Identical Particles in Ballentine's Book

[blue_leaf77]

I hope somebody is familiar with the discussion on the identical particles in Quantum Mechanics by L. Ballentine. In particular can someone help me explain how the author derive equation (17.44) from (17.41). In case your edition is different from mine, equation (17.44) is the one which looks like

$V = \frac{1}{4} \Sigma_\alpha \Sigma_\beta \Sigma_\gamma \Sigma_\delta <\alpha \beta|\upsilon|\gamma \delta> C_\alpha^+ C_\beta^+ C_\delta C_\gamma$

I have tried using the relations appearing in between those two equations mentioned above but nothing good coming out. And final question, do you know whether this formalism of creation and annihilation operators introduced in the corresponding chapter in this book will find frequent use in the typical discussions of many electron atom/molecule, that is bound systems? I don't want spend too much time on a subject which I probably encounter so often.

 

[DrDu]

This is often called the "second quantization" representation. It is fundamental in many body physics, i.e. solid state physics. It is also used in molecular physics though not so intensively. It is used exclusively in relativistic quantum field theory. So you better know how to use it.

In molecular physics, it helps already a lot, as you don't have to bother to expand Slater determinants.

 

[blue_leaf77]

Thanks DrDu, you gave me a motivation, which means I really have to understand how to derive the above equation. So how?

 

[DrDu]

I don't have Ballentine, so I can't help you with the details. Sorry.

 

[Demystifier]

Thanks DrDu, you gave me a motivation, which means I really have to understand how to derive the above equation. So how?

For this stuff, there are much better books than Ballentine.

A good example is
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

If you want something free, I recommend Simons:
http://www.tcm.phy.cam.ac.uk/~bds10/tp3.html (Lec.5)

 

[blue_leaf77]

Thanks for the references Demystifier, I think I will check the 2nd one.

 

[vanhees71]

And as we are at the topic: Although I'm not an English native speaker (but it's the same issue also in my mother tongue German), I'd never ever call it "identical particles" but "indistinguishable" particles. That's because, to my understanding, if you say "one particle is identical with a particle", you've just one particle, but what's intended is "indistinguishable particles", i.e., you have two or more particles, which are indistinguishable in a very specific sense of quantum theory, and this can be expressed only with the mathematics used to express quantum theory correctly.

The heuristic argument goes as follows: You start with classical mechanics and a point-particle model to describe a many-body system. In classical mechanics, you may have many particles with precisely the same intrinsic specifications (mass, electric charge, and other charge-like specifications identifying the type of particle described uniquely), but but in principle you can still distinguish each individual particle within the many-body system. That's, because in classical mechanics a state at a certain instant of time is a point in phase space, $(q,p) \in \mathbb{R}^{6N}$, where $N$ is the number of particles. So you can label each particle, in addition to its intrinsic specification of their type, with their positions and (canonical) momenta at some initial time $t_0$. Then the time evolution is governed by a deterministic Law, Hamilton's canonical equations, which gives a mapping from the initial phase-space positions of each particle to their phase-space positions at any later time $t$. Thus, although of the same type (determined by the intrinsic specifications) each individual particle is specified by its initial position in phase space.

This is no longer true in quantum theory, where you describe the state by a statistical operator (a complete specification of the state makes it a projection operator to the corresponding one-dimensional subspace of Hilbert space (ray) specifying the pure state), and the physics content of this state is probabilistic. Then Heisenberg's uncertainty relation forbids to specify position and momentum already at the beginning of the time evolution exactly. Within the time evolution generally this implies that, even if you have specified positions and momenta at the initial time in a way that the particles are for all practical purposes individually separated in phase space, i.e., that the probability to find two particles in the same phase-space region (specified with some accuracy way less stringent than the Heisenberg uncertainty relation, i.e., with "macroscopical accuracy") is practically zero, that's usually no longer the case after some time, but then it's no longer possible to individually distinguish any particle from another particle within the same macroscopic phase-space cell with the same intrinsic specifications (i.e., the same intrinsic quantum numbers like mass and various charges).

This in turn has profound consequences for the entire structure of the mathematical description of such indistinguishable particles in quantum theory. The impossibility to dinstinguish particles with identical intrinsic quantum numbers (here the word "identical" is really justified, and it's identity never achievable with macroscopic bodies, because an electron is always very precisely and electron with a very specific mass, electric charge, color 0 (lepton) and flavor) means mathematically that for a system of $N$ indistinguishable particles there must be a representation of the symmetric group $S_N$ (the group of permutation of $N$ entities). Each of the unitary operators representing this group must commute with the Hamiltonian, because otherwise the interaction among particles would make them distinguishable.

There are many representations of the $S_N$ (it's in fact a very important part of representation theory of finite groups). Now nature was kind to us physicists and made it easy for us. At least if the effective spatial degrees of freedom are $\geq 3$, from very general considerations, only the two most simple representations of the $S_N$ come to application. For details, see

Laidlaw, M. G. G., DeWitt, Cécile Morette: Feynman Functional Integrals for Systems of Indistinguishable Particles, Phys. Rev. D 3, 1375, 1970
http://link.aps.org/abstract/PRD/v3/i6/p1375

The only representations are given by the subspaces of the direct product $\otimes^{N} \mathcal{H}$ of single-particle Hilbert spaces $\mathcal{H}$ which are spanned by the completely symmetrized (bosons) or completely anti-symmetrized (fermions) product states made of $N$ single-particle basis states $|u_n \rangle$, i.e.,
$$|u_{n_1},u_{n_2},\ldots,u_{n_N} \rangle^{\pm} = \sum_{P \in S_N} (\pm 1)^{\sigma(P)} |u_{n_{P(1)}} \rangle \otimes |u_{n_{P(2)}} \rangle \otimes \cdots \otimes |u_{n_{P(N)}} \rangle.$$
As long as you work in non-relativistic quantum theory in an application where the total number of particles is conserved, you can deal with this "first-quantization description", but it's not so easy to keep track of all the combintatorics with the symmetrized or antisymmetrized state kets (note that the most general ket is a superposition of any of these totally symmetrized or antisymmetrized products of $N$ single-particle basis states).

To make life easier and also to generalize everything to cases, where the particle number is not conserved (and to cases, where the "particles" are not real particles but collective excitations describable as "quasi particles" in condensed-matter physics (like phonons is solids) or in relativistic quantum theory, where one destroys and creates particles in collsions at relativistic energies all the time), one uses what's named "second quantization" or (better named) "quantum field theory". You can do quantum field theory in both non-relativistic and relativistic physics. For the case that the interactions in non-relativistic QT are such that the total number of particles is conserved, the quantum-field formalism is equivalent to the above description. In the case of relativistic QT, there are no interactions known which keep the total particle number conserved (except in the low-energy regime, where a non-relativistic approximation is usually valid), and you must use QFT (or very cumbersome constructions like Dirac's hole theory, which is in fact equivalent to modern QED but to my knowledge not possible to extend to the other more complicated interactions of the Standard Model).

The trick in QFT is that you deal with bosonic or fermionic Fock spaces, which are formally the direct orthogonal sum of $N$-body bosonic or fermionic Hilbert spaces:
$$\mathcal{H}_{\text{Fock}}=\oplus_{N=0}^{\infty} \mathcal{H}^{(N)\pm}.$$
Then you can define creation operators $\hat{a}_j^{\dagger}$ and annihilation operators $\hat{a}_j$ (which are usually hermitean conjugate to each other, as written here). These have the commutator (bosons) or anti-commutator (fermions)
$$[\hat{a}_j,\hat{a}_k]_{\pm}=0, \quad [\hat{a}_j,\hat{a}_k^{\dagger}]_{\pm}=\delta_{jk}.$$
If the system has a non-degenerate ground state (i.e., no spontaneous symmetry breaking involved) then the vacuum state ("no particles") is specified uniquely by
$$\hat{a}_j |\Omega \rangle=0$$
for all $j$, and the $N$-particle basis is given by
$$|u_{n_1},u_{n_2},\ldots,u_{n_N} \rangle = \prod_{k=1}^N \hat{a}_{n_k}^{\dagger} |\Omega \rangle.$$
At the same time you can express your Hamiltonian and other observables in terms of annihilation and creation operators and evaluate expectation values, scattering matrix elements and whatever you need to describe your physical process. Last but not least, all this is expressible in terms of Feynman diagrams!

My favorite book for the classical case, using the operator technique, is

Fetter, Alexander L., Walecka, John Dirk: Quantum Theory of Many-Particle Systems, McGraw-Hill Book Company, 1971

for the path-integral formulation, see

Altland, A., Simons, B.: Condensed Matter Field Theory, 2 edition, Cambridge University Press, 2010

and last but not least, the real-time formalism (Schwinger-Keldysh contour)

Rammer, J.: Quantum Field Theory of Non-equilibrium States, Cambridge University Press, 2007

 

[blue_leaf77]

My third gratitude in this thread goes to you vanhees, although your explanation is not directly related to the problem I'm facing but you have given me a more fundamental view of the reason particles in quantum world with the same intrinsic properties are indistinguishable.

 

[samalkhaiat]

I hope somebody is familiar with the discussion on the identical particles in Quantum Mechanics by L. Ballentine. In particular can someone help me explain how the author derive equation (17.44) from (17.41). In case your edition is different from mine, equation (17.44) is the one which looks like

$V = \frac{1}{4} \Sigma_\alpha \Sigma_\beta \Sigma_\gamma \Sigma_\delta <\alpha \beta|\upsilon|\gamma \delta> C_\alpha^+ C_\beta^+ C_\delta C_\gamma$

You have the 2-body interaction operator $$V = \frac{1}{2} u_{j k , m n} C^{\dagger}_{j} C^{\dagger}_{k} C_{n} C_{m} . \ \ \ \ \ (1)$$ Since all indices are dummy, i.e. summed over, we can rewrite (1) as $$V = \frac{1}{2} u_{j k , n m} C^{\dagger}_{j} C^{\dagger}_{k} C_{m} C_{n} . \ \ \ \ \ (2)$$ Using the anti-commutation relation $C_{m} C_{n} = - C_{n} C_{m}$, we can rewrite (2) as $$V = - \frac{1}{2} u_{j k , n m} C^{\dagger}_{j} C^{\dagger}_{k} C_{n} C_{m} . \ \ \ \ (3)$$ Now add (1) to (3) and use the anti-symmetrized 2-body matrix element $$u_{j k , m n} - u_{j k , n m} = \langle j k | u ( |x_{1} - x_{2}| ) | m n \rangle .$$

 

[bhobba]

You have some excellent answers here - I don't think there is anything I can really add.

But as an interesting aside what you are taking about is actually an alternative formulation of QM:
http://susanka.org/HSforQM/[Styer]_Nine_Formulations_of_Quantum_Mechanics.pdf

Thanks
Bill

 

[Swamp Thing]

bhobba, I am looking at Styer et al.

This sign adjustment^* is not difficult for humans, but it poses a significant challenge—know as ‘‘the fermion sign problem’’—for computers. This important standing problem in quantum Monte Carlo simulation is discussed in...

* i.e. you multiply by -1 whenever two possible paths swap their destinations

Why is it hard for a computer? ..

 

[bhobba]

Why is it hard for a computer? ..

Don't know that one - sorry.

But as always - Google is your friend.

Thanks
Bill

 

[blue_leaf77]

You have the 2-body interaction operator $$V = \frac{1}{2} u_{j k , m n} C^{\dagger}_{j} C^{\dagger}_{k} C_{n} C_{m} . \ \ \ \ \ (1)$$ Since all indices are dummy, i.e. summed over, we can rewrite (1) as $$V = \frac{1}{2} u_{j k , n m} C^{\dagger}_{j} C^{\dagger}_{k} C_{m} C_{n} . \ \ \ \ \ (2)$$ Using the anti-commutation relation $C_{m} C_{n} = - C_{n} C_{m}$, we can rewrite (2) as $$V = - \frac{1}{2} u_{j k , n m} C^{\dagger}_{j} C^{\dagger}_{k} C_{n} C_{m} . \ \ \ \ (3)$$ Now add (1) to (3) and use the anti-symmetrized 2-body matrix element $$u_{j k , m n} - u_{j k , n m} = \langle j k | u ( |x_{1} - x_{2}| ) | m n \rangle .$$

You have finally ended my two days of restlessness, thanks a lot for this help.

 

[strangerep]

[Samalkhaiat has] finally ended my two days of restlessness, [...]

Ah, that was vintage Samalkhaiat. Paraphrasing Dextercioby: "he doesn't post very often, but when he does it really makes a difference".

 

[samalkhaiat]

Ah, that was vintage Samalkhaiat. Paraphrasing Dextercioby: "he doesn't post very often, but when he does it really makes a difference".

Hahaha, thanks a lot. I hope that does not imply the statement “Samalkhaiat is an old man”