Where to begin to understand the mathematics of quantum mechanics

[khemist]

I am very interested in quantum mechanics, specifically learning the math behind it. I am currently in calculus 3, or vector calculus. However, it is going a bit slower than I would like and I would like to start to study math at a quicker pace. However, I am slightly unsure of what I should learn to begin to understand different quantum mechanical equations, such as the Schrödinger equation, general wave functions, and how to define specific basis, states, and superpositions. Sorry if this is not the correct place to move it, but it seems to make since to put it in this subforum. :)

Thanks for any help. At the moment, my class is just beginning vector calculus, and will go into double/ triple integrals, partial differentiation, line integrals, and a few bit more topics. We are currently using Early Transcendentals, and we are going up to the 14th chapter I believe.

Khemist

 

[LostConjugate]

Your on the right track. You will just need Linear Algebra, Complex Algebra, and Differential Equations which usually follows Calculus. The Schrödinger equation is a Differential Wave Equation with complex numbers and usually written in a Linear Algebra based notation.

 

[khemist]

Ok cool. I am enrolled in differential equations for the fall semester and plan on taking linear algebra in the following spring.

Is there anywhere I can get either an informal education or a book that anyone recommends for complex algebra? That seems to be the one subject that my community college does not offer, and I do not really want to wait until I transfer to a UC to take that class...

 

[Pengwuino]

Is there anywhere I can get either an informal education or a book that anyone recommends for complex algebra? That seems to be the one subject that my community college does not offer, and I do not really want to wait until I transfer to a UC to take that class...

Check out the library and look for texts on complex analysis.

 

[LostConjugate]

I found it odd that almost all the courses I have taken did not go very in depth into complex numbers. Complex Analysis is the correct name for the course.

I actually learned most of what I know about complex analysis from a book called "The Road To Reality" it is a book that somewhat quickly covers a huge range of advanced mathematics required for advanced physics in general. So it may not be the best way to learn a specific subject, but I used the internet to fill in the gaps. A good book to have on hand for many mathematical references though.

https://www.amazon.com/dp/0679454438/?tag=pfamazon01-20

You will work with complex numbers somewhat in the DE course, but it doesn't cover the complex plane very well.

 

[khemist]

That is what I am concerned about. From looking at some of the higher level physics applications it appears to me that complex numbers are almost a necessity, and I find it quite odd that they have very little mention through my schooling, except for maybe a couple weeks in high school.

I will try to find a book on complex numbers. Does anyone have a web reference where I could look, in place of a book?

 

[peteratcam]

You don't need to know any complex analysis to understand basic QM.

You do need complex numbers though. If you've never met them before, you could do worse than this article:
http://nrich.maths.org/public/viewer.php?obj_id=1403&part=index

Peter

 

[LostConjugate]

There are two ways you can go about learning QM.

Start learning it with limited mathematics knowledge and spend half your time scratching your head about the mathematical notation.

or

Cram all the required mathematics into your brain so that you can spend your QM courses focusing on the physics concepts only to find that by cramming so much math, without much use for it until now, you are scratching your head trying to remember it all.

Either way you end up scratching your head which is a great way to become familiar with the study of QM.

 

[fundamentally]

QM has its foundation in quantum logic. Quantum logic has nearly the same axioms as classical logic but the modularity axiom is weakened. This has deep consequences for the structure of quantum logic. Where classical logic can be modeled with simple Venn diagrams, is the mathematical model that is congruent with quantum logic very complex. Each quantum logical proposition can be represented by a closed subspace of an infinite dimensional, separable (= has countable base) Hilbert space H. This Hilbert space can be defined over one of four number fields: the real's, the complex numbers or the quaternions. Each item in universe can be represented by a closed subspace of H. Via their eigenvectors operators can add attributes in the form of numbers to these subspaces. Unitary operators can shift vectors and thus subspaces around in Hilbert space. This means that the representation of the item can shift along eigenvectors of operators. In this way their attributes change. So, that is how dynamics is implemented. The shift occurs in a rather smooth way. This means that it is sensible to describe the movements with differential equations. The item describes a path through Hilbert space. This corresponds to a path through the attributes. The attributes may be taken from a higher dimension number field than the one that is used to define the Hilbert space. In this way the attributes may form a curved manifold in that higher dimension space. Now things become interesting, because now the toolkit of differential geometry can be used to study and characterize the path of the item. The Frenet-de Serre frames give a nice visual impression of this approach (see Wikipedia about this). If you used the quaternions to define the Hilbert space, then you will meet the situation that the unitary transform and an observation of an attribute of an item together cause a quaternion waltz. You will never experience a quaternion waltz in complex quantum mechanics. When you understand what it does and when you have analyzed what it does on micro scale, then you might understand why Minkowski metric and relativity play a role in nature.
If you want another tale without formulas, then check http://www.scitech.nl/English/Science/ATallQuantumTale.pdf
If you are prepared to see some formulas, then read http://www.scitech.nl/English/Science/Exampleproposition.pdf

 

[alxm]

Well you do may need complex analysis. It depends on what you want to do. Doing what I do, mostly dealing with the time-independent S.E. without external fields and hence real-valued wave functions I don't need it much. On the other hand, it comes in quite a bit in scattering theory, for instance.

That said I really liked complex analysis and found it very useful. It helped me think mathematically more than probably any other math course I took. Multivariate calculus is just calculus with more variables, and vector calculus is a bit like that as well. But when you get to complex analysis, many things you previously took for granted aren't necessarily true at all. Which gets you to start thinking about things in more abstract terms.

But in short, I don't think you can go wrong learning any math. You'd really have to go beyond undergrad mathematics to find math courses that had no relevance to QM at all.

 

[khemist]

Thanks for all the really helpful info. I guess what I am slowly finding out is that I might not necessarily want to major in physics, as much as I would like to be in the applied mathematics field . I know someone at UC Davis who is an applied math major, and one of his graduate research projects was working with the electron and its spin, I would guess mathematically. Quantum mechanics is something that extremely intrigues me, but I know other fields of physics and logic interest me, especially when there is math and applications that requires abstract thought.

Unfortunately neither of those links worked, though if you are able to find working ones I would be greatly appreciative. And I am always looking for more information, I will be rereading fundamentally's post over to find things that I will be researching.

I just found this quote on Wikipedia in the mathematical physics page:

The field has concentrated in three main areas: (1) quantum field theory, especially the precise construction of models; (2) statistical mechanics, especially the theory of phase transitions; and (3) nonrelativistic quantum mechanics (Schrödinger operators), including the connections to atomic and molecular physics.

This is essentially what I am trying to do, though I still cannot say for sure because I have not been introduced to the entire field of math or physics.

 

[jtbell]

Most physics students (in the USA at least) don't start learning QM by first learning all the math, then jumping into a full-bore QM textbook like Griffiths. They get their first exposure to Schrödinger's equation in a second-year "introductory modern physics" course which usually assumes only that they know basic single-variable calculus (differentiation and integration). The book introduces / reviews partial derivatives, complex numbers, and the basic concept of a differential equation. Linear algebra doesn't come into the picture at all at this level. Examples of textbooks:

Beiser, "Concepts of Modern Physics" (doesn't seem to be available from Amazon right now; 6th edition was in 2002, maybe there's a new one on the way)

https://www.amazon.com/dp/013805715X/?tag=pfamazon01-20

 

[khemist]

The thing is I don't necessarily want to wait until my first QM class. Maybe then math is not the thing I need to understand, but rather higher level physics. Could you then point me towards the physics needed to understand QM? And further, what does a further understanding of something like thermodynamics or optics help me with quantum mechanics? Does it require a higher understanding of electromagnetism? If so, what kind of EM is needed to understand the math behind QM?

P.S. Thank you for moving the post to the correct area

 

[fundamentally]

Curiosity was the biggest motivation during my studies. I wanted to know why my environment is four dimensional. Why there is time and a three dimensional space. I was unhappy with the fact that I did not understand why quantum mechanics had to be done in a different way than classical mechanics and I was particularly unhappy with the answers that my teachers gave me. I understand the formulas that Einstein gave us on how to treat relativity, but he never explained what the source of relativity is. So I dived into literature and analyzed the existing theories and came up with answers myself or I found proper explanations. This occurred in the sixties of the last century (I am born in 1941). Much of these theories were still in development. Nowadays you have internet and Wikipedia. ArXiv presents a great resource of freely accessible articles. Google is a fine tool to search all of this. Be not impatient, but stay pertinacious until YOU find all the answers. In the course you learn sufficient math. But you never learn enough. Still keep learning.

 

[khemist]

Thank you fundamentally. Essentially, that was the answer I was looking for, even though I did not necessarily frame it that way. I plan on just looking and coming up with questions to ask my teacher; it appears that he is slowly warming up to my inquisitions and hopefully he will really start to discuss some really interesting topics with me...

 

[Tedjn]

Physics cannot be summarized better than by http://abstrusegoose.com/272[/url. Don't be afraid to move slowly, or you might become overwhelmed and burned out. I agree with jtbell in that it is unlikely you will require too much math before jumping into QM. When I took my beginning QM course, I was fine with knowledge of single variable calculus and linear algebra, and the latter was not needed for some time.

I cannot emphasize enough that you cannot rush through math to get to more "interesting" stuff. That said, with your background, there's no reason not to start reading a QM book. One that I've seen recommended is that by Shankar. You might quell your impatience by jumping back and forth between your math texts and your physics texts.

 

[khemist]

Again, Tedjn, that is probably something I needed to read. While I am not necessarily "rushing" through the math (it is obviously quite complicated), that is something that I had the notion that I could do to get to more of the "meat" of the math and physics. I ordered "The Road to Reality: A Complete Guide to the Laws of the Universe" by Roger Penrose, and I have my eyes on a couple other books that people here have mentioned, unfortunately most of those other textbooks are just that, and they have a textbook price associated with it.