Second Quantization: Momentum, Kinetic & Potential Energies + 2 Particles

[Ene Dene]

Write momentum, kinetic and potential energy, and two particle interaction in second quantization.
That is the question that I need to answer for my exam, but I don't have any idea what second quantization is, except that you can solve harmonic oscilator by using ladder operators. I can't find any connection between that problem and my question.

 

[dextercioby]

Do you know what a Fock space is ?

 

[Ene Dene]

I don't know the formal definition but I know that we work in it when dealing with more than one undistinguishable particles. Than states can be symmetric for bosons and antisymmetric for fermions resulting that only one fermion can be in one state while bosons don't have that restriction.

From harmonic oscilator:

$$a=\sqrt{m\omega/2\hbar}x+ip/\sqrt{2m\omega\hbar}$$
$$a^+=\sqrt{m\omega/2\hbar}x-ip/\sqrt{2m\omega\hbar}$$

from such definition I have:

$$p=i\sqrt{\hbar m\omega/2}(a^+-a)$$ (1)

Would that momentum in second quantization?

If it is, I'm not really satisfied with such "explanation". Why did we choose a and a+ operators in such way? Why is it important?

I asume that then I could write the kinetic and potential energy in such way simply buy putting (1) in hamilton equation?

What about two particle interaction, I still don't have an idea how to write that in 2.quantization.

 

[Ene Dene]

I asume that then I could write the kinetic and potential energy in such way simply buy putting (1) in hamilton equation?

Noo... that wouldn't work H=p^2/2m+V, so I can't express kinetic energy in that way...
So the question stands...

 

[Gokul43201]

Write momentum, kinetic and potential energy, and two particle interaction in second quantization.

Is that the exact question? Please write down the question exactly as it was given to you.

Before you can attempt answering this question, you need to study at least the basics of second quantization. You will find this in any standard many-body theory text (e.g., Fetter & Walecka, Reinhardt & Greiner).

 

[Ene Dene]

That is the exact question.
I looked a few books (read whole Griffiths and most of Shiff) and they all mention lowering and rising operators when it comes to harmonic oscillator. But I haven't found any explanation why are these operators defined in why that they are, and how would I write a general hamiltonian (not just for harmonic oscillator, that I could do) or two particle interaction in terms of them.

 

[dextercioby]

I can only guess that you're not using the right book, or the lecture notes are not that illuminating. See the first chapter of F.Schwabl's book "Advanced Quantum Mechanics", 3rd Ed, Springer Verlag, 2000. You should be able to answer your question after consulting it.

 

[Ene Dene]

OK, I'll try to find that book, at least, now I have some idea where to look.
Thank you.

 

[dextercioby]

Other references:

*Section 17.4 of Ballentine's book :Quantum Mechanics. A modern development.2nd Edition.
*Section 13.6 of Galindo & Pascual's book: Quantum Mechanics (it's in the 2nd volume)
*Section 64 of Landau and Lifschitz' book: Quantum Mechanics. The Nonrelativistic theory.

 

[Ene Dene]

I found the Schwabl book and first chapeter is just what I need. Thank you

 

[azul]

Hello!
Of course Schwabl is very good! but could u help me about second quantization?
i didn't understand the result:
for bosons it was written: Ssum(alpha)of /i><j/(alpha)/i1...iN>*1/(n1!...nN!)^(-1/2)=?
Thanks in advance