Momentum conservation and decay kinematics

Shift a wavefunction bodily by a: (T(a)ψ)(x) = ψ(x − a). Expand the right side as a Taylor series, ψ(x − a) = Σ (−a)ⁿ/n! dⁿψ/dxⁿ = exp(−a d/dx) ψ, and compare with the general form of a continuous unitary family, T(a) = exp(−i a G/ħ): the generator is G = −iħ d/dx, which is P̂. Its eigenfunctions are the plane waves e^(ipx/ħ), the only functions a translation multiplies by a phase, and ⟨x|p⟩ = e^(ipx/ħ)/√(2πħ) is precisely the kernel of the Fourier transform, so the momentum representation is the Fourier transform of the position one and [X̂, P̂] = iħ follows from d/dx acting on xψ. Nothing here mentions a Hamiltonian yet: momentum is defined by what it does to space, and conservation is a separate question about H.

For H = P̂²/2m + V(X̂) the kinetic term commutes with P̂ trivially and [V, P̂] = iħ∇V, so d⟨P̂⟩/dt = −⟨∇V⟩: Ehrenfest's theorem, the quantum form of Newton's second law. A particle in an external potential therefore does not conserve momentum, because the potential singles out a place. Two particles interacting through V(x₁ − x₂) are different: ∇₁V = −∇₂V, so the total P̂ = P̂₁ + P̂₂ commutes with H even though neither P̂ᵢ does. That is the isolated-system condition in operator form. Because [H, P̂] = 0, H and P̂ share an eigenbasis, an eigenvalue p attached to a state is never changed by time evolution, and the total momentum of a decay's products equals the parent's exactly — event by event, not on average. In field theory the same statement holds at every vertex of a Feynman diagram, alongside energy: four-momentum is conserved at each vertex, and only an internal line's invariant mass is off shell.

Put the parent M at rest and let it decay to m₁ and m₂. Momentum conservation makes the two momenta equal and opposite, p₁ = −p₂, so a single magnitude p* describes both; energy conservation adds E₁ + E₂ = M with Eᵢ = √(mᵢ² + p*²). Eliminate E₂: E₂² = (M − E₁)², so m₂² + p*² = M² − 2ME₁ + m₁² + p*², which gives E₁ = (M² + m₁² − m₂²)/2M with no freedom left, and p* = √([M² − (m₁+m₂)²][M² − (m₁−m₂)²])/2M on substituting back. For K⁺ → π⁺π⁰ with masses 493.677, 139.570 and 134.977 MeV the bracket product gives p* = 205.1 MeV/c, and every such decay at rest produces exactly that momentum. Only the axis along which the pair flies is undetermined: two angles, distributed isotropically for an unpolarised parent. Measured in the parent's frame a two-body decay is a spike, a monoenergetic line whose width is the parent's lifetime broadening and the detector, nothing kinematic.

Nine momentum components, four constraints, five parameters free. Three of the five are the Euler angles that orient the decay plane and carry no physics for an unpolarised parent. Two remain, and the standard choice is the pair of invariant masses m₁₂² = (p₁ + p₂)² and m₂₃² = (p₂ + p₃)². Their allowed values fill a closed region: at fixed m₁₂ go to the (12) rest frame, where E₂* = (m₁₂² − m₁² + m₂²)/2m₁₂ and E₃* = (M² − m₁₂² − m₃²)/2m₁₂, and m₂₃² runs from (E₂* + E₃*)² − (p₂* + p₃*)² to (E₂* + E₃*)² − (p₂* − p₃*)², bodies 2 and 3 parallel or antiparallel. The phase-space element is dm₁₂² dm₂₃² times a constant, uniform across the region, so the energy of any one product is spread over a band: E₁ runs from m₁, body 1 at rest with 2 and 3 back to back, up to E₁, max = [M² + m₁² − (m₂ + m₃)²]/2M, where 2 and 3 move together with the least invariant mass they can have. That upper edge is the beta endpoint. For the free neutron it puts the electron's kinetic energy at 781.6 keV, 0.75 keV below Q = 782.33 keV, because the proton must recoil against the electron's momentum.

Projecting the flat measure onto E₁ gives the length of the Dalitz chord at that energy, 4p₁*p₂* in the (23) rest frame, which for a massless third body and a heavy recoil reduces to dΓ/dEₑ ∝ pₑ(E₀ − Eₑ). The beta spectrum students meet, pₑEₑ(E₀ − Eₑ)², is not a contradiction: Fermi's golden rule with non-covariant state normalisation counts d³pₑ d³pν without the 1/2E factors of the invariant measure, so its phase space carries one more power of the electron energy and one of the antineutrino energy; the invariant bookkeeping keeps those two powers in |A|² instead, which for an allowed decay, averaged over angles, is proportional to EₑEν, and Fermi's matrix element is then constant. Either bookkeeping gives the same rate; only the label on each factor moves. Two lessons follow. The endpoint and the band's edges are pure kinematics, trustworthy to the precision of the masses, which is why a Kurie plot extrapolates to E₀ cleanly. The shape inside the band and the total rate are dynamics — |V_ud|, g_A, the Fermi function — and momentum conservation says nothing about them, so a balanced four-momentum ledger yields no lifetime.

The counting argument becomes a program in a dozen lines. Draw m₁₂² uniformly between (m₁ + m₂)² and (M − m₃)², draw m₂₃² uniformly between the smallest and largest values the whole plot allows, keep the pair only if it lies inside the boundary computed from the formulas above, and convert each accepted pair to E₁ = (M² + m₁² − m₂₃²)/2M. Because the measure is flat in these variables, hit-or-miss sampling with equal weights is already the correct phase-space distribution: a histogram of E₁ is the kinematic spectrum, and its upper edge should land on E₁, max to within a bin. Two checks belong in the same script. Send m₃ to zero and the edge should coincide with the two-body value of E₁ against m₂; then weight the accepted events by an amplitude |A|² and watch the shape inside the band change while the edges stay put. Finally lay a Fermi function and the lepton-energy factor over the neutron case and compare with the tabulated spectrum — the unit's laboratory does exactly this, and its Kurie plot is linear because the amplitude was taken constant.