Fetter & Walecka's derivation of second-quantised kinetic term....

[nus_phy]

On page 9 of *Quantum theory of many-particle systems* by Alexander L. Fetter and John Dirk Walecka, during the derivation of the second-quantised kinetic term, there is an equality equation below:

>\begin{align}
\sum_{k=1}^{N} \sum_{W} & \langle E_k|T|W\rangle C(E_1, ..., E_{k-1}, W, E_{k+1},...,E_N, t)
\\&=
\sum_{k=1}^{N}\sum_{W}\langle E_k|T|W\rangle\times \bar{C}(n_1, n_2,...,n_{E_k}-1, ..., n_{W}+1,...,n_\infty, t)
\end{align}

Why is the number of particles with quantum numbers n_Ek decreased by 1 whereas the number of particles with quantum numbers n_W increased by 1?

Anybody know how to get this equality?

 

[Charles Link]

This derivation begins basically on p.6, (I also have a copy of Fetter and Walecka), and from what I can tell there are quite a lot of terms in the integral expansion of the product of the wave functions. The $W$ seems to replace the $E_k$ in a particular term. Then they sum over the occupation numbers rather than over particles. I don't have a complete answer for you=Fetter and Walecka's book is one of the more difficult books that I have encountered. We used in a couple of courses in graduate school back in 1978-1980, and it can often take several hours or more to try to work through a couple pages of their derivations. On some pages I was more successful at than others on being able to follow their derivations. I've spent considerable amount of time on the first 250 pages of their 500+ page book=I've studied it at length several times besides the efforts in graduate school, including the section on the Green's function expansions, but I still can't claim much proficiency with it. I think the book contains some very first-rate physics, but working through it is like climbing a mountain.

 

[nus_phy]

This derivation begins basically on p.6, (I also have a copy of Fetter and Walecka), and from what I can tell there are quite a lot of terms in the integral expansion of the product of the wave functions. The $W$ seems to replace the $E_k$ in a particular term. Then they sum over the occupation numbers rather than over particles. I don't have a complete answer for you=Fetter and Walecka's book is one of the more difficult books that I have encountered. We used in a couple of courses in graduate school back in 1978-1980, and it can often take several hours or more to try to work through a couple pages of their derivations. On some pages I was more successful at than others on being able to follow their derivations. I've spent considerable amount of time on the first 250 pages of their 500+ page book=I've studied it at length several times besides the efforts in graduate school, including the section on the Green's function expansions, but I still can't claim much proficiency with it. I think the book contains some very first-rate physics, but working through it is like climbing a mountain.

Exactly! The Green's function expansion of this book is my final target! But now I am stuck in the second quantization derivation.

 

[nus_phy]

This derivation begins basically on p.6, (I also have a copy of Fetter and Walecka), and from what I can tell there are quite a lot of terms in the integral expansion of the product of the wave functions. The $W$ seems to replace the $E_k$ in a particular term. Then they sum over the occupation numbers rather than over particles. I don't have a complete answer for you=Fetter and Walecka's book is one of the more difficult books that I have encountered. We used in a couple of courses in graduate school back in 1978-1980, and it can often take several hours or more to try to work through a couple pages of their derivations. On some pages I was more successful at than others on being able to follow their derivations. I've spent considerable amount of time on the first 250 pages of their 500+ page book=I've studied it at length several times besides the efforts in graduate school, including the section on the Green's function expansions, but I still can't claim much proficiency with it. I think the book contains some very first-rate physics, but working through it is like climbing a mountain.

It seems that I understand the coefficient on the right-hand side. What we want to do is get the coefficient with particle 1 in a certain state E₁, particle 2 in a certain state E₂ , ..., particle N in a certain state E_N, i.e. C(E₁, E₂, ..., E_N, t), which corresponds to $\bar{C}(n_1, n_2, ..., n_{E_k}, ..., n_W,..., n_{\infty}, t)$. Actually $\bar{C}(n_1, n_2, ..., n_{E_k}, ..., n_W,..., n_{\infty}, t)$ defaults the $k$th particle in state $E_k$, while what really happens on the right-hand side is that the $k$th particle is in state W running over all possible quantum states. So n_{Ek-1 in state E_k, n_W+1 in state E_W

 

[Charles Link]

It seems that I understand the coefficient on the right-hand side. What we want to do is get the coefficient with particle 1 in a certain state E₁, particle 2 in a certain state E₂ , ..., particle N in a certain state E_N, i.e. C(E₁, E₂, ..., E_N, t), which corresponds to $\bar{C}(n_1, n_2, ..., n_{E_k}, ..., n_W,..., n_{\infty}, t)$. Actually $\bar{C}(n_1, n_2, ..., n_{E_k}, ..., n_W,..., n_{\infty}, t)$ defaults the $k$th particle in state $E_k$, while what really happens on the right-hand side is that the $k$th particle is in state W running over all possible quantum states. So n_{Ek-1 in state E_k, n_W+1 in state E_W

You seem to have at least a good start to following this one along. I spent about 1/2 hour on this the other day, but was unable to verify for myself that the result of Fetter and Walecka was indeed correct. This one would be perhaps a two or three day project to try to verify completely. Much of their textbook is quite similar. It is a book where you typically work your way through it a little at a time. It is like climbing a mountain and even if you get 1/4 of the way up, it is still a very good accomplishment.

 

[vanhees71]

Unfortunately I have my copy of Fetter&Walecka at the institute. So I can't look at it. In 2nd quantization of a non-relativistic scalar particle the kinetic energy reads
$$\hat{T}=\int_{\mathbb{R}^3} \frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3} \frac{\vec{p}^2}{2m} \hat{a}^{\dagger}(\vec{p}) \hat{a}(\vec{p}),$$
where the annihilation and creation operators are normalized via
$$[\hat{a}(\vec{p}_1),\hat{a}^{\dagger}(\vec{p}_2)]=(2 \pi)^3 \delta^{(3)}(\vec{p}_1-\vec{p}_2).$$
I do not know, what the notation in the OP means.

 

[Charles Link]

Unfortunately I have my copy of Fetter&Walecka at the institute. So I can't look at it. In 2nd quantization of a non-relativistic scalar particle the kinetic energy reads
$$\hat{T}=\int_{\mathbb{R}^3} \frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3} \frac{\vec{p}^2}{2m} \hat{a}^{\dagger}(\vec{p}) \hat{a}(\vec{p}),$$
where the annihilation and creation operators are normalized via
$$[\hat{a}(\vec{p}_1),\hat{a}^{\dagger}(\vec{p}_2)]=(2 \pi)^3 \delta^{(3)}(\vec{p}_1-\vec{p}_2).$$
I do not know, what the notation in the OP means.

The form of the kinetic energy operator, where it is basically like a number operator that destroys a state and then creates the same state, makes it somewhat unclear to me how Fetter and Walecka get their result. In general, Fetter and Walecka's results are quite reliable, so I have to believe that this might be one of those that takes a little bit of work to verify.

 

[nus_phy]

You seem to have at least a good start to following this one along. I spent about 1/2 hour on this the other day, but was unable to verify for myself that the result of Fetter and Walecka was indeed correct. This one would be perhaps a two or three day project to try to verify completely. Much of their textbook is quite similar. It is a book where you typically work your way through it a little at a time. It is like climbing a mountain and even if you get 1/4 of the way up, it is still a very good accomplishment.

Actually, it took me four days to figure it out. You are right. The explanations regarding those equations somehow are not very clearly for me, a novice, which may be easy to understand for higher-leveled guys.

 

[vanhees71]

Now I have Fetter&Walecka in front of me. Obviously that's the cumbersome analysis in first quantization using the Schrödinger picture in terms of single-particle wavefunctions, $\psi_{E_j}(\vec{x})$. At the end that's of course equivalent to the 2nd-quantization formulation supposed that particle number is conserved (which is the case for many non-relativistic problems). This example shows, why you don't want to use the 1st-quantization formulation for the analysis of a many-body system, but you should go through this once since it's very instructive to see why the 2nd-quantization and 1st-quantization formulation are in fact equivalent (as long as particle number is conserved). Despite the merit of being much easier to work with the 2nd-quantization formulation, i.e., the formulation of non-relativistic many-body QFT, it also allows to extend the formalism to cases, where particle numbers are not conserved. This happens in non-relativistic many-body theory when you can treat problems in the quasiparticle picture. In solid-state physics, e.g., a solid can be described as a system of electrons interacting among themselves and with vibrations of the crystal lattice (sound waves), and often this can be formally treated as if the sound-wave modes were Bose particles in terms of annihilation and creation operators in the interaction picture. Then you can develop diagram rules in close analogy to Feynman diagrams in relativsitic QFT. The only thing that changes compared to relativistic are the expressions for the propagators and vertices to their non-relativistic version.