Quantum computing

Originally posted by adam warlock
Very good. Let me just add a few things:
1 - sometimes variables do commute (kinetic energy and momentum for instance) and that in those cases the commutator vanishes.
2 - and what is this commutator that these strange people are speaking of. The commutator is a mathematical operation that picks up two mathematicl objects and calculates AB-BA. Well AB ...[text shortened]... ises (naturally exercises that are adequate to the subject matter) than you don't understand it.

You mentioned vectors are commuting objects, only tangent vectors to flat spaces commute. Your post indicates you may know some of this. In a Lie group, which is a continuous group and therefore also a manifold (shape), the generators do not, in general, commute. Consider SU(2) which is the spin group. It is isomorphic to a three sphere - isomorphic means the same shape as and a three sphere is the three dimensional boundary of a four dimensional ball. There are three linearly independent vectors tangent to the three sphere at any given point. We can use the group operation to map those vectors around the group, so the three vectors at the identity element of the group define three vector fields on the group manifold. These three vector fields do not commute. Using the technology of what is called differential geometry one defines vectors as differential operators defined on the manifold. This works because differential operators fulfil all the axioms of a vector space and all vector spaces of the same dimension are identical up to isomorphism. It is not difficult to show that vectors tangent to a Lie group form a Lie algebra. A Lie algebra is defined by a relationship which for spin and rotation groups are identical and is [L_i, L_j] = {i, j, k} L_k where i, j, and k range from 1 to 3 and label the vectors with {1, 2, 3} = 1, {a, b, c} is anti-symmetric in each pair of entries (so {2, 1, 3} = -1). Compare this with the quantum relationship [L_i, L_j] = ih {i, j, k} L_k.

Originally posted by KazetNagorra
Theoretically, there are several options for the fundamental axioms of quantum mechanics, and it is possible to take the Heisenberg uncertainty principle as one of them. Another option is to define the momentum operator in a certain way, and derive the Heisenberg uncertainty principle from it. So in that sense, you are right. But from a more empirical p ...[text shortened]... particles is just a fact of nature, from which Heisenberg's uncertainty principle also follows.

I feel one should start at a quantum field theory level and use field creation and annihilation operators as the basic objects for an axiomatic system. The difficulty with Schrodinger equation stuff is that it treats one particle as a quantum object and everything else classically - the rest of the universe is treated as a potential well. Things like beta decay where the particle count changes are incomprehensible using one particle quantum theory. I prefer these operators since they more closely represent a real experiment - when one measures the position of a photon it is absorbed by the detector (triggering a cascade of some form) so an operator that annihilates the state is appropriate, the creation operators correspond to preparing a particle in some state in a natural way.

Originally posted by DeepThought
I feel one should start at a quantum field theory level and use field creation and annihilation operators as the basic objects for an axiomatic system. The difficulty with Schrodinger equation stuff is that it treats one particle as a quantum object and everything else classically - the rest of the universe is treated as a potential well. Things like b ...[text shortened]... riate, the creation operators correspond to preparing a particle in some state in a natural way.

Have you read Julian Schwinger's book on Quantum Mechanics? I think you'd like his vision of what Quantum Mechanics is.

Originally posted by DeepThought
You mentioned vectors are commuting objects, only tangent vectors to flat spaces commute. Your post indicates you may know some of this. In a Lie group, which is a continuous group and therefore also a manifold (shape), the generators do not, in general, commute. Consider SU(2) which is the spin group. It is isomorphic to a three sphere - isomorphic ...[text shortened]... (so {2, 1, 3} = -1). Compare this with the quantum relationship [L_i, L_j] = ih {i, j, k} L_k.

I was very sloppy in my list of objects that commute. (I should confess that I didn't remember the result about "only tangent vectors to flat spaces commute" or if I knew I forgot about it).

Originally posted by DeepThought
The difficulty is that the conceptual framework is unavoidably mathematical. The problem of measurement is quite extreme in quantum theory. This is built into the mathematical theory using things called commutator brackets - algebraic objects which encoded the difference between orderings of measurements. If one makes a position measurement followed b ...[text shortened]... lt to gain any understanding of quantum theory without understanding the mathematical framework.

Alas its a great pity that a purely conceptual representation is not possible without recourse to mathematics. I thought these principles could be established with 3D modeling or some other graphical representation and indeed i have viewed some excellent little videos with explanation which were really interesting.

Originally posted by adam warlock
1 - A thorough understanding of Physics needs math (there really is no way around it). A theory like Quantum Mechanics is very mathematical in nature and you have to dabble in it in order to fully apprehend what's going on.
2 - You know that you know something when you can apply it. That's why I said that you have to able to solve exercises (again exercises that are adequate to one's exposition to the subject matter)

I have given it some thought, and I disagree more the more I think it over.
I believe that the vast majority of physics concepts can be understood without any mathematics whatsoever. Although 'doing the math' may help to cement the concepts, - largely by forcing the student to think about it more, or seeing practical examples, I dispute that this is the only way to achieve this goal. If anything I would say doing practical experiments is just as productive if not more so, and in many cases requires no math whatsoever.
I would even go as far as to say that excessive focus on the math may lead to students that can solve the problems mathematically but do not understand why they are using certain equations, nor really understand intuitively the actual physics. So if faced with a problem they do not yet know the equation for, they may be stuck. In other words it may give a false sense of accomplishment.
To give an example: I know that when a wire passes through a magnetic field, an electric current is generated. I do not need to know the formula to understand this. In fact it would be more beneficial for my understanding to see examples of this in action than to do math problems on it.

Originally posted by robbie carrobie
Alas its a great pity that a purely conceptual representation is not possible without recourse to mathematics.

I believe the Feinman lectures I linked to earlier make the best attempt at this that I have seen to date.

Originally posted by twhitehead
I have given it some thought, and I disagree more the more I think it over.
I believe that the vast majority of physics concepts can be understood without any mathematics whatsoever. Although 'doing the math' may help to cement the concepts, - largely by forcing the student to think about it more, or seeing practical examples, I dispute that this is the ...[text shortened]... eneficial for my understanding to see examples of this in action than to do math problems on it.

This is quite an amazing statement to my mind, naturally because i have little knowledge of the subject and even less mathematical skill and although i genuinely thought what you have so admirably crystallized in words I did not dare speak it in such accomplished company for i was intimidated and I thank you for saying it.

Originally posted by robbie carrobie
This is quite an amazing statement to my mind, naturally because i have little knowledge of the subject and even less mathematical skill and although i genuinely thought what you have so admirably crystallized in words I did not dare speak it in such accomplished company for i was intimidated and I thank you for saying it.

I think twhiteheads statement applies to macroscopic physics, but microscopic physics is different. When you say that you do not understand maths what do you mean? Many people have difficulty with algebra, but that is not the whole of maths. A lot of my non-mathematical friends have no particular problem with geometry, but find algebra difficult to impossible. There's a level of abstraction with algebra which isn't needed for geometry. Certainly General Relativity can be understood by a non-mathematician - provided they get geometry. In general macroscopic physics can be understood intuitively and non-mathematically. Faraday was not very mathematically capable, but made a huge contribution to our understanding of electromagnetism.

Quantum theory is much harder, because the objects are far smaller than everyday experience it is difficult to be intuitive about. The normal approach uses calculus. But it is possible to gain an idea of some aspects using a geometrical approach - certainly Gauge Field theory is understood in terms of geometrical structures - but the quantum corrections are found in terms of fairly hard core mathematics. You have to understand that you will only get a partial understanding without getting your head around the maths.

Originally posted by DeepThought
I think twhiteheads statement applies to macroscopic physics, but microscopic physics is different. When you say that you do not understand maths what do you mean? Many people have difficulty with algebra, but that is not the whole of maths. A lot of my non-mathematical friends have no particular problem with geometry, but find algebra difficult to im ...[text shortened]... stand that you will only get a partial understanding without getting your head around the maths.

I had not much difficulty with either geometry or algebra but that was when i was at school which is hardly the level that you are talking about here. I studied computing science in college and learned to program using algebraic concepts (pascal was very like algebra because it has arrays and strings and variables) and only recently i have been learning after a gap of about twenty years, javascript and jquery which is also reliant upon algebraic principles especially with regard to creating functions which use multiple variables. Html5 and Css3 to an extent is also expressed in algebraic terms, although much less than javascript.

If a partial understanding is all that can be achieved then so be it, I do not mind, something is better than nothing, but i don't want to actually measure anything, or to know how to calculate the trajectory of particles, simply to know why they behave in a certain manner is enough.

Originally posted by twhitehead
I believe the Feinman lectures I linked to earlier make the best attempt at this that I have seen to date.

Its fascinating to be honest and the correlation between experimental physics and theoretical physics i was unaware of. This man is an excellent teacher, his teaching is full of illustration and he is an excellent public speaker.

Originally posted by robbie carrobie
I had not much difficulty with either geometry or algebra but that was when i was at school which is hardly the level that you are talking about here. I studied computing science in college and learned to program using algebraic concepts (pascal was very like algebra because it has arrays and strings and variables) and only recently i have been lear ...[text shortened]... ulate the trajectory of particles, simply to know why they behave in a certain manner is enough.

If you are a computer scientist then you should not have any problems with the maths associated with introductory quantum theory. Quantum field theory is a little tougher and by the time it gets to string theory you need to know your way around the maths. To get the basic ideas you need to know what a partial derivative is and to understand what linearly independent means. If you had a course on Fast Fourier Transforms then you should have no real problems. There's no particular need to be able to calculate things like spherical harmonics, provided you understand what they are.

Originally posted by DeepThought
IQuantum theory is much harder, because the objects are far smaller than everyday experience it is difficult to be intuitive about.

I think it was Richard Feynman - correct me if I am wrong, who claimed that nobody understood quantum theory. I think what he said, was that although we can do the math, we will never fully understand it intuitively - because a lot of it is not intuitive.

Originally posted by twhitehead
I think it was Richard Feynman - correct me if I am wrong, who claimed that nobody understood quantum theory. I think what he said, was that although we can do the math, we will never fully understand it intuitively - because a lot of it is not intuitive.

I have just watched the first lecture, honestly, he is a wonderful teacher, he makes you feel that you can actually grasp the conceptual basis.

Originally posted by robbie carrobie
Its fascinating to be honest and the correlation between experimental physics and theoretical physics i was unaware of. This man is an excellent teacher, his teaching is full of illustration and he is an excellent public speaker.

Actually I think physicists have actually done a really poor job of publicizing where we are with quantum mechanics. I think that the vast majority of people do not realise just how central it is to modern physics and just how successful it is at matching experiment. They also don't realise how long it has been around. This is largely the fault the education system. Not once in my formal education was I ever introduced to quantum mechanics - and I think this is true for the majority of people. Almost all of us were introduced to Newtons Laws, and most of us were given at least hints of what relativity is all about, but quantum mechanics is left for only those who chose to specialise in it at university level. I find this odd given that it underpins almost all of physics and even a large part of chemistry.