Quantum tunnelling in space vs. time

Question

In the Gamov model of alpha decay they use the WKB approximation to find the magnitude of the stationary state wavefunction of an alpha particle with a given fixed energy $Q$ that has tunnelled through the potential barrier of the nucleus, then say the rate of alpha decay $R = \frac{1}{\tau} = fe^{-2G}$ where $e^{-2G}$ is the probability of tunnelling to a certain position and $f$ is the frequency of the alpha particle 'hitting the side of the nucleus'.

My question is, why does this model work so well? In reality we have an alpha particle wavefunction superposition of many energy eigenstates which dynamically flows from being located within the nucleus to being an approximately free state. The dynamic evolution will depend on the relative phase evolution of the coefficients of each of these energy eigenstates.

Why would some parameter $f$ multiplied by the probability of a stationary state being detected outside the nucleus give a similar result to a time dependent calculation?

This question may relate to how, in general, we can use information from time-independent stationary states to derive information about the time evolution of a dynamic wavefunction.

Answer 1

1. Firstly, we assume for simplicity $s$-waves (zero angular momentum), so that we can replace the 3D geometry with a 1D radial geometry.

2. Secondly, the main point is that the time-independent incoming and outgoing waves inside and outside the barrier can be viewed as approximations to time-dependent wavetrains/wavepackets narrowly peaked in $k$-space (and hence spread out in $r$-space), cf. e.g. this related Phys.SE post.

3. Thirdly, we assume that the usual semiclassical approximations apply, such as, e.g., $T=e^{-2G}\ll 1$.

References:

1. D. Griffiths, Intro to QM, 1995; section 8.2.

Answer 2

TL; DR: the two approaches give the same answer in the long-time limit

When dealing with time-independent perturbations, there is a great deal of similarity between Fermi-Golden-Rile-like calculations and scattering theory. To be more precise, scattering theory calculates the transition probability under the assumption of the adiabatic switching (i.e., pertirbation is absent at $t\rightarrow -\infty$), whereas FGR usually assumes sudden switching (and is usually presented only for time-dependent perturbation and finite orders in perturbation). Both however assume that we look at the result in a long-time limit (i.e., long time after switching on the perturbation in the case of FGR), which makes them mathematically indistinguishable.

Specifically, if we were to consider the spread of a wave packet initially localized at the nucleus, we would expand its wave function in terms of the true (outgoing) eigenstates of the Hamiltonian: $$ |\varphi(0)\rangle = \sum_nc_n(0)|\phi_n\rangle, $$ and apply the time evolution of the expansion coefficients $$ c_n(t)=e^{-iE_n t/\hbar}. $$ at small times the behavior will be very different from the exponential decay, but at long times it will indeed resemble a series of exponential terms.

Furthermore, out of these exponential terms we will choose only the one with the smallest exponent $\Gamma$, since the other decay faster.

Let me further point out that the value of the Gamow's equation is mainly empirical, as predicting the correct functional form of the decay. In many practical situations the coefficients $f$ and $G$ may be not calculated, but taken from the experiment, to better fit the observations.

As a supplementary reading I could recommend the papers by Shmuel Gurvitz, who has been extensively using this approach for describing tunneling in seiconductor quantum dots, e.g., this one. (I recommend looking through more of his papers, since some of them may contain more relevant details.)

Remark: Let me bring up a seemingly very different classical problem of diffusion escape of a particle from a potential well, due to noise/Brownian motion (Kramers rate). Although the setting and the methods of solution (e.g., using Fokker-Planck equation) may appear very different, the long-time approximation resulting in the exponential decay is virtually the same. Gamow's formula is essentially the extension of this result from chemical reactions to nuclear domain.