Background
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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
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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=N∑i=1ˆhi+12N∑i≠ji,j=1ˆvi,j↔∞∑n=1ena†nan+12∞∑m,n,p,q=1(∫∫d3r1d3r2um(→r1)∗un(→r2)∗ˆv12um(→r1)up(→r2))a†ma†napaq
where en i the nth energy eigevalue, ui are the one-particle eigenstates and a†i 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,…nN≡1√N!∑Permutations of r1,…,rNun1(→r1)…unN(→rN)↔(a†1)n1(a†2)n2…(a†N)nN|0⟩
The potential corresponds for one body operator (Page 14):
N∑i=1ˆBi↔∞∑n,p=1(∫d3r1u∗n(→r1)B1up(→r1))a†nap
Now, let's say I suddenly turn on the one body operator (potential) in the assembly of quantum mechanical systems. Then:
H′N=N∑i=1ˆhi+12N∑i≠ji,j=1ˆvi,j+N∑iˆBi
Let the new energy eigenstates be ˜un with the original state be un1,n2,…,nN. Then the final state being in ˜un′1,n′2,…,n′N has probability |∫˜un′1,n′2,…,n′Nu∗n1,n2,…,nNd3r|2. Now using the dictionary we can find the corresponding 2nd quantised state. This is nothing but the **sudden approximation**.
Question
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Is there a way to extend the sudden approximation to potentials which do not conserve particle number?