Is Using Complex Numbers Necessary for Understanding Quantum Mechanics?

[guillefix]

Hello, I've chosen to do my Ib extended essay, a piece of research of about 4000 words, on mathematics because I really like physics but I think I'm not very good at expreimental physics, and in order to do it on physics you needed an experiment or some kind of data. So I chosed to od it on maths, but applied to physics. Finally I've found two possible topics which I like and think they are good for an extended essay. One is the mathematics used in determining the location of a GPS receiver, and the other is about why should complex numbers be used in quantum mechanics. My supervisor and other teacher said both of them were good topics, but they were inclined for the GPS one, saying that the other one might be too complicated.
I decided to do it in GPS following their advice, altough I would continue doing research on the other topic for my own interest. Doing this research, I found this webpage http://www.scottaaronson.com/democritus/lec9.html" and since then I've thinking that I'd like to do my Extended essay on the use of complex number and the mathematical foundations of quantum mechanics, I'm researching right now, looking at the Standford lectures on Quantum mechanics on youtube, and even thought it is quite complicated, I think I'm starting to understand it. I really like the explanation on that webpage, saying that quantum mechanics is basically a generalization of classical probability to include negative numbers, and that complex numbers are necessary in order to time evolution to be continuous.
However, I'm still afraid that it will end up being too complicated, so I will like someone's advice, if you think it is suitable for an extended essay, and if you can recommend some sources for information it would be great.

Oh that was long Thank you for your help!

 

[Mépris]

Paragraphs. Use them.

Good luck with your EE. That's precisely one of the things which put me off IB. That and sub-par teaching at my school.

 

[guillefix]

Basically what I want to know is which mathematics are harder those involved in the explanation of why should complex numbers be used in quantum mechanics or those involved in the GPS? Thank you!

 

[uman]

The GPS thing should only use basic geometry and trigonometry; the other is based on complex analysis. The former should be much easier. The only issue is that I'm not sure if you could stretch it to anywhere near 4000 words.

I did my extended essay on history, which you may want to consider if you're any good at reading/library research. Even though I'm a math guy, it's just way easier that way.

 

[jowens1988]

I agree with uman; the mathematics involved in GPS is going to be much simpler than that used in quantum mechanics.

First, I would say that, if your science and math teachers are anything like mine were, they probably don't know a thing about quantum mechanics, and therefore will not be able to provide much help.

I read a little on Aaronson's website, and I think he makes some interesting points. On the other hand, he himself says he is not discussing quantum mechanics from the physicist's point of view, or even the mathematician's point of view. Though he seems to be much smarter than me, I disagree with him when he says quantum mechanics isn't a physical theory in the same way as electromagnetism and general relativity are. Anyway, I don't want to get a debate started about that.

One simple reason that quantum mechanics involves complex numbers is because some of the solutions to the Schrödinger Equation involve complex numbers. Is there something deeply philosophical about this, or is it just how our system of mathematics made everything work out?

I think, in general, mathematics has an interesting role in quantum mechanics, not just the portion dealing with complex numbers. (Note that we can only measure operators which have real expected values. See: http://mathworld.wolfram.com/HermitianOperator.html.) Unfortunately, to really even begin a coherent discussion about the role of mathematics, you would need to at least understand quantum mechanics at the level of an advanced undergraduate course; meaning, you would need a lot of prerequisite work (basic physics, differential and integral calculus of multiple variables, differential equations, linear algebra, some probability theory wouldn't hurt, etc.) I am not trying to discourage you, but you are treading into deep waters, and it will be easy to get lost or discouraged.

These are certainly great questions to think about, but I don't know that you know enough about quantum mechanics (or have the time to learn enough quantum mechanics) to write a strong essay about them.

Have you thought about discussing the general effectiveness of mathematics in the physical sciences? It is a broad topic, yes, but there are many different points-of-view on the topic, and you could pick one and analyze it. Also, it would be much less technical in nature than any discussion of mathematics in quantum mechanics.

Writing it over a historical topic isn't a bad idea, either. I also did that.

I hope this helps.

 

[guillefix]

Thank you very much for your answers. I have also read Aaronson's lecture 9 in quantum mechanics and that actually was what encouraged me to do my extended essay about it. Without touching the discussion of wether it is a physical theory, I think that way of thinking about it is the most mathematically intuitive one. Assuming that what Aaronson says is true, I don't see why you would need very advanced differential and integral calculus with multiple variables and differential equations, which I think is the part which I would find the hardest. I can only see linear algebra in Aaronson's arguments. Schordinger equation does need knowing differential equations, as well as lots of other aspects in quantum mechnics. However do you really need that in order to discuss the necessarity of complex numbers in quantum mechanics or other fundamental mathematical aspects of it? Both in Aaronson's article and other I found like http://www.arxiv.org/abs/quant-ph/0101012" they seem to explain the foundations of quantum mechanics using only linear algebra. Is this incomplete then?

About the GPS I don't think that the mathematics are going to be that easy, altough surely easier than in quantum mechanics. Altought linear algebra and geometry are the basics, there is also a bit of calculus involved. I think both of them would make a good 4000 words extended essay but I think that I could only do the quantum mechanics one if Aaronson's way of seeing it is enough and you don't need much more.

 

[jowens1988]

I guess it all boils down to what topic you're really trying to address. If you're trying to discuss why there are complex numbers in quantum mechanics, in a more philosophical sense, then I don't think I can provide much insight. From a conceptually mathematical point of view, the article you posted gives a good idea of some of the reasons for complex numbers.

I will say that, personally, I don't read too much into the fact that there are complex numbers in quantum mechanics. We cannot physically observe quantities with complex numbers. As an example, we impose the probability restrictions on the norm-squared of the wavefunction, because it is real. Measurable quantities must have real expectation values, from which we get the hermitian property. We can use hermitian operators to derive the general form of the uncertainty principle, which has a complex number. But, that complex number rids us of the other complex numbers in the expectation of the commutator between the hermitian operators. Point being, we cannot make much physical sense out of complex numbers.

Take a look at this derivation, for example: http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/GenUncertPrinciple.htm.

So, maybe the above is a case in which you do not need to know calculus or differential equations to understand.

If, however, say you were trying to discuss where the complex number comes into the time dependent portion of the Schrödinger equation (for a time-independent potential) then I think you need to understand the solutions to the Schrödinger equation, which involve differential equations and calculus. I don't think there is any way around that.

It's been years since I wrote my extended essay, but don't you have to have multiple sources? Make sure that you can find enough sources so that you're not just rehashing the discussions of the two sources you've posted.