Question about a passage in Blundell's QFT

• I
• WWCY
In summary, the second-quantised interaction potential in a momentum-space representation is defined as:1) The usual expression is ##V(\vec{x},\vec{y})=V(\vec{x}-\vec{y})##, because you want to fulfill Newton's 3rd Law, i.e., if there's an interaction and particle 2 excerts a force ##\vec{F}_{12}=-\vec{\nabla}_1 V(\vec{x}_1,\vec{x}_2)## then necessarily particle 1 excerts a force on particle 2, ##\vec{F}_{21}=-\vec{F}_{12}=-
WWCY
TL;DR Summary
On the second quantisation of potential term associated with inter-particle interactions
In Bundell's QFT book, the second-quantised, pairwise interaction potential was defined as:

in a later step, this was re-written in a momentum-space representation

1. I don't understand why it goes from ##V(x, y)## to ##V(x-y)##, why do we consider only this form of interaction potentials?

2. Also, could someone show how the expression for ##\hat{V}## derive from the general form of the second-quantised two particle operator below?

ad 1) The usual expression is ##V(\vec{x},\vec{y})=V(\vec{x}-\vec{y})##, because you want to fulfill Newton's 3rd Law, i.e., if there's an interaction and particle 2 excerts a force ##\vec{F}_{12}=-\vec{\nabla}_1 V(\vec{x}_1,\vec{x}_2)## then necessarily particle 1 excerts a force on particle 2, ##\vec{F}_{21}=-\vec{F}_{12}=-\vec{\nabla}_2 V(\vec{x}_1,\vec{x}_2).##
So you have
$$\vec{\nabla}_1 V(\vec{x}_1,\vec{x}_2)=-\vec{\nabla}_2 V(\vec{x}_1,\vec{x}_2).$$
Now introduce center-of-mass and relative coordinates for the particle pair,
$$\vec{R}=\frac{1}{2} (\vec{x}_1+\vec{x}_2), \quad \vec{r}=\vec{x}_1-\vec{x}_2.$$
and consider ##V## as a function of ##\vec{R}## and ##\vec{r}##. Then you have
$$\partial_{1j} V= \frac{\partial R_k}{\partial x_{1j}} \frac{\partial}{\partial R_k} V + \frac{\partial r_k}{\partial x_{1j}} \frac{\partial}{\partial r_k} V=\frac{1}{2} \frac{\partial}{\partial R_k} V +\frac{\partial}{\partial r_k} V$$
and in the same way
$$\partial_{2j} V=\frac{1}{2} \frac{\partial}{\partial R_j} V -\frac{\partial}{\partial r_j} V.$$
Now Newton's 3rd law tells us that
$$\partial_{1j} V + \partial_{2j} V=0 \; \Rightarrow \; \frac{\partial}{\partial R_j} V=0 \; \Rightarrow \; V=V(\vec{r})=V(\vec{x}_1-\vec{x}_2).$$
And that's what you plug into your Hamiltonian. Note that the factor 1/2 comes from overcounting the sum over all particles (in the field formalism that's the integration over ##\vec{x}## and ##\vec{y}##) since the total potential energy of one pair has to count only once and not twice. So summing over all particle indices gives a factor 2 too large contribution of the two-body contributions. And that's why you have to compensate by the factor ##1/2##.

ad 2) Just go on with the calculation as it stands! Obviously the author wisely uses first a quantization volume (with single-particle wave functions periodic, leading to discrete momenta ##\vec{p} \in \frac{2 \pi}{L} \mathbb{Z}^3##, where ##L## is the length of the cube making up the quantization volume). Then you get
$$\hat{V}=\frac{1}{2 \mathcal{V}} \sum_{\vec{p}_1,\ldots \vec{p}_4} \hat{a}_1^{\dagger} \hat{a}_2^{\dagger} \hat{a}_3 \hat{a}_4 \int_{\mathbb{R}^3} \mathrm{d}^3 x \int_{\mathbb{R}^3} \mathrm{d}^3 y \exp[\mathrm{i} \vec{x}\cdot (\vec{p}_4-\vec{p}_1) + \mathrm{i} \vec{y} \cdot (\vec{p}_3-\vec{p}_2)] V(\vec{x}-\vec{y}).$$
From this you get
$$V_{1234}=\int_{\mathbb{R}^3} \mathrm{d}^3 x \int_{\mathbb{R}^3} \mathrm{d}^3 y \exp[\mathrm{i} \vec{x}\cdot (\vec{p}_4-\vec{p}_1) + \mathrm{i} \vec{y} \cdot (\vec{p}_3-\vec{p}_2)] V(\vec{x}-\vec{y}).$$
Now you should go on and use that ##V## depends only on the relative coordinates ##\vec{r}=\vec{x}-\vec{y}## to simplify the matrix elements.

Hint: Introduce ##\vec{R}## and ##\vec{r}## as integration variables (note that you need the Jacobian of the corresponding coordinate transformation!).

WWCY
Cheers, I think i get it now. I'll try and work through the details!

1. What is Blundell's QFT?

Blundell's QFT refers to the textbook "Quantum Field Theory" written by Stephen Blundell. It is a comprehensive introduction to the mathematical and theoretical foundations of quantum field theory, a branch of physics that describes the behavior of subatomic particles and their interactions.

2. Who is Stephen Blundell?

Stephen Blundell is a Professor of Physics at the University of Oxford. He is an expert in condensed matter physics and has authored several textbooks, including "Quantum Mechanics" and "Concepts in Thermal Physics".

3. What is the purpose of studying quantum field theory?

The main purpose of studying quantum field theory is to understand the fundamental laws that govern the behavior of particles at the subatomic level. It is also essential for developing theories and models to explain phenomena such as particle interactions, quantum entanglement, and the behavior of matter at extreme conditions.

4. Is Blundell's QFT suitable for beginners?

While Blundell's QFT is a comprehensive textbook, it is not recommended for beginners. It assumes prior knowledge of quantum mechanics, special relativity, and classical mechanics. It is better suited for advanced undergraduate or graduate students in physics.

5. Are there any resources available to supplement Blundell's QFT?

Yes, there are various resources available to supplement Blundell's QFT, such as lecture notes, online tutorials, and problem sets. Additionally, the textbook itself contains numerous exercises and examples to aid in understanding the material.

Replies
13
Views
2K
Replies
18
Views
4K
Replies
4
Views
1K
Replies
15
Views
2K
Replies
12
Views
2K
Replies
4
Views
2K
Replies
2
Views
1K
Replies
2
Views
1K
Replies
2
Views
1K
Replies
87
Views
6K