What mathematics should I know to study quantum mechanics?

Discussion in 'Physics & Math' started by Saint, Mar 12, 2020.

1. SaintValued Senior Member

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What mathematics should I know to study quantum mechanics, particle physics?
Tensor analysis?
Is it as difficult as Finite Element Method?

3. CptBorkValued Senior Member

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Depends how far you want to go. Most aspects of the Bohr atomic model can be understood with high school algebra, but if you want to deal with topics like the Schrodinger equation and beyond, you'll definitely want to have a good background in multivariable and vector calculus, partial differential equations and linear algebra. The maths can be just as difficult as solving PDE's with the Finite Element method, and even more so, but again it depends on how deep you want to go in your understanding and application. Tensor analysis isn't really necessary but it can be useful in understanding certain results such as the Wigner-Eckart theorem or quantum electrodynamics form factors, but these are really fine details that aren't important if your goal is just to gain a general philosophical understanding of the theory.

Last edited: Mar 12, 2020

5. SaintValued Senior Member

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Statistical calculus necessary?

7. arfa branecall me arfValued Senior Member

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You can learn how to write quantum software at IBM's cloud computing site. This is superficially just object-oriented programming, but with qubits. There are Python libraries and a developer's kit (Qiskit). On the other hand you could learn about the underlying hardware, how a superconducting Josephson junction is a 2-state qubit given the right design.

The statistics in quantum programming is down to how many times a program is run (maybe a few hundred times). Running it once would mean you have a high confidence in the output, say, which would be a somewhat unusual situation.

8. CptBorkValued Senior Member

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Just a basic understanding of probability and how to work with continuous probability distributions should be sufficient for your needs. I forgot to mention some basic knowledge of complex variables is necessary too, and a more detailed background in complex analysis will help down the road when dealing with particle propagators and other more advanced concepts.

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11. arfa branecall me arfValued Senior Member

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In my opinion, which I tend to keep to myself, studying quantum mechanics spans a lot of other disciplines, but mainly these are logic (and what quantum logic is or isn't), solid-state physics, optics, and information science.

Modern digital computers use 2-valued logic and Boolean algebra, one correspondence with quantum information science is that it's more useful to have a 2-state quantum logic (roughly, spin-up or spin-down) than try to engineer access to higher eigenstates of individual qubits. One quite striking non-correspondence is that it's generally more utilitarian to use semiconductor technology which is relatively constrained (in the solid-state context, a handful of elements), but a qubit can be just about anything (which is small enough), it doesn't even have to be a single quantum particle like say, an electron. An electron is just one of the very many things that a qubit can be.

Look also, at what a universal computer is, or a universal set of gates/operations is, compared to non-universality. Quantum information appears to be something we can only exploit if we don't look at it in any way, including letting the environment do this. This is explained in terms of inputs and outputs, and two important quantum principles: unitarity and locality.

But that's just the start of your study, and, what you want to know might depend on what you want to do with that. These days you can teach yourself quantum logic and programming online, if you can demonstrate your knowledge of this to someone who cares (maybe they work for IBM), maybe they'll offer you a job. You don't really need to know a lot of mathematics to do this.

Why did IBM make its very expensive machine open to the public? It wants to attract innovators, and it wants to make cloud quantum computing into a commodity (why else?).

Last edited: Mar 16, 2020
12. James RJust this guy, you know?Staff Member

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Basic Algebra. Basic Calculus. Linear algebra will be very useful. Vector calculus. Some group theory could be useful. Differential equations, particularly partial differential equations.

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13. Write4UValued Senior Member

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Not only what is Quantum Mechanics, but also how do Quantum Mechanics work. How is a quantum event modelled?
We know there is a threshold event involved which triggers the quantum change. At this threshold event the system is in a state of "quantum suspension" or "in superposition" before collapsing into a single state.

If we are ever going to find an absolute universal time constant, it should be the duration of a "superposition" before the quantum collapse itself.

AFAIK quantum events have a value, but I don't know of any equation for the duration of "superposition" before collapse of the function.

14. CptBorkValued Senior Member

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I don't think it's been completely resolved to everyone's satisfaction, but there is a theoretical mechanism governing the process of wave collapse in the quantum decoherence model, and the transition is continuous (albeit extremely rapid) as opposed to an instantaneous *poof* wave collapse just because a measurement was performed. I think the main sticking point is that it either leads into the many worlds parallel universe interpretation, or else some as yet unknown force is needed to make sure our universe is the only one to manifest.

15. pluto2Valued Senior Member

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There is basically tons of mathematics you need to master to become really good at doing mathematical and theoretical physics.

http://math.ucr.edu/home/baez/books.html
http://www.staff.science.uu.nl/~hooft101/theorist.html

That's why I see that for someone who is not good at math studying theoretical physics can be a waste of time and money.

For starters physics and math textbooks are highly expensive and so obviously not everyone can buy them.

A better alternative to studying physics for someone who isn't good at math is to study economics and business administration and try to find a way to become rich quickly.

Physics is a waste of time. Since money is everything these days I think that rather than studying physics I believe it is more productive to just find a way to become very rich.

16. AlphaNumericFully ionizedRegistered Senior Member

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Is it more difficult? Depends your preferences for mathematics. I personally preferred it to FEM.

You need calculus, linear algebra (as much as you can do!), functional analysis, tensor calculus, basic probability theory. You need to be comfortable with vector spaces, duals (everything is a Hilbert space!), inner products, eigen-values/vectors/spaces/functions/states, variational methods, multi-dimensional integrals/calculus, group theory. Differential geometry wouldn't hurt either, especially if you lean towards anything related to space-time (general relativity, string theory etc).

A lot of non-relativistic QM is about linear operators (finite or infinite matrices usually) acting on Hilbert spaces. Almost everything then depends on operator spectral decompositions - ie energies and eigen-states. If you're comfortable with matrices, vectors, calculus and index notation you'll get up to speed pretty quickly and a lot of that does come up with FEM.

17. CptBorkValued Senior Member

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This may be off-topic but it's awesome to see you back, hope you're doing well! I've actually been back to studying a lot of QM lately because I'm trying to understand the historical roots, things like BKS theory and Thomson's raisin bun model, these sorts of models were far more detailed than I was led to believe when I first learned about them. Here's one of the main sources I'm reading through to get the history:
https://publishing.cdlib.org/ucpressebooks/view?docId=ft4t1nb2gv&brand=ucpress

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18. AlphaNumericFully ionizedRegistered Senior Member

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Thanks, I just randomly thought "I wonder how SciForums is doing" and took a look. Occasionally in the past I've checked up on how people like Farsight are/were doing, for a chuckle, and stuff like that but not felt compelled to post. Someone actually wanting to learn some QM seemed a more worthwhile reason to actually post.

19. exchemistValued Senior Member

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Farsight disappeared long ago, I'm pleased to say and Reiku seems to have stopped. I think Reiku, who is on the autistic spectrum, has finally discovered words, which must be a big help to him. Perhaps we are some of the beneficiaries of that.

We still get cameo appearances from Theorist and GaiaGirl/Frank Baker from time to time, and we still have the hardy perennials of Rover and Magical Realist to contend with. The main issue is lack of new people interested in genuine science. I suspect they have moved to social media mostly - which I avoid on principle.

20. CptBorkValued Senior Member

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Yeah, I've been picking up lots of bits and pieces lately. If there's ever enough interest I might eventually want to go back to that thingy I was doing about deriving everything in Special Relativity directly from classical electromagnetism and some basic physical reasoning, I think the Lie Symmetry stuff someone else brought up was an unnecessary tangent. I took a course in String Theory a long time ago and want to eventually come back to learning it, but for now I'm more interested in the historical foundations of QM and Relativity, how we can confidently rule out the alternatives from basic historical experiments before we go into more advanced details etc. Found some good sources on the historical development of General Relativity, Wiens and Planck dealing with blackbodies, Arago's stellar aberration experiments and Fresnel's ether drag theory, lots of cool stuff.

Are you much familiar with historical developments, in case I might have questions to ask?

21. exchemistValued Senior Member

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I got taught the historical foundations of QM as part of my undergrad studies, (Ultraviolet Catastrophe, Photo-Electric Effect, Geiger-Marsden Experiment etc) but my understanding may be a bit basic for you.

22. CptBorkValued Senior Member

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I'm getting into pretty esoteric details with a lot of math involved, and it's especially surprising how much of the modern narrative is partially or completely wrong. For instance Niels Bohr derived his atomic model based on a combination of the correspondence principle (quantum mechanics needing to reduce to classical mechanics at large scales) and fitting to Rydberg's hydrogen spectrum formula which was already known from experiments, finding that quantization of angular momenta was the key. However at the same time Bohr was publishing his paper, a man named Paul Ehrenfest was deriving this same model by showing that quantization of angular momentum (and other adiabatic invariants) is an absolute necessity in order for Planck's radiation law to be consistent with statistical mechanics, not just an easy option. In the case of the photoelectric effect, it actually only formed a tiny portion of Einstein's photon argument and was used mainly as an example of it working in practice, because most of his argument was actually based on pure thermodynamics.

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23. Write4UValued Senior Member

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Max Tegmark claims that just 32 numbers (relative values) and a handful of equations are all that is necessary to explain the properties (potentials) of the Universe as sets of mathematical patterns.
Is the Universe Made of Math? [Excerpt]
In this excerpt from his new book, Our Mathematical Universe, M.I.T. professor Max Tegmark explores the possibility that math does not just describe the universe, but makes the universe. By Max Tegmark on January 10, 2014