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Aninda Saha

IA (HL)
MATH HL IA - Directions please.

Hey everyone,

 

Okay so we've been asked to get started on our Math IAs and I was quite interested at first to explore the minimum speed required to run on water but there were some complications so now I have decided to do mine on the Schrödinger equation. Could anyone please help me out in deciding which direction to go with this topic? Like, what are the expectations of the math IA and which approach to the aforementioned topic would have a solid mathematical approach?

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Hey everyone,

 

Okay so we've been asked to get started on our Math IAs and I was quite interested at first to explore the minimum speed required to run on water but there were some complications so now I have decided to do mine on the Schrödinger equation. Could anyone please help me out in deciding which direction to go with this topic? Like, what are the expectations of the math IA and which approach to the aforementioned topic would have a solid mathematical approach?

 

Are you sure that you will be capable of doing this topic? I know it's quite tempting to do a Math IA on Schrödinger equation because it is so significant and absolutely central to the whole of quantum physics. However, you must realize that it is DIFFICULT. Let me remind you what a standard time-dependent Schrödinger equation looks like:

gif.latex? \hbar \frac{\partial}{\partia

  1. The symbol gif.latex? \frac{\partial}{\partial t} tells you that differential equation is involved. In fact it's partial differentiation that you must deal with here (but that is a fairly easy concept to learn).
  2. The symbol gif.latex? H stands for the so-called 'Hamiltonian' (i.e. the total energy of the system). It's one of those that you can measure & observe, which means that it is a Hermitian operator. So basically, this tells you that matrix is involved.
  3. The symbol gif.latex? i tells you that complex number is involved.
  4. The symbol  gif.latex? \left| \Psi \right> is the so-called Dirac's bra-ket notation for vector space, which is not a normal vector in our normal 3D-space, but it is a vector that exists in an imaginary Hilbert's space (which is the reason why complex number is necessary in the first place). You might even need some knowledge in probability to understand this.

So basically, this short and elegant equation requires you to have the basic understanding of 3 big fields within mathematics: Calculus (which covers differential equation), Linear Algebra (which covers matrix & vectors in general), and Abstract Algebra (which deals with vector space & complex number), not to mention probability. Not only that, you do need some sort of background in quantum physics to even comprehend things like vector spaces, discrete vs continuous, superposition, probabilistic waves, etc.

 

A common approach to a differential-equation-related IA is to maybe derive the equation by yourself, which is very difficult in the case of Schrödinger equation. Another approach is to supply the equation with some initial conditions, which then allows you to solve the equation. For example, you can try to solve the equation for an electron in some atoms in order to find out its probability of being at some position in time, etc. However, doing this would also require you to understand all the mathematical concepts that I've mentioned above. That would take an incredibly huge amount of time. Besides, with little experience with these types of mathematics, I realized that it would be extremely easy to make mistakes in the mathematical reasoning.

 

So I guess I would strongly recommend you to reconsider your choice. You mentioned "the minimum speed required to run on water". What do you mean by that? What were the complications? Why did you stop with that exploration?

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Hey everyone,

 

Okay so we've been asked to get started on our Math IAs and I was quite interested at first to explore the minimum speed required to run on water but there were some complications so now I have decided to do mine on the Schrödinger equation. Could anyone please help me out in deciding which direction to go with this topic? Like, what are the expectations of the math IA and which approach to the aforementioned topic would have a solid mathematical approach?

 

Are you sure that you will be capable of doing this topic? I know it's quite tempting to do a Math IA on Schrödinger equation because it is so significant and absolutely central to the whole of quantum physics. However, you must realize that it is DIFFICULT. Let me remind you what a standard time-dependent Schrödinger equation looks like:

gif.latex? \hbar \frac{\partial}{\partia

  1. The symbol gif.latex? \frac{\partial}{\partial t} tells you that differential equation is involved. In fact it's partial differentiation that you must deal with here (but that is a fairly easy concept to learn).
  2. The symbol gif.latex? H stands for the so-called 'Hamiltonian' (i.e. the total energy of the system). It's one of those that you can measure & observe, which means that it is a Hermitian operator. So basically, this tells you that matrix is involved.
  3. The symbol gif.latex? i tells you that complex number is involved.
  4. The symbol  gif.latex? \left| \Psi \right> is the so-called Dirac's bra-ket notation for vector space, which is not a normal vector in our normal 3D-space, but it is a vector that exists in an imaginary Hilbert's space (which is the reason why complex number is necessary in the first place). You might even need some knowledge in probability to understand this.

So basically, this short and elegant equation requires you to have the basic understanding of 3 big fields within mathematics: Calculus (which covers differential equation), Linear Algebra (which covers matrix & vectors in general), and Abstract Algebra (which deals with vector space & complex number), not to mention probability. Not only that, you do need some sort of background in quantum physics to even comprehend things like vector spaces, discrete vs continuous, superposition, probabilistic waves, etc.

 

A common approach to a differential-equation-related IA is to maybe derive the equation by yourself, which is very difficult in the case of Schrödinger equation. Another approach is to supply the equation with some initial conditions, which then allows you to solve the equation. For example, you can try to solve the equation for an electron in some atoms in order to find out its probability of being at some position in time, etc. However, doing this would also require you to understand all the mathematical concepts that I've mentioned above. That would take an incredibly huge amount of time. Besides, with little experience with these types of mathematics, I realized that it would be extremely easy to make mistakes in the mathematical reasoning.

 

So I guess I would strongly recommend you to reconsider your choice. You mentioned "the minimum speed required to run on water". What do you mean by that? What were the complications? Why did you stop with that exploration?

 

Wow. well first of all, thank you for that insight Vioh. I guess I would have to admit that I was more intimidated, rather than tempted, into choosing some of these complex topics having seen some of the exemplar explorations. But today I had a talk with my teacher and he seemed to be advising me to change my topic as well. To be honest, I just don't have a clear idea of the levels of difficulty of math that IB wants to see. There are some rather advanced math incorporated IAs which did not seem to score very well but on the other hand, there are others of lower math difficulty which scored higher. 

 

Ohhh, that previous idea I got from watching The Flash and wondered whether it was possible for us to run on water given that we were able to build up enough speed. I hypothesized at first that lifting your feet off the surface of the water before the surface tension of the water broke would enable us to maintain balance on water. However, I found out that the surface tension only works for much much smaller insects and the only way for us to run on water would be if we could emulate the way the Basiliscus lizard runs on water. This deals with complicated hydrodynamics which I cannot quite comprehend or find interest in. 

I guess my question to you would be: can we score well in an IA without making it super complicated? I am willing to do it if that's what it takes but an general idea would be very helpful.

 

Thanks.

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I guess my question to you would be: can we score well in an IA without making it super complicated? I am willing to do it if that's what it takes but an general idea would be very helpful.

Thanks.

 

No, it doesn't have to be super complicated. But it has to be on something that is within the HL syllabus but beyond the SL syllabus, things like vector in 3D-space (e.g. cross products) or integration by parts, or complex numbers, etc.

If you are interested in physics, perhaps you would want to do something about the problem of least time? You've probably heard about the fact that light travels along the path that takes the least amount of time. You can use that as the starting point. Or you can investigate the principle of least potential energy in order to understand the catenary.

These topics are all valid because they all require HL maths to understand.

Good luck!

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Hi,

 

I know how hard the maths IA is. For me, it was - by far - the worst IB assignment I had during the whole 2 years!  :P

 

What I would recommend is to choose a very personal topic, from which you can start using maths. For example, if you wanna use 3D vectors, you might wanna introduce this topic with a personal experience, like wanting to build a pool, or finding the best way possible for a European trip. Or if you wanna use matrices, you might wanna invent your own matrix code, and try to compare it to a more complex, real-life example. It doesn't matter if you tell the truth and all the truth, you must show the IB that you can connect maths to the real life. Furthermore, if maths can't apply perfectly, you can reflect on it and explain why the results you get might not be a distinctive representation of the reality.

 

These are just ideas, but connecting your topic to a personal experience is an easy way to score high in the "personal experience" mark.

 

Hope this helps!

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