Calculating group representation matrices from basis vector/function

[pellis]

TL;DR Summary

This is a carefully-worded question, from a chemist, about calculating group representation matrices from basis vectors/functions, both in real space and, correspondingly, for the related Hilbert space of solutions.

Being myself a chemist, rather than a physicist or mathematician (and after consulting numerous sources which appear to me to skip over the detail):

1) It’s not clear to me how one can go generally from a choice of basis vectors in real space to a representation matrix for a spatial symmetry group (ignoring the later issue of finding irreps).

Let’s consider a rigid pyramidal ammonia molecule (ignoring internal dynamics, such as vibration modes, etc.) whose symmetry group is C3v. [NB The C3v case is simple enough that the matrix reps can be identified without using a general procedure – but it’s the general procedure this question is asking about]

Presumably, one can choose a real-space basis a-priori (orthogonal unit vectors along the three axes with z vertical; or, better perhaps, two unit vectors at 120-degrees plus the vertical z-unit vector).

How does one then calculate the representation matrix for each symmetry operation of the group?

This is, presumably, pure classical physics – no quantum mechanics required.

2) Having subsequently solved some semi-empirical approximations to the Schrödinger equation for the ammonia molecule, one obtains a set of molecular orbitals (eigenfunctions) in a Hilbert space.

The eigenfunctions should display the same symmetries as the classical molecule (and, anyway, we’ve likely used the ‘classical’ spatial symmetries already, in setting up the calculation).

If we now choose the molecular orbital eigenfunctions/eigenvectors that form a basis in Hilbert space as basis functions/vectors for symmetry purposes, then:

How does one then calculate the representation matrix for each symmetry operation of the group using the molecular orbital basis?

(and in general, how do the resulting eigenvectors (molecular orbital functions) and representations of the symmetry operations relate to those in the answers to (1) above?

NB This is NOT a question about irreducible representations, but about the stage before they are considered.

An equally carefully-worded reply will be greatly appreciated.

Thank you for reading this far, and Happy New Decade to all.

 

[vanhees71]

I'm not sure that I understand the question correctly. First of all a group representation in the sense usually used in quantum theory is homomorphism from the group to be represented to the group of unitary (or anti-unitary) automorphisms on Hilbert space. Given a basis in Hilbert space leads to a representation in terms of matrices (finite- or infinite-dimensional depending on which (sub-)space you define the representation) or in terms of integration kernels (if there are "generalized bases" with continuous "eigenvalues").

Let's consider the case, where the group can be represented by a unitary representation (the case of anitunitary transformations occurs in practice only when considering time-reversal symmetry). If you have some group $G$ with elements $g$ the mapping is $\hat{U}:G \rightarrow U(\mathcal{H})$, where $U(\mathcal{H})$ is the group of unitary automorphisms on the Hilbert space, this mapping is a representation if $\hat{U}(g_1) \hat{U}(g_2)=\hat{U}(g_1 g_2)$.

If you have a complete orthonormal set vectors $|n \rangle$ then the matrix elements of the unitary operators are
$$U_{n_1n_2}(g)=\langle n_1 |\hat{U}(g)|n_2 \rangle.$$
For your concrete example you need the representation of the rotations and reflections making up the group $C_{3v}$. To get the corresponding matrix elements you only need to know, how your wave functions defining the basis in the position representation transform under rotations and reflections. This can be found in any standard textbook on quantum mechanics, e.g.,

J.J. Sakurai, Modern Quantum Mechanics, Addison-Wesley

 

[pellis]

Thankyou for your reply, vanhees71

“I'm not sure that I understand the question correctly.”

I was hoping to ask a fairly simple question at a rather general level, to try to plug the gaps in my available texts. In one text I see the three matrix reps of symmetry operations for an object (not necessarily a quantum object – no wave functions mentioned) with C3v symmetry, but no immediate explanation of how the matrix reps were generated – until I found an appendix, a short while ago, that provides the answer, by working out, for the basis vectors listed in character tables, matrices showing how points on the x,y and z axes, and rotations around each axis, transform under each sym op (though I don’t see the exact reason for choosing the quadratic terms listed in a second column of character tables).

For the quantum case, for molecular symmetry, I take it that “To get the corresponding matrix elements you only need to know, how your wave functions defining the basis in the position representation transform under rotations and reflections” means simply: see how the orbitals (as functions of space and spin) transform under rotations and reflections, by the method you outlined

Thanks and regards - P

 

[PeterDonis]

In one text

Which text? Please give a specific reference.

 

[pellis]

Reply to PeterDonis:

The text is Eyring, Walter & Kimball's Quantum Chemistry , 1944 (sic!); Ch 10, which left me puzzled as regards constructing mat reps without using QM, as do other sources; but I found an Appendix on pp. 376-388, which looks like it will answer my first part - I'll need to work through a couple examples. (It's an early classic.)

But I suspect the 4th ed. of Peter W Atkins & Ronald Friedman's Molecular Quantum Mechanics (2010), a PDF of which I've found online, might also have the detail I sought. I have a hardcopy of the 1st ed., by PWA alone, but that's harder going. (I've done more searches since I wrote the post, so my concern will shift to better understanding of approaches for the QM cases, from the 2nd part of my original post).

Ultimately (longer term), I'm interested to work towards achieving some sort of intuitive appreciation of the meaning of the phrase "An elementary particle is an irreducible representation" (Poincare'/Wigner etc.). For example, how does Wigner's Classification deal with internal symmetries such as isospin, color, flavor etc., when it's supposed to only cover mass and spin... (I've read about the Coleman Mandula Theorem, but haven't reached that level yet). Should Wigner's Classification encompass Dark Matter?

Thanks for your interest

 

[vanhees71]

The Wigner classification in HEP QFT only deals with the necessary constraints implied by the symmetries of Minkowski space. For a relativistic QT to be consistent with this relativsitic spacetime model it must necessary be symmetric under proper orthochronous Poincare transformations, and from this assumption alone you already get very general features about particles. If you assume on top that the Hamiltonian must be bounded from below, i.e., that there's a stable ground state and that relativistic QT should be expressed as a local (microcausal) QFT, which is the only successful kind of relativistic QT known today, you find out that (a) for any sort of particle there must be a corresponding antiparticle (which can also be the same as a particle, which is called a strictly neutral particle then), (b) there are massive particles and massless particles; tachyons don't work for interacting theories; (c) massive and massless particles can have additional intrinsic degrees of freedom, which are the spin in the massive case and the helicity in the massless case; spin and helicity come in integeger and half-integer values (it's similar to angular momentum in non-relativistic QT); (d) half-integer spin/helicity particles are necessarily fermions, integer spin/helicity particles are necessarily bosons; (e) The "grand reflection" PCT is also a symmetry, i.e., the combination of the three discrete symmetries involving spatial reflection, charge conjugation (exchanging each particle with its antiparticle), and time-reversal, is a symmetry, though P, C, T, CP,... do not need and in nature in fact are not symmetries.

The other (exact or approximate) symmetries you mentioned come on top to these spacetime symmetries. They are inferred from observed conservation laws or symmetries like the mass pattern found in the zoo of hadrons (for the approximate strong isospin symmetry in the light-quark sector of QCD).

 

[pellis]

Thank you for picking up on the Wigner Classification (W.C.) point.*

All of your first para is familiar (conceptually, if not yet – to me – in its full mathematical glory**), except for the underlying justification in terms of the boundedness of Hamiltonians, for which thanks. ***

Carelessly, perhaps, I long ago gained the false impression that W.C. covers all possible fundamental particles:

“Ever since the fundamental paper of Wigner on the irreducible representations of the Poincaré group, it has been a (perhaps implicit) definition in physics that an elementary particle ‘is’ an irreducible representation of the group, G, of ‘symmetries of nature’ (Ne’eman and Sternberg 1991, pp. 327.) ”

But if I correctly understand your final paragraph:

“The other (exact or approximate) symmetries you mentioned come on top to these spacetime symmetries. They are inferred from observed conservation laws or symmetries like the mass pattern found in the zoo of hadrons (for the approximate strong isospin symmetry in the light-quark sector of QCD).”

… it means that properties such as isospin, color and flavour etc – being other that mass and spin/helicity – are not taken into account by W.C., which is … disappointing.

But that raises the question of why, in principle, does WC take account of one internal degree of freedom, quantum spin (which Bruce Schumm’s “Deep Down Things” (2004, p. 187) argues “We [physicists] don’t really have a clue about the physical origin of spin”) but not other positive-energy internal degrees of freedom…?

As we take account of spin by using multi-component (spinor) wavefunctions, which WC does seem to cover, why shouldn't other internal degrees of freedom (handled in a similar fashion) also be recognisable to WC?

Perhaps I can get a bit more of a handle on this from T Takabayasi’s 1961 article “Internal Degrees of Freedom and Elementary Particles. I”: (free PDF) via https://academic.oup.com/ptp/article/25/6/901/1861451

And finally, it's clear that Wigner’s Classification is unlikely to provide a handle on dark matter. * Wigner's Classification is a topic often asked about on scientific social media and other places, like https://physics.stackexchange.com/search?q=wigner's+classification and even Quora, as well as P.F.

** As a retired chemist with a strong visual intuition I’m still filling in the gaps in my understanding of the relevant maths

*** Clarified via https://www.physicsforums.com/threads/can-a-hamiltonian-be-unbounded.406651/
https://physics.stackexchange.com/q...-the-hamiltonian-to-not-be-bounded-from-below and a chemistry-oriented article https://www.themathcitadel.com/energy-levels-of-molecules-are-bounded-below/ inspired by Tosio Kato’s “Fundamental properties of Hamiltonian operators of Shrödinger type” (google the title to get the PDF, if interested )

PS Though I'll be very interested in any further replies, I must start on my UK tax return for an upcoming deadline, so may not be free to reply in any detail for a while.