12.2 The Klein–Gordon equation

Last section’s diagnosis was written plainly: the disease is $E=p^2/2m$ . The prescription seems ready-made — chapter 2’s translation rules for building the Schrödinger equation have not expired:

E\;\to\;\ii\hbar\frac{\partial}{\partial t},\qquad \vec p\;\to\;-\ii\hbar\nabla\tag{12.2.1}

Just plug them into the genuine relativistic relation $E^2=p^2c^2+m^2c^4$ , and done? Schrödinger himself tried exactly this route first, in 1926 (before he even published his non-relativistic equation), and Klein and Gordon published it formally the same year. In this section we walk the route ourselves and see where it succeeds and where it stumbles — and the way it stumbles is highly instructive.

Constructing the equation

From dispersion relation to the Klein–Gordon equationbasic~5 min

Step 1: direct translation. Substitute the translation rules into $E^2=p^2c^2+m^2c^4$ and let both sides act on a wavefunction $\phi(x,t)$ :

-\hbar^2\frac{\partial^2\phi}{\partial t^2} =-\hbar^2c^2\nabla^2\phi+m^2c^4\phi\tag{12.2.2}

Step 2: tidy into standard form. Divide by $-\hbar^2c^2$ and rearrange:

\boxed{\;\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2}-\nabla^2\phi +\Bigl(\frac{mc}{\hbar}\Bigr)^2\phi=0\;}\tag{12.2.3}

This is the Klein–Gordon equation. Identify the parts: the first two terms together are the wave-equation operator (the d’Alembertian $\Box$ ) — second derivatives in both time and space, paired in the correct $1/c^2$ ratio: Lorentz covariant, box ticked. The coefficient of the third term is the inverse square of the Compton wavelength we just met, $1/\lambda_C^2$ : the mass’s sole role in the equation is to endow the wave equation with an intrinsic length scale.

Step 3: two limiting self-checks.

$m=0$ : it degenerates to the wave equation $\Box\phi=0$ , whose solutions travel at the speed of light — the equation a photon should satisfy.
Insert the plane wave $\phi=\ee^{\ii(px-Et)/\hbar}$ and the equation demands

E^2=p^2c^2+m^2c^4\;\Longrightarrow\;E=\pm\sqrt{p^2c^2+m^2c^4}\tag{12.2.4}

The dispersion relation returns exactly to our starting point — but note the $\pm$ : every momentum comes paired with a negative-energy solution. File that away; this section is trying a different case first.

The crime scene: where did the probability go?

A candidate “quantum-mechanical equation” must hand over two things: a probability density $\rho$ and a probability current $\vec j$ satisfying the continuity equation $\partial_t\rho+\nabla\cdot\vec j=0$ (probability neither appears from nowhere nor vanishes — recall section 2.2). The Schrödinger equation hands over $\rho=|\Psi|^2\geq0$ , qualified by birth. What about KG?

Deriving the KG conserved current — and the autopsyadvanced~9 min

Step 1: the standard trick for a conservation law. Exactly the move of section 2.2: multiply the equation by $\phi^*$ and subtract $\phi$ times the conjugate equation. The KG equation and its complex conjugate are

\frac1{c^2}\partial_t^2\phi-\nabla^2\phi+\frac{m^2c^2}{\hbar^2}\phi=0,\qquad \frac1{c^2}\partial_t^2\phi^*-\nabla^2\phi^*+\frac{m^2c^2}{\hbar^2}\phi^*=0\tag{12.2.5}

$\phi^*\times$ (first) $-\;\phi\times$ (second) — the mass terms cancel:

\frac1{c^2}\bigl(\phi^*\partial_t^2\phi-\phi\,\partial_t^2\phi^*\bigr) -\bigl(\phi^*\nabla^2\phi-\phi\nabla^2\phi^*\bigr)=0\tag{12.2.6}

Both brackets can be written as total derivatives:

\frac{\partial}{\partial t}\Bigl[\frac1{c^2}\bigl(\phi^*\partial_t\phi-\phi\,\partial_t\phi^*\bigr)\Bigr] -\nabla\cdot\bigl(\phi^*\nabla\phi-\phi\nabla\phi^*\bigr)=0\tag{12.2.7}

Step 2: normalise the coefficient and recognise the current. Multiply by $\frac{\ii\hbar}{2m}$ so the spatial part matches the Schrödinger probability current:

\rho=\frac{\ii\hbar}{2mc^2}\Bigl(\phi^*\frac{\partial\phi}{\partial t}-\phi\frac{\partial\phi^*}{\partial t}\Bigr), \qquad \vec j=\frac{\hbar}{2m\ii}\bigl(\phi^*\nabla\phi-\phi\nabla\phi^*\bigr)\tag{12.2.8}

$\vec j$ is identical to Schrödinger’s probability current — familiar and reassuring. But $\rho$ has changed: it is no longer $|\phi|^2$ but a combination involving the time derivative.

Step 3: the autopsy. Insert the plane wave $\phi=N\ee^{\ii(\vec p\cdot\vec x-Et)/\hbar}$ (so $\partial_t\phi=-\ii E\phi/\hbar$ ):

\rho=\frac{\ii\hbar}{2mc^2}\cdot\Bigl(-\frac{2\ii E}{\hbar}\Bigr)|N|^2 =\frac{E}{mc^2}\,|N|^2\tag{12.2.9}

Positive-energy solutions ( $E>0$ ): $\rho>0$ , all is well, and at low speed $E\to mc^2$ , $\rho\to|N|^2$ , joining smoothly onto Schrödinger — good. Negative-energy solutions ( $E<0$ ): $\rho<0$ . The probability density is negative.

And as we just established, the negative-energy solutions make up fully half of the solution space and cannot be thrown away (discard half the solutions and the remaining family is incomplete — you cannot even assemble an arbitrary initial wave packet).

To the root: negative probability is the inevitable child of 'second order in time'

This is not a flaw in the derivation technique; it is structural necessity. Seeing this clearly matters most of all:

The Schrödinger equation is first order in time: given $\Psi(x,0)$ the future is uniquely determined, and $\rho=|\Psi|^2$ contains no time derivative — it could not go negative if it tried.

The KG equation is second order in time — like Newton’s equation, it needs two pieces of initial data: $\phi(x,0)$ and $\partial_t\phi(x,0)$ , independent of each other. The conserved density $\rho$ is precisely a combination of these two, and a combination of two independent quantities can of course take either sign.

Now the knot is visible: relativity forces you to choose the same derivative order for time and space. Choose second order (KG) — a positive-definite probability density cannot be salvaged. That leaves only one untried road: first order in both time and space. This is exactly Dirac’s entrance in the next section.

Rehabilitation: not wrong, just misused

The KG equation was later fully rehabilitated — the problem was never the equation, but the reading ” $\phi$ is a single-particle probability amplitude”.

The picture

Change spectacles. Multiply $\rho$ by the charge $q$ and stop calling it a probability density: call it the charge density. A charge density is supposed to take either sign — positive particles contribute positively, antiparticles negatively. The scandal of “negative probability”, under its new name, becomes a prediction: the negative-energy solutions of the KG equation are holding a seat for a particle of opposite charge. The equation was telling the truth all along; we were just listening in the wrong language.

In quantum field theory, $\phi$ is recast as a field (one degree of freedom at each spacetime point, whose quantised excitations are the particles), and the KG equation is the field equation for spin-0 particles. The pion (discovered 1947) and the Higgs boson (discovered 2012) satisfy exactly this “failed” equation of 1926.

The mathematics

What is conserved is charge, not particle number.

Q=q\int\rho\,\dd^3x=\text{const}\tag{12.2.10}

A typical relativistic process: a photon creates a particle-antiparticle pair,

\gamma\;\to\;\pi^+ +\pi^-\tag{12.2.11}

Particle number $1\to2$ (not conserved); charge $0\to(+1)+(-1)=0$ (conserved). The KG conservation law keeps precisely this ledger.

A clean corroborating clue: a real $\phi$ ( $\phi^*=\phi$ ) inserted into the expression for $\rho$ gives identically zero — a real field describing an electrically neutral particle (such as $\pi^0$ or the Higgs) has no such density to speak of, and the “probability interpretation” never applied to it in the first place.

Key formulas

Klein–Gordon equation
$\frac1{c^2}\frac{\partial^2\phi}{\partial t^2}-\nabla^2\phi+\Bigl(\frac{mc}{\hbar}\Bigr)^2\phi=0$
Second derivatives in both space and time, Lorentz covariant; mass term = 1/λ_C²
Energy spectrum
$E=\pm\sqrt{p^2c^2+m^2c^4}$
Negative-energy solutions are half the solution space; completeness forbids discarding them
Conserved density
$\rho=\frac{\ii\hbar}{2mc^2}\bigl(\phi^*\partial_t\phi-\phi\,\partial_t\phi^*\bigr)$
Plane waves give ρ=(E/mc²)|N|²; negative-energy solutions have ρ<0
Modern reading
$q\rho=\text{charge density},\qquad Q=q\int\rho\,\dd^3x=\text{const}$
KG is the spin-0 field equation: pions, the Higgs boson

Self-check4 questions

What comes next

Cause of death established: the second time derivative killed positive-definite probability. Dirac’s strategy strikes at the root — build an equation with only first derivatives in both time and space, which upon squaring returns to $E^2=p^2c^2+m^2c^4$ . With ordinary numbers this cannot be done; Dirac discovered that with matrices it can. And the algebraic structure introduced purely to “take a square root” will hand over, free of charge, the most mysterious thing in quantum mechanics — spin.

The Klein–Gordon equation

Recommended first

After this section you should be able to

Constructing the equation

The crime scene: where did the probability go?

Rehabilitation: not wrong, just misused

The picture

The mathematics

Key formulas

Self-check4 questions

What comes next

Constructing the equation

The crime scene: where did the probability go?

Rehabilitation: not wrong, just misused

◑The picture

∑The mathematics

∎Key formulas

What comes next

The picture

The mathematics

Key formulas