Question about the use of group theory in QM

In summary, the book "Groups and Symmetries. From Finite Groups to Lie Groups." by Yvette Kosmann-Schwarzbach is a good book to start from if you want to learn more about group theory and representations in quantum mechanics.
  • #1
AlexChandler
283
0
I am currently in my second undergraduate quantum course and just finished studying the addition of angular momenta. I am also in my third abstract algebra course and am now covering product groups and group actions. In my QM book (griffiths) there was a reference made to group theory. it said " what we are talking about is the decomposition of the direct product of two irreducible representations of the rotation group into a direct sum of irreducible representations" I am familiar with direct products and sums and representations and read a bit about the rotation group on wikipedia, but am still not really understanding what groups we are talking about. for example, for a spin 3 particle and a spin 3/2 particle, we would have

3X3/2=3/2+6/2+9/2

the left hand side representing the spins of the particles and the right side representing the total spin of the system. Now from my understanding, direct products and sums are done on groups, not numbers. So I am guessing that each number here represents a subgroup of the group of rotations SO3. Would this be somewhat accurate? If yes then what subgroups are they, and if no then what groups are they? or am i completely mistaken as to what this notation means? Thanks!
 
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  • #2
The numbers denote representations.

For example, the fundamental rep of SO(3) is 3-dimensional, so it is denoted 3. If you combine two spin-1 particles (i.e., particles in the fundamental rep of SO(3)), then you get

[tex]3 \otimes 3 = 1 \oplus 3 \oplus 5[/tex]
what this means is instead of a 3-component column vector, you now have a 3x3 matrix. It can be decomposed into three parts:

1. The trace, which transforms under the trivial 1 rep of SO(3),
2. The antisymmetric part, which transforms under the 3 rep of SO(3), and
3. The symmetric, traceless part, which transforms under the 5 rep of SO(3) (count that there are 5 components left to make up this part).

In this notation, half-integer spins are even numbers (technically, half-integer spins are reps of SU(2), but SU(2) and SO(3) have the same Lie algebra). So, for example, if we have two spin-1/2 particles, we can combine them via

[tex]2 \otimes 2 = 1 \oplus 3.[/tex]
That is, two spin-1/2 particles combine to make a singlet (spin-0) state, and a triplet (spin-1) state.

The example you gave, combining spin-3 and spin-3/2, would go like

[tex]7 \otimes 4 = 4 \oplus 6 \oplus 8 \oplus 10.[/tex]
That is, the combination of spin-3 and spin-3/2 would decompose into the irreps spin-3/2, spin-5/2, spin-7/2, and spin-9/2. (Note that you can't get spin-6/2, because the product of a boson and a fermion is still a fermion).

These things are easy to calculate in SU(2), because all irreps are built out of symmetric tensor products of the fundamental rep. The trivial 1 rep has no spinor index; the 2 rep has 1 spinor index, the 3 rep has 2 spinor indices, and so on. For the example above, spin-3 is the 7 rep, which has 6 spinor indices, and spin-3/2 is the 4 rep with 3 spinor indices. Their product looks like

[tex]G_{abcdef} \psi_{ghi}.[/tex]
Now you just see which indices you can trace over, and just write down all the possibilties. Since both tensors are fully symmetric, this is easy.

It looks like your quantum book is using a different notation where they write down the weight of the representation rather than its dimension. I think it's easier using the dimension.
 
  • #3
Thanks so much for your reply! I don't really know anything about Lie algebra yet, but this definitely helps me understand a bit of what I need to learn in order to pursue the topic further. Could you possibly recommend a good book or books that I could buy that would cover these types of things? (tensors, lie algebra, and representations)
 
  • #4
Unfortunately, I don't have the slightest idea what book to recommend.
 
  • #5
Try Wu Ki Tung's group theory text (World Scientific, 1984). It explains the basics pretty well without delving into complicated mathematics.
 
  • #6
You can also take a look at "Groups and Symmetries. From Finite Groups to Lie Groups." by Yvette Kosmann-Schwarzbach.

It is very introductory and accessible. Also in the Lie theory part follows the physics notations.
 
  • #7
A classic is

H. Lipkin, Lie groups for pedestrians
 
  • #8
vanhees71 said:
A classic is

H. Lipkin, Lie groups for pedestrians

Actually I just bought this book. It is very good, however I feel it is a bit above my current level in my understanding of quantum mechanics.
 

Related to Question about the use of group theory in QM

What is group theory and how is it used in quantum mechanics?

Group theory is a branch of mathematics that studies the properties and symmetries of objects under transformations. It is used in quantum mechanics to describe the symmetries of physical systems and the behavior of particles within those systems.

What are the advantages of using group theory in quantum mechanics?

Group theory allows us to simplify complex problems in quantum mechanics by identifying patterns and symmetries within the system. It also allows us to make predictions about the behavior of particles and understand the fundamental principles of quantum mechanics.

Can group theory be applied to all quantum mechanical systems?

Yes, group theory can be applied to all quantum mechanical systems. It is a universal mathematical language that can be used to describe the symmetries and dynamics of particles in any physical system.

How does group theory help in solving Schrödinger's equation?

Group theory provides a systematic approach to solving Schrödinger's equation by breaking down the problem into smaller, more manageable parts. It helps identify symmetries and conservation laws that can simplify the equation and provide insights into the behavior of particles.

Are there any practical applications of using group theory in quantum mechanics?

Yes, there are many practical applications of group theory in quantum mechanics, such as predicting the properties of new materials, understanding the behavior of complex molecules, and developing algorithms for quantum computing.

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