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Derivation

Klein-Gordon Equation and the Yukawa Potential

D-277 Home PU-304 Threads waves · fields · force Depends on four-vector-lorentz-invariants, greens-function-helmholtz
Statement

Canonical quantization of the relativistic energy–momentum relation \(E^2 = p^2c^2 + m^2c^4\) yields the Klein–Gordon equation \(\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 + \mu^2\right)\phi = 0\) with \(\mu \equiv mc/\hbar\); its static Green's function for a point source, \((-\nabla^2 + \mu^2)\phi = g\,\delta^3(\vec r)\), is the Yukawa potential \(\phi(r) = \dfrac{g}{4\pi}\dfrac{e^{-\mu r}}{r}\), a force of finite range \(R = 1/\mu = \hbar/mc\).

Why it matters

The Klein–Gordon equation is the simplest relativistic wave equation and the field equation for any spin‑0 (scalar) field; it is the template on which the Dirac equation, quantum electrodynamics, and all of relativistic quantum field theory are built. Its static solution answers a physical question that classical field theory cannot: what does the force between two charges look like when the mediating field has mass?

The answer — an exponentially screened Coulomb potential with range set by the mediator's Compton wavelength — is the quantitative heart of Yukawa's 1935 prediction of the pion. Reading the ~1 fm range of the nuclear force backwards through \(R=\hbar/mc\) forecast a new particle of mass \(\sim 100\ \mathrm{MeV}/c^2\), discovered a decade later. The same relation explains why electromagnetism and gravity are long‑ranged (massless mediators) while the weak interaction is confined to sub‑femtometre scales.

Assumptions
The particle is spin‑0 (a scalar field).If dropped, the correct relativistic equation is Dirac's (spin‑½) or Proca's (spin‑1); the mediator's spin changes the sign structure and tensor form of the potential, though the exponential range \(e^{-\mu r}\) survives for any massive mediator. Minimal coupling via the correspondence \(E\to i\hbar\partial_t,\ \vec p\to -i\hbar\nabla\).If a different operator ordering or a nonminimal coupling is used, extra derivative terms appear; the free equation is fixed only by demanding Lorentz covariance and the classical dispersion relation. The field is a classical c‑number field, not a second‑quantized operator.If dropped one confronts the negative‑energy solutions honestly: \(\phi\) cannot be a single‑particle probability amplitude (its conserved density is not positive‑definite), and consistency requires reinterpreting \(\phi\) as a quantum field creating and annihilating particles and antiparticles. Static, point source: \(\partial_t\phi=0\) with source \(g\,\delta^3(\vec r)\).If the source moves or oscillates the retarded, time‑dependent Green's function is needed; the pure Yukawa form is the zero‑frequency (Born) limit describing a stationary "charge."
Derivation
1
\[ E^2 = p^2 c^2 + m^2 c^4 \]
The relativistic dispersion relation is the mass‑shell condition \(p_\mu p^\mu = m^2 c^2\) written in a frame, using the four‑momentum invariant established previously. A
2
\[ E \;\longrightarrow\; i\hbar\frac{\partial}{\partial t}, \qquad \vec p \;\longrightarrow\; -i\hbar\,\nabla \]
Canonical quantization: energy and momentum become the operators that generate time and space translations on a field \(\phi(\vec r,t)\); this is fixed by the de Broglie plane wave \(\phi\sim e^{i(\vec p\cdot\vec r - Et)/\hbar}\). B
3
\[ \left(i\hbar\frac{\partial}{\partial t}\right)^2\!\phi = \left[\left(-i\hbar\,\nabla\right)^2 c^2 + m^2 c^4\right]\phi \]
Promote the scalar relation of step 1 to an operator identity acting on \(\phi\); squaring \(E\) keeps only even powers of the operators, so no square root of the Laplacian is needed (the source of the equation's simplicity). B
4
\[ -\hbar^2\frac{\partial^2 \phi}{\partial t^2} = -\hbar^2 c^2\,\nabla^2\phi + m^2 c^4\,\phi \]
Evaluate the squared operators: \((i\hbar\partial_t)^2 = -\hbar^2\partial_t^2\) and \((-i\hbar\nabla)^2 = -\hbar^2\nabla^2\). Pure algebra of the derivative operators. A
5
\[ \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} - \nabla^2\phi + \left(\frac{mc}{\hbar}\right)^{2}\phi = 0 \]
Divide through by \(-\hbar^2 c^2\) and collect terms. Define the inverse length \(\mu \equiv mc/\hbar\); symbols are rearranged before any number is inserted. B
6
\[ \left(\Box + \mu^2\right)\phi = 0,\qquad \Box \equiv \partial_\mu\partial^\mu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 \]
Recognize the d'Alembertian with the mostly‑minus metric \((+,-,-,-)\). Manifest Lorentz covariance: \(\Box\) and \(\mu^2\) are both scalars, so the equation holds in every inertial frame. A
7
\[ \partial_t\phi = 0 \;\Rightarrow\; \left(-\nabla^2 + \mu^2\right)\phi(\vec r) = g\,\delta^3(\vec r) \]
Take the static limit and drive the field with a point "charge" of strength \(g\) at the origin. This is the screened Poisson (Helmholtz‑type) equation whose solution is the potential of a stationary source. B
8
\[ G(\vec r) = \int\!\frac{d^3k}{(2\pi)^3}\,\frac{e^{\,i\vec k\cdot\vec r}}{k^2 + \mu^2} = \frac{1}{4\pi^2 r}\int_{-\infty}^{\infty}\!\frac{k\sin(kr)}{k^2+\mu^2}\,dk \]
Fourier‑transform the operator: \((-\nabla^2+\mu^2)\to (k^2+\mu^2)\), so the Green's function is the inverse transform of \((k^2+\mu^2)^{-1}\). The angular integral over \(d\Omega_k\) collapses to the sine kernel (this is the Helmholtz Green's function established previously). C
9
\[ \int_{-\infty}^{\infty}\!\frac{k\sin(kr)}{k^2+\mu^2}\,dk = \pi\,e^{-\mu r} \;\Rightarrow\; G(\vec r) = \frac{e^{-\mu r}}{4\pi r} \]
Close the contour in the upper half‑plane (\(r>0\)); the pole at \(k = +i\mu\) has residue giving \(e^{-\mu r}\), while \(k=-i\mu\) would give the discarded growing solution. The exponential decay is the mass \(\mu\ne 0\) at work. C
10
\[ \phi(\vec r) = g\,G(\vec r) = \frac{g}{4\pi}\,\frac{e^{-\mu r}}{r} \]
By linearity, the response to the source \(g\,\delta^3(\vec r)\) is \(g\) times the unit Green's function. The Yukawa potential. A
Result
\[ \phi(r) = \frac{g}{4\pi}\,\frac{e^{-\mu r}}{r},\qquad \mu = \frac{mc}{\hbar},\qquad R = \frac{1}{\mu} = \frac{\hbar}{mc} \]

Reading. A massive mediator produces a Coulomb‑like \(1/r\) potential multiplied by an exponential screening factor \(e^{-\mu r}\). The field is effectively "used up" beyond one screening length \(R=\hbar/mc\), the mediator's reduced Compton wavelength. The heavier the exchanged quantum, the shorter the reach of the force; a massless mediator (\(\mu\to 0\)) restores the infinite‑range Coulomb law. The static potential energy between two such charges is \(V(r) = -\dfrac{g^2}{4\pi}\dfrac{e^{-\mu r}}{r}\) (attractive for like scalar charges), the Fourier transform at zero energy transfer of the scalar propagator \((k^2+\mu^2)^{-1}\).

Units check. \(\mu = mc/\hbar\) has units \(\dfrac{\mathrm{kg}\cdot(\mathrm{m/s})}{\mathrm{J\cdot s}} = \dfrac{\mathrm{kg\,m\,s^{-1}}}{\mathrm{kg\,m^2\,s^{-1}}} = \mathrm{m^{-1}}\), so \(\mu r\) is dimensionless and \(R=\hbar/mc\) is a length. Convenient practical form: \(R = \dfrac{\hbar c}{mc^2}\) with \(\hbar c = 197.327\ \mathrm{MeV\cdot fm}\), so \(mc^2\) in MeV gives \(R\) directly in fm.

Limiting cases
  • \(m\to 0\) (\(\mu\to 0\)): \(e^{-\mu r}\to 1\), recovering the Coulomb/Poisson potential \(\phi = g/(4\pi r)\) of infinite range — the photon and graviton limit.
  • \(r \ll R\) (\(\mu r\ll 1\)): \(e^{-\mu r}\approx 1\), so at short distance the force is indistinguishable from Coulomb; the mass matters only in the tail.
  • \(r \gg R\) (\(\mu r\gg 1\)): exponential cutoff dominates; the potential is negligibly small beyond a few screening lengths — an effectively finite‑range force.
  • Non‑relativistic reduction \(E\approx mc^2\): writing \(\phi = e^{-imc^2 t/\hbar}\psi\) and dropping \(\partial_t^2\psi\) relative to \(mc^2\partial_t\psi\) returns the free Schrödinger equation for \(\psi\).
Breaks when
  • Spin‑½ sources (nucleons): the physical nuclear‑force charges are fermions, not scalars. One‑pion exchange between spin‑½ nucleons gives a spin‑ and tensor‑dependent potential (with a \(\vec\sigma_1\!\cdot\!\vec\sigma_2\) contact term and a non‑central tensor part); only the overall \(e^{-\mu r}/r\) envelope is captured by the naive scalar result.
  • Strong coupling / non‑perturbative regime: the pion–nucleon coupling \(g^2/4\pi \approx 14\) is not small, so the single‑exchange (Born) potential is not quantitatively reliable. Two‑pion and multi‑pion exchange reshape the intermediate range, and QCD confinement, not meson exchange, governs the very short distance where quark substructure is resolved.
  • Single‑particle probability interpretation: the KG density \(\rho \propto i(\phi^*\partial_t\phi - \phi\,\partial_t\phi^*)\) is not positive‑definite, so \(\phi\) cannot be a one‑particle wavefunction; at energies \(\gtrsim mc^2\) pair creation makes the field‑theoretic description mandatory.
  • Time‑dependent or relativistic sources: a rapidly moving or radiating source needs the full retarded Green's function of \(\Box+\mu^2\); the static Yukawa form is only the zero‑frequency limit.
Failure modes
  • Dropping \(c\) and \(\hbar\): working in natural units \(\hbar=c=1\) and forgetting to restore them, so \(\mu\) is quoted as "the mass" \(m\) rather than \(mc/\hbar\), giving a range wrong by factors of \(\hbar c\).
  • Wrong pole / growing exponential: choosing the pole \(k=-i\mu\) (or the \(e^{+\mu r}\) homogeneous solution) and reporting a potential that blows up at infinity instead of decaying.
  • Reduced vs full Compton wavelength: writing the range as \(h/mc\) instead of \(\hbar/mc\); these differ by \(2\pi\). The screening length is the reduced Compton wavelength \(\hbar/mc\).
  • Missing \(4\pi\): omitting the \(1/4\pi\) from the point‑source Green's function, which propagates into a wrong coupling normalization.
  • Positive‑definite density: asserting \(|\phi|^2\) is a probability density as in Schrödinger theory — the conserved KG current is a charge density that can be negative.
  • Sign of the potential: assuming like scalar charges repel (Coulomb intuition); scalar exchange gives a universally attractive potential between like charges.
Discussion

The physical origin of the range is illuminated by the energy–time uncertainty relation. A static charge can emit a virtual mediator of rest energy \(mc^2\) only by "borrowing" that energy for a time \(\Delta t \sim \hbar/mc^2\). Travelling at most at speed \(c\), the quantum reaches a maximum distance \(\Delta x \sim c\,\Delta t = \hbar/mc = R\) before it must be reabsorbed. The exponential \(e^{-\mu r}\) is precisely the amplitude for a virtual quantum of mass \(m\) to propagate a distance \(r\); the mass sets a hard scale below which force can be exchanged and above which it is exponentially unlikely. This is why Yukawa could read a particle mass off a force range.

Formally, the Yukawa potential is the three‑dimensional Fourier transform of the scalar propagator \(\tilde D(\vec k) = (k^2+\mu^2)^{-1}\) evaluated at zero energy transfer. In Born approximation the scattering amplitude for two charges is \(g^2\tilde D(\vec q)\) with \(\vec q\) the momentum transfer, and the potential is its transform. The pole of the propagator at \(k^2=-\mu^2\), i.e. \(p^2 = m^2c^2\) in covariant notation, is the mass shell of the exchanged particle; a massless propagator \(1/k^2\) transforms to \(1/4\pi r\), the Coulomb law, exhibiting electromagnetism as the \(\mu\to 0\) member of the same family.

The Klein–Gordon equation is also the field equation for the free scalar Lagrangian \(\mathcal L = \tfrac12(\partial_\mu\phi)(\partial^\mu\phi) - \tfrac12\mu^2\phi^2\), and its plane‑wave solutions \(e^{-ip\cdot x/\hbar}\) come in both positive and negative frequency. Historically these negative‑energy modes were read as a fatal flaw, driving Dirac to his first‑order equation; the modern resolution (Pauli–Weisskopf) keeps the KG equation but quantizes \(\phi\) as an operator whose negative‑frequency part creates antiparticles. The Higgs boson is a physical, fundamental scalar obeying exactly this equation.

The metric‑signature convention deserves care: with \((+,-,-,-)\) the operator is \(\Box+\mu^2\) and the equation reads \((\Box+\mu^2)\phi=0\); with \((-,+,+,+)\) it becomes \((\Box-\mu^2)\phi=0\) because \(\Box\) flips sign. The physics — a decaying tail of range \(\hbar/mc\) — is convention‑independent, but the relative sign of the mass term and the location of the propagator pole \((k^2+\mu^2)^{-1}\) versus \((k^2-\mu^2)^{-1}\) track the signature and must be kept consistent throughout, including the \(i\epsilon\) prescription that selects the retarded or Feynman Green's function.

Common misconceptions. The Yukawa force has no hard cutoff at \(r=R\): the potential is nonzero at all \(r\), merely exponentially small beyond a few \(R\). The "range" is the \(1/e\) screening length of the exponential, not a sharp edge. The reduced Compton wavelength \(\hbar/mc\) — not \(h/mc\) — is the correct range. And \(\phi\) is a field amplitude, not a probability amplitude: its squared modulus is not a probability density.

Worked examples
1
Range of the nuclear force from the charged pion (\(mc^2 = 139.6\ \mathrm{MeV}\)).
Use \(R = \hbar/mc = \hbar c/(mc^2)\) with \(\hbar c = 197.327\ \mathrm{MeV\cdot fm}\); symbols first, then numbers. A
\[ R = \frac{\hbar c}{mc^2} = \frac{197.327\ \mathrm{MeV\cdot fm}}{139.6\ \mathrm{MeV}} = 1.413\ \mathrm{fm} \]
\[ R_\pi \approx 1.41\ \mathrm{fm} \]

Reading. This matches the observed \(\sim 1\!-\!2\ \mathrm{fm}\) range of the strong nuclear force, the empirical fact Yukawa inverted in 1935 to predict a mediator of mass \(\sim 100\ \mathrm{MeV}/c^2\). Units: MeV·fm divided by MeV gives fm, as required.

2
Range of the weak force from the \(W\) boson (\(mc^2 = 80.38\ \mathrm{GeV} = 80380\ \mathrm{MeV}\)).
Same relation \(R=\hbar c/(mc^2)\); the large mass will give a tiny range. A
\[ R = \frac{197.327\ \mathrm{MeV\cdot fm}}{80380\ \mathrm{MeV}} = 2.455\times 10^{-3}\ \mathrm{fm} \]
Convert to metres: \(1\ \mathrm{fm} = 10^{-15}\ \mathrm{m}\). A
\[ R = 2.455\times 10^{-3}\ \mathrm{fm}\times 10^{-15}\ \mathrm{m/fm} = 2.46\times 10^{-18}\ \mathrm{m} \]
\[ R_W \approx 2.5\times 10^{-18}\ \mathrm{m} \]

Reading. The weak interaction reaches only a few thousandths of a femtometre — far shorter than the atomic nucleus — because its mediator is nearly a thousand times heavier than the pion. This is why the weak force appears as an almost point‑like, feeble interaction at nuclear scales despite an intrinsic coupling comparable to electromagnetism. Units check: MeV·fm / MeV = fm, then fm → m.

Problems
  1. Show explicitly that as \(m\to 0\) the Yukawa potential reduces to the Coulomb form, and state the physical mediator for which this limit applies.
    Solution As \(m\to 0\), \(\mu = mc/\hbar \to 0\), so \(e^{-\mu r}\to e^0 = 1\) for every fixed \(r\). Then \(\phi(r) = \dfrac{g}{4\pi}\dfrac{e^{-\mu r}}{r} \to \dfrac{g}{4\pi}\dfrac{1}{r}\), the \(1/r\) Coulomb/Poisson potential of infinite range (no screening). Physically this is the massless mediator limit: the photon (electromagnetism) and, at the classical level, the graviton, giving inverse‑square forces of unlimited range.
  2. Compute the range \(R\) of a force mediated by the \(\rho\) meson, \(mc^2 = 775\ \mathrm{MeV}\), and compare with the pion range of Worked Example 1.
    Solution \(R = \dfrac{\hbar c}{mc^2} = \dfrac{197.327\ \mathrm{MeV\cdot fm}}{775\ \mathrm{MeV}} = 0.255\ \mathrm{fm}\). This is about \(0.255/1.413 \approx 0.18\) times the pion range, i.e. roughly \(1/5.6\). The heavier \(\rho\) mediates a much shorter‑ranged component of the nucleon–nucleon interaction, dominating the intermediate/short‑range region while the light pion controls the long‑range tail.
  3. A hypothetical new force is observed to have a range of \(R = 1\ \mu\mathrm{m}\). Estimate the mass of its mediator in \(\mathrm{eV}/c^2\).
    Solution Invert \(R = \hbar c/(mc^2)\Rightarrow mc^2 = \hbar c/R\). Convert the range: \(1\ \mu\mathrm{m} = 10^{-6}\ \mathrm{m} = 10^{9}\ \mathrm{fm}\). Then \(mc^2 = \dfrac{197.327\ \mathrm{MeV\cdot fm}}{10^{9}\ \mathrm{fm}} = 1.973\times 10^{-7}\ \mathrm{MeV} = 0.197\ \mathrm{eV}\). So \(m \approx 0.20\ \mathrm{eV}/c^2\). A micron‑range force requires an extraordinarily light mediator — such "fifth‑force" searches constrain sub‑eV bosons.
  4. Verify by dimensional analysis that every term in the static Klein–Gordon equation \((-\nabla^2 + \mu^2)\phi = g\,\delta^3(\vec r)\) has consistent units, and hence confirm that \(\mu\) has dimension of inverse length.
    Solution \(\nabla^2\) carries dimension \(\mathrm{length}^{-2}\), so \(\nabla^2\phi\) has units \([\phi]\,\mathrm{m^{-2}}\). For \(\mu^2\phi\) to match, \([\mu^2] = \mathrm{m^{-2}}\), hence \([\mu] = \mathrm{m^{-1}}\), consistent with \(\mu = mc/\hbar\) (shown in the Units check). The source term: \(\delta^3(\vec r)\) has units \(\mathrm{m^{-3}}\) (it integrates to 1 over volume). Matching \([g]\,\mathrm{m^{-3}} = [\phi]\,\mathrm{m^{-2}}\) fixes the coupling dimension as \([g] = [\phi]\,\mathrm{m}\), consistent with \(\phi = (g/4\pi)e^{-\mu r}/r\) where \(g/r\) restores \([\phi]\).
  5. Starting from the full Klein–Gordon equation, substitute \(\phi(\vec r,t) = e^{-imc^2 t/\hbar}\,\psi(\vec r,t)\) and show that, in the regime \(\left|\hbar\,\partial_t\psi\right| \ll mc^2\,|\psi|\), the free Schrödinger equation for \(\psi\) emerges. Identify the term you neglect.
    Solution Compute the time derivatives of \(\phi = e^{-imc^2t/\hbar}\psi\): \(\partial_t\phi = e^{-imc^2t/\hbar}\!\left(-\tfrac{imc^2}{\hbar}\psi + \partial_t\psi\right)\), and \(\partial_t^2\phi = e^{-imc^2t/\hbar}\!\left(-\tfrac{m^2c^4}{\hbar^2}\psi - \tfrac{2imc^2}{\hbar}\partial_t\psi + \partial_t^2\psi\right)\). Insert into \(\tfrac{1}{c^2}\partial_t^2\phi - \nabla^2\phi + \tfrac{m^2c^2}{\hbar^2}\phi = 0\) and cancel the common exponential. The \(-\tfrac{m^2c^2}{\hbar^2}\psi\) from the first term cancels the \(+\tfrac{m^2c^2}{\hbar^2}\psi\) mass term exactly. What remains is \(-\tfrac{2im}{\hbar}\partial_t\psi + \tfrac{1}{c^2}\partial_t^2\psi - \nabla^2\psi = 0\). In the non‑relativistic regime the second‑order time derivative is negligible: \(\left|\tfrac{1}{c^2}\partial_t^2\psi\right| \sim \tfrac{1}{c^2}\tfrac{|E_{\rm kin}|}{\hbar}|\partial_t\psi| \ll \tfrac{m}{\hbar}|\partial_t\psi|\) when \(E_{\rm kin}\ll mc^2\). Dropping it gives \(-\tfrac{2im}{\hbar}\partial_t\psi - \nabla^2\psi = 0\), i.e. \(i\hbar\,\partial_t\psi = -\tfrac{\hbar^2}{2m}\nabla^2\psi\), the free Schrödinger equation. The neglected term is \(\tfrac{1}{c^2}\partial_t^2\psi\), the relativistic correction that repackages rest energy back into the dynamics.