Survival Amplitude and Decay of Unstable States

The three-era anatomy is now measured physics at both ends. The quadratic era was pinned in cold-atom tunnelling (Wilkinson 1997) — the same platform where Zeno and anti-Zeno control of decay was later demonstrated. The Khalfin tail took until 2006: Rothe and co-workers watched dissolved organic molecules luminesce over more than 20 lifetimes and saw the exponential hand over to a clean power law. In hadron physics the second-sheet pole is the definition of an unstable particle: the Particle Data Group quotes pole positions $E_R - i\Gamma/2$ for broad resonances, because peak-fitting becomes convention-dependent when $\Gamma$ is not small. In quantum simulation the survival amplitude works under aliases — Loschmidt echoes and fidelity decay are $|a(t)|^2$ by other names. And false-vacuum decay in cosmology is the same formalism scaled up: a metastable vacuum's fate is a survival-amplitude problem.

Think of the initial state as a chord struck on a piano: a bundle of pure tones $e^{-iEt/\hbar}$, one per energy component, with weights $n(E) = |\langle E|\psi(0)\rangle|^2$. At $t=0$ every tone is in phase: $a(0) = 1$. Then the tones drift apart — higher energies turning faster — and the overlap with the original chord decays. Nothing is "lost"; evolution stays unitary. Decay of survival is dephasing, and its speed and shape are dictated entirely by which tones are present at what strength. One curve, $n(E)$, is the whole score.

A sharp, isolated resonance is a chord whose tones cluster around $E_R$ with spread $\Gamma$: the narrower the cluster, the longer the coherence, the longer the life — $\tau = \hbar/\Gamma$. The exponential law appears when the cluster has one particular shape, the Lorentzian, whose tails reach to arbitrarily negative energy. Nature only approximates that shape, because real spectra have a floor — and the floor is what forbids eternal exponential decay.

The universal flat start was proved next door: over infinitesimal times the phases have barely spread, so the overlap loss is second order — $\dot P(0) = 0$ always. That is Quantum Zeno Effect's quadratic law with $\tau_Z = \hbar/\sigma_H$; we cite it and build the other two eras around it.

Prepare $|\psi(0)\rangle$, evolve freely under a time-independent $\hat H$, and ask for the amplitude that it is still — coherently — its old self:

where: $a(t)$ is a complex number with $a(0) = 1$ — the diagonal matrix element of The Time-Evolution Operator's $\hat U(t)$ in the initial state. For an energy eigenstate $a(t) = e^{-iEt/\hbar}$: pure phase, no decay ever — the fixed-point physics of Stationary States and Energy Eigenstates, which warned that true stationary states of unstable systems do not exist. This article is the promised account of what replaces them. The measurable object is its square:

where: $P(t)$ is the survival probability — the chance an ideal projective check at time $t$ finds the system in its initial state. Read over an ensemble it becomes "the fraction not yet decayed": $N(t) = N_0 P(t)$, with $t_{1/2} = \tau\ln 2$ in the exponential era.

Everything else here is one theorem. Resolve the initial state over the energy eigenbasis — the expansion craft of Spectral Decomposition of Time Evolution, powered by Completeness Relation and Resolution of Identity — and the amplitude collapses into a Fourier integral:

where: $n(E)$ is the energy distribution (spectral density) of the initial state — nonnegative, normalized, supported only on the actual spectrum of $\hat H$. Discrete spectrum: $n(E) = \sum_n |c_n|^2\delta(E - E_n)$, a sum of phases. Continuum-coupled decaying state: a smooth line shape. The theorem is stark — the entire decay law is the Fourier transform of the line shape; spectroscopy and decay kinetics are the same data in conjugate variables.

Feed the master formula the most famous line shape in physics — a resonance of central energy $E_R$ and FWHM $\Gamma$, the Breit–Wigner (Lorentzian) density:

where: the normalization integrates to 1 over the whole real line and $\Gamma$ has units of energy. It is the shape Resonant Transmission and Tunnelling Resonances fitted to a double-barrier transmission peak; here it plays its other role, as the energy content of a decaying state. Fine print that matters later: the support runs down to $-\infty$. Its Fourier transform is an exact exponential — the only line shape with that property:

where: the phase turns at the resonance energy while the modulus dies with lifetime $\tau = \hbar/\Gamma$ — the width–lifetime dictionary ($t_{1/2} = \tau\ln 2$). Read the amplitude whole: $a(t) = e^{-i(E_R - i\Gamma/2)t/\hbar}$ — evolution under a complex energy. That is the Gamow vector previewed by Rigged Hilbert Space, made quantitative as the pole of the line shape; and it is the same pole Green Functions in Quantum Mechanics watched from the resolvent side — completed in the doctrine card below.

The dictionary is not new to this track: Conservation of Normalization's optical-potential card put $-i\Gamma/2$ into the Hamiltonian by hand and got $\tau = \hbar/\Gamma$ ($\Gamma = 6$ MeV → $1.1\times10^{-22}$ s). That was the shortcut; equations (4)–(5) are its microscopic license.

The opening act is owned and derived by Quantum Zeno Effect: every finite-variance state starts as $P(t) = 1 - (\sigma_H t/\hbar)^2 + \mathcal O(t^4)$, flat at $t=0$, with Zeno time $\tau_Z = \hbar/\sigma_H$. In master-formula language it is Derivation A's moment dictionary: the first two moments of $n(E)$ fix $\dot a(0)$ and $\ddot a(0)$, and the linear term of $P$ cancels because the first moment only turns the phase. Note the tension with equation (5): a pure exponential has $\dot P(0) = -\Gamma/\hbar \ne 0$. The resolution: the Lorentzian's second moment diverges, so the quadratic theorem does not apply to it. Any physical line shape — finite moments, bounded below — obeys the quadratic law first, then crosses into the exponential era once components a few $\Gamma$ off resonance have dephased.

The floor of the spectrum eventually speaks. Physical Hamiltonians are bounded below; the integral in (3) runs over $[E_0, \infty)$, not the whole line. Khalfin's argument (1958), built on the Paley–Wiener theorem of Fourier analysis, says: if $n(E)$ vanishes below $E_0$, then

where: the condition caps how fast $|a|$ may fall. A pure exponential has $|\ln|a|| = \Gamma t/2\hbar$, growing linearly — the integrand falls only like $1/t$ and the integral diverges. So exponential decay cannot persist forever; $|a(t)|$ must eventually fall slower than any exponential. The theorem forbids; the threshold behaviour of $n(E)$ dictates what replaces it.

where: $\alpha$ is the threshold exponent and $\Gamma(1+\alpha)$ is Euler's gamma function (a standard symbol collision with the width — context disambiguates). For the generic 3D density-of-states threshold $\alpha = 1/2$: $|a| \sim t^{-3/2}$, $P \sim t^{-3}$.

How long does the exponential era last? Take the honest model: a Breit–Wigner line truncated at the floor $E = 0$, with $E_R = 10\,\Gamma$. Numerical transform (article verifier): the exponential holds to about 3% at 5 lifetimes; the tail first doubles the exponential prediction at $t \approx 19\,\tau$, and by $40\,\tau$ exceeds it by $\sim4\times10^{8}$ — while absolute survival there is $P \sim 10^{-9}$. The generic verdict: the deviation is mathematically inevitable and experimentally buried under decades of signal loss, which is why it took until 2006 to see.

Green Functions in Quantum Mechanics left a preview parked: real-axis resolvent poles are bound states, the continuum is a branch cut, and a resonance is a pole that has "slipped through the cut". This article completes the sentence. On the physical sheet $\hat G(E)$ is analytic off the axis — a complex pole there would violate unitarity. But continue $\langle\psi|\hat G(E)|\psi\rangle$ through the cut from above, onto the second Riemann sheet, and a Breit–Wigner density delivers a pole at

where: the real part locates the resonance, the imaginary part is half the width. Three faces, one object: in the line shape, a Lorentzian peak of FWHM $\Gamma$ — what a spectrometer sees, and what Resonant Transmission and Tunnelling Resonances fitted from the scattering side; in the time domain, exponential decay $e^{-\Gamma t/\hbar}$ (equation 5); in the resolvent, the second-sheet pole, completing the Green-function article's analytic-structure card. The Gamow vector of Rigged Hilbert Space is the generalized eigenvector attached to this pole — legal on the rigged space's distribution floor, never in the Hilbert space proper.

The why-Lorentzian honesty card. No physical spectrum is exactly Lorentzian: the true $n(E)$ has a floor at $E_0$ (killing the left tail) and finite moments (taming the right). "Lorentzian" means: near resonance, over the window that dominates the transform at intermediate times, the density is well-approximated by (4) — one second-sheet pole close to the axis, far from thresholds. Superb for narrow isolated resonances ($\Gamma \ll E_R - E_0$); and its failures are exactly the other two eras — the floor manufactures the Khalfin tail, the finite moments the Zeno top. Where $\Gamma$ comes from microscopically — the Weisskopf–Wigner construction, Fermi's golden rule — is owned downstream by the time-dependent chapter.

Example 1 — the muon's width. The muon lives $\tau = 2.197\ \mu$s. What energy width does quantum mechanics assign it?

Step 1. The dictionary: $\Gamma = \hbar/\tau = 6.582\times10^{-16}\ \text{eV}\,\cdot\,\text{s}\,/\,2.197\times10^{-6}\ \text{s} = \textbf{3.0} \times \textbf{10}^{-10}\ \textbf{eV}$.

Step 2. Scale it: against the 105.7 MeV rest energy that is a part in $3.5\times10^{17}$ — an extraordinarily narrow resonance; "the muon mass" is well-defined to seventeen places.

Step 3. Half-life: $t_{1/2} = \tau\ln 2 = 1.523\ \mu$s.

Example 2 — the Mössbauer standard. The 14.4 keV level of ⁵⁷Fe has mean life $\tau = 141$ ns. Width and quality factor?

Step 1. $\Gamma = \hbar/\tau = 6.582\times10^{-16}/1.41\times10^{-7} = \textbf{4.7} \times \textbf{10}^{-9}\ \textbf{eV}$.

Step 2. $Q = E/\Gamma = 1.44\times10^{4}/4.67\times10^{-9} = \textbf{3.1} \times \textbf{10}^{12}$.

Step 3. Read it: sharp enough that recoil-free Mössbauer geometry resolves gravitational red-shifts over a building's height (Pound–Rebka rode this line). Line width is lifetime — one measurement, two languages.

References. Sakurai & Napolitano, Modern Quantum Mechanics, 2nd/3rd ed., §5 (time evolution) and resonance discussion · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, complement on unstable states and Fermi's golden rule · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 18 (time-dependent perturbation theory) · Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., decay and golden rule sections · L. A. Khalfin, Sov. Phys. JETP 6, 1053 (1958) — nonexponential long-time decay · L. Fonda, G. C. Ghirardi, A. Rimini, Rep. Prog. Phys. 41, 587 (1978) — decay theory of unstable systems · Misra & Sudarshan, J. Math. Phys. 18, 756 (1977) — quantum Zeno effect. Key identities worth keeping in view: the Lorentzian Fourier transform $\int (\Gamma/2\pi)e^{-iEt/\hbar}/[(E-E_R)^2+(\Gamma/2)^2]\,\mathrm{d}E=e^{-iE_Rt/\hbar-\Gamma t/2\hbar}$ (by residues); $\ddot P(0)=-2(\Delta E)^2/\hbar^2$ (from expanding $|a|^2$); the Paley–Wiener integrand diverges for a pure exponential, establishing Khalfin's theorem; and $\Gamma\tau=\hbar$, $t_{1/2}=\tau\ln2$. Experimental claims (cold-atom Zeno/anti-Zeno: Fischer et al. PRL 87, 040402 (2001); long-time deviation: Rothe et al. PRL 96, 163601 (2006)) are cited as reported.

Builds on — The Time-Evolution Operator · Spectral Decomposition of Time Evolution · Quantum Zeno Effect · Rigged Hilbert Space · Green Functions in Quantum Mechanics. Leads to — Constants of the Motion and Conservation Laws · The Interaction (Dirac) Picture · the golden rule and Weisskopf–Wigner machinery · scattering resonances.