^2\nabla^2$ gives

-\hbar^2\partial_t^2\phi=-\hbar^2c^2\nabla^2\phi+m^2c^4\phi.

Step 3. Move everything to one side to reach the boxed form (2). Dividing by $-\hbar^2c^2$ and using $\hbar=c=1$ collapses it to the covariant $(\partial_\mu\partial^\mu+m^2)\phi=0$ . Why the covariant form matters: each factor — $\Box$ (a scalar), $m^2$ (a scalar), $\phi$ (a scalar field) — is Lorentz invariant, so the equation looks the same in every inertial frame. *Verified: sympy confirms that acting the operator on a plane wave returns $m^2+\mathbf p^2-E^2$ , which vanishes precisely on the mass shell (verify script, "KG operator on plane wave").*

Written covariantly, the equation wears its symmetry on its sleeve. This is the form you will meet in field theory.

\left(\Box+\frac{m^2c^2}{\hbar^2}\right)\phi=0,\qquad \Box\equiv\frac{1}{c^2}\partial_t^2-\nabla^2 \qquad (3)

where: $\Box$ is the d'Alembertian $\partial_\mu\partial^\mu$ (signature $+---$ ); $\partial_\mu=(\tfrac1c\partial_t,\nabla)$ . In natural units this reads simply

(\partial_\mu\partial^\mu+m^2)\,\phi=0 \qquad (4)

where: $m$ now has units of inverse length. Equation (4) is the free scalar field equation, the Euler–Lagrange equation of $\mathcal L=\tfrac12(\partial_\mu\phi)(\partial^\mu\phi)-\tfrac12 m^2\phi^2$ . The plane-wave solutions are where the trouble — and the physics — first appears. Feeding $e^{\mp ip\cdot x}$ into (4) forces the four-momentum onto the mass shell, and the mass shell has two energy roots.

\phi_\pm(x)=N\,e^{\mp ip\cdot x/\hbar},\qquad p^0=\pm\frac{1}{c}\sqrt{\mathbf p^2c^2+m^2c^4} \qquad (5)

where: $N$ is a normalization; $p\cdot x=Et-\mathbf p\cdot\mathbf x$ is the invariant phase; the two signs of $p^0$ are the positive- and negative-energy branches drawn in the figure above. Unlike Schrödinger's equation, which fixes $E$ uniquely for a given state, the relativistic equation offers both signs — a direct consequence of $E$ appearing squared. This two-branch structure is owned in full by the next articles; here it is a fact to note. Because the equation is second order in time, $\phi(\mathbf x,0)$ alone does not determine the future — you must also give $\partial_t\phi(\mathbf x,0)$ . This is precisely where the contrast with Schrödinger dynamics bites. The article Time-Dependent Schrödinger Equation states that "the law is first order in time, so the present wavefunction alone determines the future"; the Klein–Gordon equation forfeits exactly that property, needing two initial fields per mode instead of one. That extra freedom is what wrecks the naive probability interpretation. The conserved density built from the equation, $\rho\propto i(\phi^*\partial_t\phi-\phi\,\partial_t\phi^*)$ , depends on $\partial_t\phi$ independently and is therefore not positive-definite — evaluate it on a plane wave and $\rho\propto E$ , negative for the lower branch. The construction of that current, its conservation law, and the crisis it triggers are the subject of Klein–Gordon Probability Current and Negative Densities; the full reckoning with the negative-energy branch is Negative-Energy Solutions and the Interpretive Crisis. The best evidence that the equation is right, despite its interpretive troubles, is that it contains ordinary quantum mechanics. If you peel off the fast rest-mass oscillation and keep only the slow envelope, the free Schrödinger equation drops out exactly — the structural result that anchors this article.

i\hbar\,\partial_t\psi=-\frac{\hbar^2}{2m}\nabla^2\psi \qquad (6)

where: $\psi$ is the slowly varying amplitude defined by $\phi=e^{-imc^2t/\hbar}\psi$ . Equation (6) is the free Schrödinger equation; recovering it fixes the normalization of the time-derivative term and shows the relativistic law reduces continuously to the nonrelativistic one.

Go deeper · Derivation B · 1 minNonrelativistic reduction to the Schrödinger equation

Derivation BNonrelativistic reduction to the Schrödinger equation

We want to show that the Klein–Gordon equation becomes the free Schrödinger equation when the kinetic energy is small compared to the rest energy $mc^2$ . The strategy is to factor out the dominant rest-mass phase $e^{-imc^2t/\hbar}$ , whose frequency $mc^2/\hbar$ is enormous, leaving a slow amplitude $\psi$ ; substituting and comparing the sizes of the surviving terms, one term is negligible and Schrödinger's equation is what remains. Watch for the near-cancellation of the mass term — it is what makes the reduction work.

Step 1. Write $\phi(\mathbf x,t)=e^{-imc^2t/\hbar}\psi(\mathbf x,t)$ , stripping off the frequency $mc^2/\hbar$ carried by the rest energy. Why: even a slow massive particle has a huge phase-oscillation rate $mc^2/\hbar$ ; that fast piece is not the dynamics we care about nonrelativistically, so we remove it.

Step 2. Differentiate twice in time. The second derivative brings down three terms, and inserting into the full-units equation $-\hbar^2\partial_t^2\phi=-\hbar^2c^2\nabla^2\phi+m^2c^4\phi$ , the bare mass term $m^2c^4\psi$ cancels on both sides — leaving

-\hbar^2\partial_t^2\psi+2i\hbar mc^2\partial_t\psi=-\hbar^2c^2\nabla^2\psi.

Why the cancellation matters: the huge mass term is what would otherwise dominate; its removal is exactly what the rest-mass phase was engineered to do.

Step 3. Apply the nonrelativistic approximation $|\partial_t^2\psi|\ll(mc^2/\hbar)|\partial_t\psi|$ (kinetic energy $\ll mc^2$ ), dropping the $\partial_t^2\psi$ term. Dividing the survivor $2i\hbar mc^2\partial_t\psi=-\hbar^2c^2\nabla^2\psi$ by $2mc^2$ gives $i\hbar\partial_t\psi=-(\hbar^2/2m)\nabla^2\psi$ — equation (6). Verified: sympy confirms the mass term cancels identically, the remaining equation is $-\hbar^2\psi_{tt}+2i\hbar mc^2\psi_t+\hbar^2c^2\psi_{xx}=0$ , and the surviving-term coefficients are exactly proportional to the free Schrödinger equation. Independently, the dispersion $\omega=\sqrt{k^2+m^2}=m+k^2/2m+O(k^4)$ reproduces the rest energy plus the nonrelativistic kinetic energy (verify script).

Limiting cases

Every limit of the Klein–Gordon equation lands on a familiar piece of physics, which is the best evidence that the equation is correct even where its interpretation is subtle.

case	result	meaning
massless $m\to0$	$\Box\,\phi=0$	The relativistic wave equation; every mode rides the light cone at $c$ . Each Lorenz-gauge component of the electromagnetic potential obeys exactly this.
nonrelativistic $\|\mathbf p\|c\ll mc^2$	$i\hbar\partial_t\psi=-\tfrac{\hbar^2}{2m}\nabla^2\psi$	Writing $\phi=e^{-imc^2t/\hbar}\psi$ and dropping $\partial_t^2\psi$ against $mc^2\partial_t\psi$ recovers the free Schrödinger equation.
static, sourced	$(-\nabla^2+m^2)\phi=\rho_{\text{src}}$	The Green's function is the Yukawa potential $e^{-mr}/4\pi r$ of range $\hbar/mc$ — Yukawa's 1935 picture of the nuclear force.
high-spin component	$(\Box+m^2)\psi_a=0$	Each component of a Dirac spinor or massive vector separately satisfies Klein–Gordon; the mass shell is universal.

Common trapThe field

\phi

is not a one-particle wavefunction and

|\phi|^2

is not a probability

The instinct carried over from Schrödinger's equation — that $|\phi|^2$ is a probability density — fails here. The conserved Klein–Gordon density $\rho=\tfrac{i\hbar}{2mc^2}(\phi^*\partial_t\phi-\phi\,\partial_t\phi^*)$ is not positive-definite, because the equation is second order in time and $\phi,\dot\phi$ are independent. On a negative-energy plane wave $\rho<0$ , which no probability can be. The resolution is that $\phi$ is a field; only after second quantization does a consistent particle picture (with antiparticles) emerge, and $\rho$ becomes a charge density that may legitimately carry either sign.

Worked examples

Example 1 — The reduced Compton wavelength as the equation's built-in length scale. The mass term in the Klein–Gordon equation sets a length. Let us compute it for the electron and see what it controls.

Step 1. In natural units the equation is $(\Box+m^2)\phi=0$ , so the only scale is $1/m$ . Restoring units, $1/m\to\hbar/mc$ , the reduced Compton wavelength $\lambdabar_C=\hbar/mc$ .

Step 2. Insert the electron mass $m_ec^2=0.511\ \text{MeV}$ . Using $\hbar c=197.3\ \text{MeV}\,\cdot\,\text{fm}$ ,

\lambdabar_C=\frac{\hbar c}{m_ec^2}=\frac{197.3\ \text{MeV}\,\cdot\,\text{fm}}{0.511\ \text{MeV}}=386.2\ \text{fm}=3.862\times10^{-13}\ \text{m}.

Step 3. Interpret it. This is the scale below which localizing a particle costs energy $\gtrsim mc^2$ , enough to create particle–antiparticle pairs — the length at which single-particle quantum mechanics gives way to field theory. The answer is $\boxed{\lambdabar_C=3.86\times10^{-13}\ \text{m}}$ , confirmed to CODATA precision in the verifier.

Example 2 — Yukawa's mass estimate for the nuclear force. Yukawa reasoned that a short-range force is mediated by a massive scalar, whose static Klein–Gordon Green's function is $e^{-mr}/r$ with range $\hbar/mc$ . Turn the measured range into a mass.

Step 1. The nuclear force acts over roughly $r_0\approx1\ \text{fm}$ . Identify this with the Yukawa range: $\hbar/mc=r_0$ , so $mc^2=\hbar c/r_0$ .

Step 2. Plug in $\hbar c=197.3\ \text{MeV}\,\cdot\,\text{fm}$ and $r_0=1\ \text{fm}$ :

mc^2=\frac{197.3\ \text{MeV}\,\cdot\,\text{fm}}{1\ \text{fm}}=197\ \text{MeV}.

Step 3. Compare with reality. The predicted quantum should weigh a couple hundred electron masses; the charged pion, discovered in 1947, has $m_{\pi^\pm}c^2=139.6\ \text{MeV}$ , giving a range $\hbar/m_\pi c=1.41\ \text{fm}$ . The estimate is right to within the sloppiness of " $r_0\approx1$ fm". The answer is $\boxed{mc^2\sim200\ \text{MeV}}$ , the mass scale that predicted the pion.

Origins · 1 minOskar Klein and how this was discovered

Oskar Klein

1894–1977 · Swedish theoretical physicist who gave the relativistic scalar wave equation its covariant form and, with the Klein paradox, first glimpsed the pair-creation physics hidden in it

Oskar Klein studied under Arrhenius and Bohr and moved through the great centres of the new quantum theory. In 1926 he published the relativistic wave equation that now bears his name jointly with Walter Gordon; the same year he wrote down what became the Kaluza–Klein idea of extra dimensions. His 1929 "Klein paradox" — apparent over-unity transmission of a relativistic wave through a steep potential step — was the first sign that the negative-energy solutions demanded a many-particle, pair-creating interpretation. Modest and deeply physical in his thinking, Klein spent most of his career in Stockholm.

History

1926

— Schrödinger derives the relativistic (Klein–Gordon) equation before his nonrelativistic one, then discards it for predicting the wrong hydrogen fine structure (it lacks spin)

1926

— Oskar Klein (Z. Phys. 37, 895) and Walter Gordon (Z. Phys. 40, 117) publish the equation independently; Fock, de Donder, and Kudar reach it too

1929

— Klein's paradox exposes pair-creation physics in the negative-energy solutions of a steep step

1935

— Yukawa uses the static Klein–Gordon Green's function to predict a nuclear-force quantum of mass ~200 electron masses

1947

— the charged pion, a spin-0 Klein–Gordon field of mass 140 MeV, is discovered in cosmic rays

2012

— the Higgs boson, a fundamental scalar obeying the Klein–Gordon equation with self-interaction, is observed at the LHC

Going further

References. Griffiths, Introduction to Elementary Particles, 2nd ed., §7.1 · Sakurai & Napolitano, Modern Quantum Mechanics, relativistic-QM chapter · Shankar, Principles of Quantum Mechanics, 2nd ed., ch. 20 · Bjorken & Drell, Relativistic Quantum Mechanics, §1.3–1.4 · Peskin & Schroeder, An Introduction to QFT, §2.1–2.3, §4.7.

Provenance. Every equation here is standard and machine-checkable: the canonical-quantization derivation and covariant form (Griffiths §7.1; Bjorken & Drell §1.3), the plane-wave mass shell and the two energy roots (Sakurai; Bjorken & Drell §1.3), the indefinite density on a plane wave $\rho\propto E$ (Bjorken & Drell §9.2), the nonrelativistic reduction to the Schrödinger equation (Bjorken & Drell §1.4), the Lagrangian and its Euler–Lagrange recovery (Peskin & Schroeder §2.2), and the Yukawa Green's function via contour integration (Peskin & Schroeder §4.7). The numeric anchors — $m_ec^2=0.511$ MeV, $\lambdabar_C=3.86\times10^{-13}$ m, $\hbar c=197.3$ MeV·fm, $m_{\pi^\pm}c^2=139.6$ MeV and its 1.41 fm range — are standard CODATA/PDG values reproduced in `verify_aq-qm-klein-gordon-equation.py`. No base physics error was found: the served skeleton's derivations agree with the sympy verification (the `### Derivation` headings, missing history/figures/worked-examples were upgraded to V3 anatomy; no equation was changed).

Builds on: Special relativity and four-vectors · Time-Dependent Schrödinger Equation · Plane waves and Fourier analysis. Leads to: Klein–Gordon Probability Current and Negative Densities · Negative-Energy Solutions and the Interpretive Crisis · The Dirac Equation.

Didn't Schrödinger himself find this equation? Why is it not the "Schrödinger equation"?

He did — in 1926, before his famous nonrelativistic equation, Schrödinger wrote down the relativistic version. He discarded it because, applied to hydrogen, it gave the wrong fine structure: the relativistic scalar equation has no spin, so it misses the spin–orbit splitting. He then retreated to the nonrelativistic $E=p^2/2m$ , which fit the gross spectrum. Klein and Gordon (and Fock, de Donder, Kudar) published the relativistic equation later in 1926, so it carries their names. The moral is that the equation was right — it just describes a spin-0 field, not the electron.

Why can't I just throw away the negative-energy solutions and keep the physical ones?

Two reasons. First, completeness: the positive-energy plane waves alone do not span the solution space of a second-order equation, so a generic initial condition $(\phi,\dot\phi)$ cannot be built from them. Second, covariance: a Lorentz boost mixes positive- and negative-frequency parts, so a "positive-energy only" rule is not frame-independent. The modern resolution keeps both branches and reinterprets the negative-frequency part as antiparticle content on second quantization. The article on negative-energy solutions works this through.

What exactly does "second order in time" cost me physically?

It means the field at one instant does not determine its future — you must specify both $\phi(\mathbf x,0)$ and $\partial_t\phi(\mathbf x,0)$ . That is fine for a classical field (like the electromagnetic field, also second order), but it destroys the Schrödinger-style probability story, where knowing $\psi$ now fixes everything. Because $\phi$ and $\dot\phi$ are independent, the conserved density can be negative, and there is no positive $|\phi|^2$ interpretation. In short, second order in time is exactly the feature that makes the equation covariant and the feature that makes it a field equation rather than a one-particle law.

How is the mass related to the range of the force a scalar mediates?

Freeze the time dependence: the equation becomes $(-\nabla^2+m^2)\phi=\text{source}$ , whose point-source Green's function is the Yukawa potential $e^{-mr}/4\pi r$ . The exponential decays over a length $1/m$ , which restoring units is $\hbar/mc$ — the reduced Compton wavelength. So a heavier mediator gives a shorter-range force. Setting $m=0$ removes the exponential and returns the infinite-range Coulomb $1/r$ . Yukawa ran this backwards: a ~1 fm nuclear force implies a mediator of ~200 MeV, which is how the pion was predicted.

Does the Klein–Gordon equation still matter if the "real" particles have spin?

Yes, in two ways. First, genuine spin-0 particles exist — pions, kaons, the Higgs boson — and Klein–Gordon is literally their free field equation. Second, and more subtly, every relativistic field obeys the Klein–Gordon equation component by component, because the mass-shell condition $p^2=m^2$ is universal. Squaring the Dirac operator $(i\gamma^\mu\partial_\mu-m)$ produces $(\Box+m^2)$ , so each of the four components of a Dirac spinor separately solves Klein–Gordon. It is the substrate on which the spinning theories are built.

What is the Klein paradox and why is it famous?

Apply the Klein–Gordon (or Dirac) equation to a potential step taller than $2mc^2$ and you find apparently strange transmission and reflection coefficients — a wave seemingly penetrating the barrier with "negative" reflected current. The paradox dissolves once you recognize that a step that steep can create particle–antiparticle pairs: the "transmitted" flux is antiparticles produced at the barrier. It was the first concrete signal that the single-particle reading of these equations breaks down, and it was seen directly in graphene, whose carriers obey a massless relativistic equation.

In natural units mass, momentum and energy all look the same — how do I get back real numbers?

In $\hbar=c=1$ units, energies, momenta, and masses are all measured in the same unit (say MeV), and lengths and times in its inverse (MeV $^{-1}$ ). To restore SI numbers, reinsert factors of $\hbar c=197.3$ MeV·fm and $c$ until the dimensions work out. For example a mass $m$ (in MeV) gives a length $1/m\to\hbar c/(mc^2)$ : for the electron, $197.3/0.511=386$ fm. The rule of thumb is that $\hbar c=197.3$ MeV·fm converts an energy scale into a length scale, and that single constant handles almost every conversion in this chapter.