In section 134 of Vol. 3 (Quantum Mechanics), Landau and Lifshitz make the energy complex in order to describe a particle that can decay:
$$ E = E_0 - \frac{1}{2}i \Gamma. $$
The propagator $U(t) = \exp(-i H t)$ then makes the wavefunction die exponentially with time. But also, $H$ is non-Hermitian.
My question: Do we have to modify the basic postulates of quantum mechanics (as described by Shankar, say, or the earlier sections of Landau & Lifshitz) to describe unstable particles?
We don't have to modify the basic laws of quantum mechanics to describe unstable particles. The full state of the system is includes the state of the decay products, and what you really have is a coupling from one state to another. No imaginary energies are required to describe this, but you do need to include the states of the decay products in your calculation.
This coupling is symmetric (and the total hamiltonian is therefore still hermitian). Still, it is frequently unlikely that the decay products will re-form the original particle because the decay products are usually more than one particle. This means that the entropy of the products is greater than the entropy of the original particle. It is unlikely that this entropy will decrease, so the part of the quantum state that corresponds to the products is in some sense "lost".
Besides the larger state space, the products have less rest mass than the parent particle, which means that for energy to be conserved they have to have more kinetic energy. This makes the product particles fly far away from the location where they formed and from each other; it is unlikely for them to recombine when separated by a great distance.
What Landau is describing is a trick to calculate certain observables without including the dynamics of the decay products. The imaginary part of the Hamiltonian makes the wavefunction decay in a manner similar to a one-way coupling to another state. Since there are many more possible product states, each of them is nearly empty and this is a reasonable approximation.
I think it may also be fruitful to not think of the Hamiltonian as the energy. The Hamiltonian is the generator of time translations, and so an eigenvalue of the Hamiltonian that is complex tells you that there is some decay, as was pointed out by Edoot. This is an important distinction. When we compute the "energy spectrum" of a Hamiltonian, what we are really computing is the frequency spectrum of the time evolution of the system.
One of the better discussions I've seen on HOW you obtain these effective Hamiltonians by directly computing them is in Wen's Quantum Field Theory of Many-Body Systems. In one of the early chapters, he talks about a "quantum RLC circuit". The book is hit or miss, but I think this discussion is clear and a good example of effective field theories.
We don't need modify postulates.
$ E = E_0 - \frac{1}{2}i \Gamma $ only describes partially the system. $x$ particle disappear, and $y$ particle appears (one or more).
Initialy: $E_x = E_0$ and $E_y = 0$
Finally: $E_x = 0$ and $E_y = E_0$
Add $- \frac{1}{2}i \Gamma$ because of we need a decay:
$$ \exp(-iHt) = \exp(-iE_0t)\exp(-\frac{1}{2} \Gamma t) $$
$\exp(-\frac{1}{2} \Gamma t)\rightarrow 0$ fast = particle disappear.
