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  Extending the Sudden Approximation to 2nd quantised systems?

+ 2 like - 0 dislike
1610 views

Background
---

So basically there is a dictionary/correspondence between first and second quantised quantum mechanical theories. When I suddenly turn on a potential in potential in first quantised theories I can use the sudden approximation and find the probability of the final state. I can use the dictionary to find a result in the second quantised theory as well. My question is what happens when the potential turned on is such that the particle number is not conserved? Is there a way to extend the approximation?

Calculations
---

Let us restrict our discussion to bosons and adopt the convention First Quantised Second Quantised Theory (We are following these Ashok Sen's [Quantum Field Theory I][1] of HRI institute also on my [google drive][2]). Consider a single particle system's hamiltonian ˆh and energy operator ˆE:

ˆhψ=ˆEψ

with energy eigenstates ui and eigenvalues en

ˆhun=enun

Now moving to an assembly of quantum mechanical Hamiltonians (many body system)  ihi with an interacting potential ˆvik corresponds to a second quantised version (Page 16):

ˆHN=Ni=1ˆhi+12Niji,j=1ˆvi,jn=1enanan+12m,n,p,q=1(d3r1d3r2um(r1)un(r2)ˆv12um(r1)up(r2))amanapaq

where en i the nth energy eigevalue,  ui are the one-particle eigenstates and ai is the creation operator of the i'th particle and they obey the energy eigenvalue system hiun=enun. The symmetric wave function for HN corresponds as a second quantised version (Page 6):

un1,n2,nN1N!Permutations of r1,,rNun1(r1)unN(rN)(a1)n1(a2)n2(aN)nN|0

The potential corresponds for one body operator (Page 14):

Ni=1ˆBin,p=1(d3r1un(r1)B1up(r1))anap

Now, let's say I suddenly turn on the one body operator (potential) in the assembly of quantum mechanical systems. Then:

HN=Ni=1ˆhi+12Niji,j=1ˆvi,j+NiˆBi

Let the new energy eigenstates be ˜un with the original state be un1,n2,,nN. Then the final state being in ˜un1,n2,,nN has probability |˜un1,n2,,nNun1,n2,,nNd3r|2. Now using the dictionary we can find the corresponding 2nd quantised state. This is nothing but the **sudden approximation**.

Question
---
Is there a way to extend the sudden approximation to potentials which do not conserve particle number?
 

asked Oct 18, 2020 in Theoretical Physics by Asaint (90 points) [ revision history ]

Yes, there is, and it is the same.

The real problem is not in a final state being represented in terms of (quasi) particles of the new Hamiltonian, but in the degree of suddenness. If the perturbation takes a smal,, but a finite time and if the spectrum of the final Hamiltonian contains high-energy excitations, then for some high-energy excitations (hard modes) the perturbation may be not sudden and the approximation will not be good.

Thanks any reference where I can read more about the second approximation in the context of second quantised systems? 

I do not have any, sorry.

@Vladimir Kalitvianski any idea of a physical system where this might be applicable?

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