Exploring Vector Space: A Guide to Understanding its Role in Quantum Physics

• blue_leaf77
In summary: I don't feel the need to learn it at the moment. I am familiar with what a vector is and what the properties of a vector are, so I don't feel the need to learn about vector space specifically.

blue_leaf77

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The thing is in my undergrad I haven't gone into a class that includes discussion about vector space, and the related stuffs, simply because they were not offered in the syllabus. As I have seen in some Quantum physics courses in some universities, they did talk about vector space in a certain chapter. I have tried to take a peek in their lecture notes, I didn't go through it until the end of this chapter though, but I noticed in the beginning of this chapter that it will talk about properties of vector space like
(i) u+v = v+u
(ii) a(u+v) = au + av
and so on, where u and v are vectors and a is number (real or complex). And I can predict it will mainly talk about mathematical abstraction. But they are just things I already learned in high school, as you guys did. What is so special about u+v = v+u, isn't it obvious. So why do they bother reviewing those things. Well I know there might appear some things rather new later on if I keep reading through it, but just:
1) what good will it do me to study vector space if I know how to deal with vectors?
2) which parts of quantum physics that rely heavily on understanding of vector space, is it so crucial that without having learned vector space I won't be able to get around those parts?
3) is worth for me to spent days to learn this matter?

I think it's safe to assume that "all of it" is the right answer to point 2). That should also help with the answers to the other 2 questions.

But I can still understand basic things, like what operator and its eigenvalues and eigenvectors are, how to compute them, I am also familiar with stuffs, like expansion of wavefunction into bases, hydrogen atom, and some others, and I did well in the exam. I have even taken quantum optics course, although my score was not perfect I still got the basic idea of light quantization. I didn't feel any urgent need in learning vector space up to now.

Maybe you shouldn't worry so much about those exact axioms at the moment, but the problem you are having is that you don't see the full context that made mathematicians write down those axioms. It isn't "obvious" that u + v = v + u because it's an axiom that is being postulated, rather than a claim about some specific thing like R^3 that's obvious in that context. If you want to write proofs like mathematicians do, you have to be very pedantic and spell out all the details like this. Part of the point is that you don't want to just prove your theorem for R^3. You want it to apply whether it's R^3 or C^3 or maybe some other gadgets that obey the same rules. In quantum mechanics, in particular, you want to think of wave functions as being vectors. So, that's a different thing from 2 or 3-dimensional vectors from high school. It's not really deep or anything, but it's just to spell things out. Intuitively, you can think or a vector space as a place where you can add vectors together and multiply by scalars. The vector space axioms are just a way of pinning down exactly what that means. Sounds to me like you are just over-thinking it.

As you get farther in math, people start assigning formal terminology to things you've basically already been using all along (and took for granted). The purpose of that is to generalize some concepts and also to contrast with new algebras that are different from what you have been using before.
For example, you probably learned about positive numbers in early grade school or kindergarten, but you called them numbers. What's the point of calling them positive numbers until you learn what a negative number is? At some point you learn about complex numbers and you probably thought, those aren't real numbers. But we still call them numbers because we can use them in most of the same ways, and so we generalize the definition of number. We can also include various other extensions to numbers, such as hypercomplex numbers, quaternions, etc.

Vector, in the context of vector space, is a much more general concept than the vectors in geometry. Of course vectors in geometry are vectors, so you shouldn't be surprised that they follow those properties such as commutativity and distributive... But there are other vectors. The properties of vectors is basically a list of requirements of what kind of extensions or new mathematical objects can still properly be called a vector. You might be surprised to find out that a function is a vector.

Thanks homeomorphic that sounds relieving for me.

Well I was surprised when people tell me that a function is a vector. But anyway I don't actually mind that mathematicians extend the definition of vectors to cover broader properties, and I have also learned from my reading of the lecture note I mentioned before that a matrix is a vector (correct me if I am wrong). But my original question is why should someone studying quantum physics go into certain depth in the matter of vector space?
To me, when I heard for the first time people talking about vector and wavefunction at the same time, I directly speculated this is because I can regard wavefunction as a vector of infinite dimension because it can be composed of a set of infinite number of orthogonal eigenstates of certain operator. This is just like how we can express geometrical vector in x,y, and z components. Other similarities, is that two wavefunction is said to be orthogonal if their inner product is zero, much like dot product of orthogonal vectors But to me, that's all there is to it about the term vector applied to wavefunction.

The reason it's useful to recognize a wavefunction is a vector is that it's possible to use all the theorems that were proven in linear algebra on wavefunctions, without having to re-prove them for this new object.

blue_leaf77 said:
But my original question is why should someone studying quantum physics go into certain depth in the matter of vector space?

Check out the following for the linear algebra needed:
http://quantum.phys.cmu.edu/CQT/chaps/cqt03.pdf

The reason is the correct mathematical treatment of QM as detailed in Von-Neumanns classic on the subject requires it - in fact it requires Hilbert Spaces - which is a vector space.

For example the wave-function is not a systems state - its a representation in terms of the position observable. Indeed even system states do not form a vector space - only so called pure states do. In general a state is a positive operator of unit trace. To understand what I just said you need linear algebra.

Thanks
Bill

But my original question is why should someone studying quantum physics go into certain depth in the matter of vector space?

Khashishi answered this, but I would add that mathematicians have a wide range of vector space, but physicists still have enough that it's a useful concept for them.

To me, when I heard for the first time people talking about vector and wavefunction at the same time, I directly speculated this is because I can regard wavefunction as a vector of infinite dimension because it can be composed of a set of infinite number of orthogonal eigenstates of certain operator. This is just like how we can express geometrical vector in x,y, and z components. Other similarities, is that two wavefunction is said to be orthogonal if their inner product is zero, much like dot product of orthogonal vectors But to me, that's all there is to it about the term vector applied to wavefunction.

Those are all good motivations as to why we might think of functions as vectors, but in the end the reason why they are vectors is that they satisfy the vector space axioms (or informally, you can add them and you can multiply them by scalars, such that those operations behave in a similar way to vectors in R^3). So, now you see what the point of that is. The vector space axioms characterize what we actually mean when we say that they are vectors. Without the vector space axioms, all you can say is that wishy-washy stuff you are talking about.

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and a set of operations that can be performed on those vectors, such as addition and scalar multiplication. It is a fundamental concept in linear algebra and is used to model real-world phenomena in fields such as physics, economics, and engineering.

2. What are the properties of a vector space?

There are several key properties of a vector space, including closure under addition and scalar multiplication, existence of a zero vector and additive inverses, and the distributive and associative properties. These properties ensure that the operations on vectors behave in a consistent and predictable manner.

3. How is a vector space different from a matrix?

A vector space is a set of vectors, while a matrix is a rectangular array of numbers. Vectors are used to represent individual points or objects in a vector space, while matrices are used to represent linear transformations between vector spaces. Additionally, the operations performed on vectors (such as addition and scalar multiplication) are different from those performed on matrices.

4. What are the applications of vector spaces?

Vector spaces have many applications in various fields of science and engineering. They are used in physics to represent physical quantities such as forces and velocities, in economics to model supply and demand relationships, and in computer graphics to manipulate and transform 3D objects. They are also used in machine learning and data analysis to represent and analyze high-dimensional data.

5. Can a vector space have infinite dimensions?

Yes, a vector space can have infinite dimensions. In fact, most commonly used vector spaces (such as the real numbers, complex numbers, and Euclidean space) have infinite dimensions. This means that there are an infinite number of vectors that can span the space, making it a powerful tool for representing and analyzing complex systems and phenomena.