# Can Half Lives (hypothetically) be Measured by Wave-functions?

I understand that half-lives are measured over several days/months/years of observing a certain amount of an element and seeing how long it takes to decay a certain amount, but I'm curious as to whether or not knowing the wavefunctions of the particles would give us a more accurate half life, given the fact that quantum tunneling is responsible for $\alpha$-decay.

• The wavefunction of quantum mechanics is not an observable. – ACuriousMind Oct 29 '14 at 18:06
• Yes, I'm aware of this :) It's more of a thought experiment. – Goodies Oct 29 '14 at 18:07
• @Goodies: If you change "measuring the wavefunctions" to "knowing the wavefunctions" would it still capture the essence of your question? – BMS Oct 29 '14 at 18:18
• Half-lives can be computed in certain models by considering the wavefunction, but wavefunctions are not themselves observable: you don't (can't) measure them. – dmckee Oct 29 '14 at 18:18
• @BMS Yes, that is a more accurately worded form. – Goodies Oct 29 '14 at 18:28

If we were able to have a quantum mechanical model for nuclei that needed no experimental input, then the half lives of unstable nuclei would be computed utilizing the fully known wavefunctions.

An approximation to this ideal is the shell model, and there are papers in the literature using the wavefunctions of the model to calculate lifetimes that fit the data. Still, searching the literature the whole field heavily depends on experimental data. An example is here

We have performed calculations of all the involved wave functions by using the nuclear shell model with the GXPF1A interaction in the full f-p shell. The computed half-life of the EC branch is in good agreement with the measured one.

• What, exactly, is the half life contingent on if not the wavefunction? – Goodies Oct 29 '14 at 20:33
• this becomes philosophy. Physics is the process of gathering observations on one hand and creating mathematical models which and some postulates which prescribe how the solutions will relate to the data. Half lives have been observed experimentally by fitting the decay distributions with simple exponential functions and calculating how much time it will take to lose half of the sample. One does not have to wait for hours or days or years, a high statistics curve will give an accurate lifetime even with small time interval. The exponential is the simplest mathematical model, and nobody would – anna v Oct 30 '14 at 4:56
• say that the lifetime is "contigent" to the exponential in any existential way. Wave functions are much more complicated mathematical models that have been arrived at by inspiration while fitting experimental data, and wavefunctions squared will give the lifetime (i.e. exponential functions) but as I say in my answer there does not exist a complete mathematical model from which all observations can be predicted. Suppose it existed , it is there that philosophy comes in : mathematics creates reality ( platonic ideas) or is a useful tool to model reality? – anna v Oct 30 '14 at 5:02

The half life can in principle be determined using Fermi's Golden Rule. Well, this calculates transition probabilities per unit time, but the half life is simply derived from the transition probability. So if you know the initial and final wavefunctions and the appropriate operator then yes you can calculate the half life.

However in practice nuclei are far too complicated to do this except in special cases and/or in an approximate way. It works very well in simple cases like calculating the half life of excited states of atoms.

• It is worth noting that Fermi's Golden Rule is itself not exact, but only a first-order perturbative result. – ACuriousMind Oct 30 '14 at 13:20