![]() ![]() Wave function we must add together “plane waves in time” e − i E t / ℏ corresponding to different energies. Now, in the present situation the wave function decays in time rather than space, but the argument Wave packets reaches essentially the same conclusion. ![]() The more precise derivation based on Gaussian Waves, e i p x / ℏ and e i ( p + Δ p ) x / ℏ , and noticing that they fall out of phase inĪ distance Δ x ∼ ℏ / Δ p. Spread in momentum space Δ p turns out to be given by taking just two Of p components, but the right expression for the A true localized wave packet has a continuum Were necessary in order for the wave packet to die away from its center over aĭistance of order Δ x. Recall we introduced the p, x uncertainty principle by finding what spread Δ p in the Fourier components of a wave packet Ψ ( t ) = ψ ( 0 ) e − i ( E 0 − i Γ / 2 ) t / ℏĪt this point, the analogy with Δ p ⋅ Δ x ≥ ℏ emerges. The time dependences together in one exponential factor: This is a far slower time dependence than that of the e − i E 0 t / ℏ term, so it is an excellent approximation to put The same shape, but gradually decrease in amplitude: The wave function inside the well will stay Particle in the well is decreasing with time, the time dependence of ψ can no longer be just the e − i E 0 t / ℏ of the original “bound state”. Obviously, if the modulus of the wave function of the Period - the half-life - fixed for that nucleus. The exponential decay law for radioactive elements isĬompletely confirmed experimentally, it is the basis of the “half-life” rule:įor any given amount of a radioactive nucleus, half of it will decay in a time In other words, Γ / ℏ is an inverse lifetime. The standard notation is to introduce a variable Γ = ℏ ε / τ having the dimensions of energy ( ε being dimensionless), in terms of which Write P as a function of time by using the formula e − x = lim ε → 0 ( 1 − ε ) x / ε.įrom this, the probability of the particle being in the well Small for α -decay (less than 10 -12), we can conveniently Is still in the well after a time t = n τ is P ( n τ ) = ( 1 − ε ) n. Therefore, the probability that the particle Walls backwards and forwards inside, time τ between hits, and at each hit probability of Particle escaping - so no longer a true bound state, but for a thick barrierĪs with the α -decay analysis, we’ll look at this semi-classically, picturing the particle as bouncing off the True bound state having energy E 0 , and for E 0 well below V 0 , having approximately an integral number ofīarrier of finite thickness, there is of course some nonzero probability of the If the barrier thickness were increased to infinity (keeping High enough and thick enough that there is a small probability per unit time of To illustrate the meaning of the equation Δ E ⋅ Δ t ≥ ℏ, let us reconsider α -decay, but with a slightly simplifiedĬombined nuclear force/electrostatic repulsion barrier with a square barrier, Kind of relationship to the momentum-position one, because t is not a dynamical variable, so this can’t have Evidently, though, this must be a different The momentum-position uncertainty principle Δ p ⋅ Δ x ≥ ℏ has an energy-time analog, Δ E ⋅ Δ t ≥ ℏ. ![]() The Energy-Time Uncertainty Principle: Decaying States and Resonances ![]()
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