Electron spin entanglements - time-energy uncertainty

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Ferrus
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I've been watching Susskind's lectures on quantum mechanics and he mentions that the time taken for two electrons to entangle their spin is a function of their distance, which in itself determines the energy that is released when two electrons fall into a singlet state. Does anyone here know how to derive the equations which determine what the expectated value for the time is for a given distance?
 

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Ferrus
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So I have been thinking through this. I am an amateur so bear with me but...

Two opposite spin electrons will have a hamiltonian given by p^2/2m + mu(S1.S2)

Or some equivalent constant in from of that. Clearly there are two energy states referring to opposite and same spin, in fact, the difference should be equal to 2mu(S1.S2). Now, I know a triplet state is made up of:

a[++> + b [+-> + c [-+> + d[--> s.t. the absolute value squared of all is equal to 1.

And the entangled singlet state is:

1/Sqrt(2)([-+> - [+->)

So, is it the case that the Hamiltonian's energy is depedent on the probability amplitudes of the triplet state?

And then, how does one generate the probability amplitudes of each individual state.

I know each electron can be modelled as a gaussian wavepacket in r^3, is it the case that one simply uses the energy eigenbras on the ket of the wave function for both states and then deduces relative probability amplitudes (normalised in both cases) from both?

Sorry for the confusion, I have only really started to learn this in the last week or so.
 
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