exchemist
Valued Senior Member
I only knew Ehrenfest from his paradox. Interesting.
What mathematics should I know to study quantum mechanics?
- Thread starter Saint
- Start date
It's just that you need about ten years of focused mathematical accomplishment to understand, and handle for yourself, those 32 numbers and "handful" of equations. About ten thousand hours of focused effort is the standard rule of thumb requirement for human mastery of high level skill sets like that.
There is no realm or branch of mathematics that is sure to have no bearing on modern fundamental physics - the unique role of mathematics in assembling virtual sensory organs for the perception of aspects of the world our somatically founded ones cannot register involves all major branches of mathematics, and the unknown in physics, the boundary of theory or experiment, has largely expanded beyond what our somatic senses can present for analysis.
It has got to the point that no one mathematician can handle even just the math (knot theory and topology, probability theory and combinatorics, higher dimensional logic and computer "science", on top of all the real analysis? People don't live that long, and their highest mental capabilities in that arena degrade even faster than their cardiovascular systems).
Which means that the answer to the OP question is kind of a happy one: you need a very high level of comprehension and ability in whatever branch(es)of mathematics you find most congenial (or immediately necessary for something that has struck you), coupled with the willingness to spend years applying it to the patterns careful observation has found in the physical world at the scale of quantum phenomena.
Write4U
Valued Senior Member
"Mastery" is a different level than "understanding" mathematics.
I agree, but fundamentally mathematics is a purely logical and consistent discipline. Max Tegmark's own words;
"Modern mathematics is the formal study of structures that can be defined in a purely abstract way, without any human baggage. Think of mathematical symbols as mere labels without intrinsic meaning. It doesn't matter whether you write “two plus two equals four”, “2 + 2 = 4” or “dos mas dos igual a cuatro”.
"The notation used to denote the entities and the relations is irrelevant; the only properties of integers are those embodied by the relations between them. That is, we don't invent mathematical structures – we discover them, and invent only the notation for describing them".
https://www.scientificamerican.com/article/is-the-universe-made-of-math-excerpt/
This pdf may be of interest;
0.1 A taste of quantum mechanics
https://uwaterloo.ca/institute-for-...ng/files/uploads/files/mathematics_qm_v21.pdf
Last edited:
I've got dozens of doozies like that one, stuff being done before it was officially done and in a completely different way. Many different topics to cover both in QM and Relativity. I've seen around 5 or so completely different ways to derive/explain Planck's radiation law, trying to learn the dimensional scaling argument method now.
exchemist
Valued Senior Member
Could be interesting if you start a thread on some of these little known alternative reasons, which don’t feature in undergraduate treatments of the origins of QM and relativity. We could do with a bit more proper science
.But you won’t want to get into all the maths I realise. Just some links would be good.
P.S. I’m on hols in Brittany at present and the weather is great, so I won’t be on line much for the next week or so,
But I’d be interested when I’m back.
Write4U
Valued Senior Member
OP title; "What mathematics should I know to study QM" ? AFAIK there is only one kind of mathematics, but their application is divided into "branches" of mathematics depending on the desired area of study.
Of course the study of QM requires special areas (branches) of study, each which involves the application of mathematics.
The University of Waterloo seems to get by with that conceptualization.
I guess it depends on your interpretation of the term mathematics. Mathematical functions are not difficult to understand. How to apply the maths is a different story......
What are the Branches of Mathematics?
Mathematics can be broadly grouped into the following branches:
- Arithmetic: It is the oldest and the most elementary among other branches of mathematics. It deals with numbers and the basic operations- addition, subtraction, multiplication, and division, between them.
- Algebra: It is a kind of arithmetic where we use unknown quantities along with numbers. These unknown quantities are represented by letters of the English alphabet such as X, Y, A, B, etc. or symbols. The use of letters helps us to generalize the formulas and rules that you write and also helps you to find the unknown missing values in the algebraic expressions and equations.
- Geometry: It is the most practical branch of mathematics that deals with shapes and sizes of figures and their properties. The basic elements of geometry are points, lines, angles, surfaces, and solids.
- Trigonometry: Derived from two Greek terms trignon (meaning a triangle) and metron (meaning a measure), it is the study of relationships between angles and sides of triangles.
- Analysis: It is the branch that deals with the study of the rate of change of different quantities.
- Calculus forms the base of analysis.
https://byjus.com/maths/branches-of-mathematics/
So, question is; "which branches of mathematics applies to QM".
p.s.
has an on-site advisor, to answer questions.https://byjus.com/maths/branches-of-mathematics/
Last edited:
Really.
That would explain the ease with which the enlightened comprehend "quantum mechanics", intuitively - after "experiencing" it/them, of course.
And the answer I posted, as noted above, was: "probably: all or any of them".
Last edited:
Write4U
Valued Senior Member
I am happy you qualified that statement.
quantum mechanics, Physics
- The branch of mechanics that deals with the mathematical description of the motion and interaction of subatomic particles, incorporating the concepts of quantization of energy, wave–particle duality, the uncertainty principle, and the correspondence principle.
‘According to the laws of quantum mechanics, these electrons may exist only in certain states.’
More example sentences- ‘In fact, it is arguably the most important fundamental concept behind all of quantum mechanics.’
- ‘On the other hand, in quantum mechanics you will deal more with algebraic techniques.’
- ‘One of the bizarre paradoxes of quantum mechanics is that elementary particles can exist in two or more states at the same time.’
- ‘Even a biologist must trust what a physicist says about quantum mechanics.’
- ‘This implies that we will always have uncertainty in any system, not just in quantum mechanics or in mathematics.’
- ‘For Doppler cooling, we need another detail from quantum mechanics, and a bit of relativity.’
- ‘The theory that describes atoms and their constituents is quantum mechanics.’
- ‘Firstly, it consistently embodies both special relativity and quantum mechanics.’
- ‘The entire field of quantum mechanics owes much of its existence to the study of angular momentum.’
- ‘String theory's claim thus allows quantum mechanics to incorporate gravity and do so successfully.’
- ‘His official courses were on quantum mechanics, classical mechanics, and complex function theory.’
- ‘His interests range from astrophysics and quantum mechanics to mathematical puzzles and games.’
- ‘It can be said that Heisenberg's quantum mechanics has made possible a systemization of spectra of atoms.’
- ‘In the answer to this question lies the whole key to quantum mechanics.’
- ‘So with growing trepidation, I searched through my past writings on quantum mechanics.’
- ‘We know from quantum mechanics that nothing is real, except for the observations themselves.’
- ‘The second question, of course, was rendered questionable by quantum mechanics.’
- ‘In 1976 I began investigating what quantum mechanics might have to say.’
- ‘What happens when we add quantum mechanics to the analysis of classical black holes?
exchemist
Valued Senior Member
Quite. Posts 2 and 9 give pretty good answers.
As a chemist, I was pleased to see James mentioned group theory.
That sounds really interesting, what applications does group theory have in chemistry? Like symmetry groups in crystallography, perhaps? On a somewhat unrelated topic, this reminds me of a course that was being offered at some university which went along the lines of "group theory for aboriginals" and was trying to push mathematics from a different cultural perspective with a different set of applications relevant to cultural traditions. I found the premise of the course somewhat racist though because I figure aboriginals already need math for the same reasons everyone else needs it.
As to your comments about me starting another thread with "the real history of modern physics", I'll definitely have a think about it. I'd like to start with J.J. Thomson's raisin bun model if I had a thread like that, I find it actually to be a very impressive classical attempt that reproduced a lot of the qualitative features of quantum behaviour while staying within the bounds of classical electromagnetism. There were also lots of reasons to rule it out or express strong skepticism even before the Geiger-Marsden experiments. I'd probably also want to write a bit about the origins of Fresnel's ether drag model, its success in anticipating the results of Fizeau's experiments with the speed of light in moving fluids, and why it's not a coincidence that Fresnel's velocity addition formula matches the one given by Relativity to first order in Taylor expansions, even though Fresnel's hypothesis was eventually falsified. There's also a lot of classical atomic theory developed by Hendrik Lorentz which carried over into the quantum realm, such as his highly successful explanation for the non-anomalous Zeeman effect. Lorentz's explanation for this effect also led to an alternative means of measuring an electron's charge/mass ratio and was actually discovered shortly before Thomson published the results of his cathode ray experiments.
Last edited:
exchemist
Valued Senior Member
Yes, while I understand crystallography uses space groups, bonding in molecules (i.e. molecular orbitals ) can be analysed by point symmetry groups. Given that exact solutions are impractical for multi-body problems, one needs ways of determining the QM properties of molecules in a qualitative way. Group theory enables this. I’m very rusty on this now and away from my books (on holiday in Brittany) so I dare not try to describe examples from memory alone, but group theory is very powerful in theoretical chemistry. Essentially, the electron is in a potential well of complex geometry, due to the various atomic nuclei of the atoms making up the molecule, and the form and the relative energies of various molecular orbitals it can occupy are determined by the symmetry.
Well, actually, the group theory of interest in QM is that of the continuous symmetry groups. Also known as the Lie groups (say "Lee"). This is quite a difficult subject, but fascinating and beautiful.
As a taster - the Lie group SO(3) describes the symmetry of spin-1 particles, whereas the Lie group SU(2) describes the symmetry of spin-1/2 particles.
Any wiser? No, I thought not
As a taster - the Lie group SO(3) describes the symmetry of spin-1 particles, whereas the Lie group SU(2) describes the symmetry of spin-1/2 particles.
Any wiser? No, I thought not
There is so much abstract algebra I have yet to learn or once learned and completely forgot after the exam. Woe is me, I wish I could know more about Lie groups and Clifford algebras and all this stuff without having to read a 500 page book.
I'm trying to understand the Wikipedia article on Clifford algebras, and I can't for the life of me understand how $$Cl_{0,0}$$ is isomorphic to the real numbers, rather than the set consisting solely of a zero vector?
Or rather I should say $$Cl_{0,0}(R)$$
Well, I am not familiar with this notation. I am guessing it means something like this:
Given a Clifford algebra Cliff(V,Q), where V is a vector space and Q is a symmetric bilinear form, then if $$Q = 0$$, then Cliff(V,Q) becomes the exterior algebra on V. Now for any exterior form in $$\Lambda^p(V_n)$$ (n is the dimension of the vector space), then p cannot be greater than n,
So if dim(V) = 0, then n=0 and p=0. But dim Cliff(V,Q) = $$2^n$$ which for n =0 is 1.
Moreover, $$\Lambda^0(V_n)$$ is just the set of scalar functions on a $$V_n$$ which in this case is obviously isomorphic to R .
Given a Clifford algebra Cliff(V,Q), where V is a vector space and Q is a symmetric bilinear form, then if $$Q = 0$$, then Cliff(V,Q) becomes the exterior algebra on V. Now for any exterior form in $$\Lambda^p(V_n)$$ (n is the dimension of the vector space), then p cannot be greater than n,
So if dim(V) = 0, then n=0 and p=0. But dim Cliff(V,Q) = $$2^n$$ which for n =0 is 1.
Moreover, $$\Lambda^0(V_n)$$ is just the set of scalar functions on a $$V_n$$ which in this case is obviously isomorphic to R .
I think the issue was that I was trying to picture a Clifford algebra as simply a vector space with some extra properties added to it, and now upon further reading from a proper source I see that there's a lot more to it. Thanks for your help.
Last edited:
Don't the symmetries also give the spin numbers of the gauge particle of a field? Or, the field symmetries are 'captured' by the gauge particle (a photon for the EM field).