Complex Vector Space Analogy To Quantum Mechanics

[CrazyNeutrino]

Guys I am having a little trouble understanding how and why we use complex vector spaces
to describe the quantum states of a particle. Why complex vector spaces, and how is a complex vector space defined. Also are the 'vectors' in the field of quantum mechanics simply elements of a vector space like real numbers, or are the vectors actually analogous to actual vectors. Leonard susskind at Stanford already said that they are ONLY elements of a set of elements but i don't seem to understand how we can do STANDARD vector operations if they are not STANDARD vectors and are only ABSTRACT vectors. Can someone please help me with basic intuition behind the mathematics of quantum mechanics.

 

[George Jones]

Have you studied linear algebra?

 

[CrazyNeutrino]

Yeah, but I haven't learned complex vector spaces and how can we perform real vector operations on abstract vectors?

 

[George Jones]

I am not sure what you mean by "real vector operations." Did you take the definition of "vector space" in linear algebra?

 

[dextercioby]

Did you study mathematics in high school ? If so, then the curriculum ought to have included that bit of abstract algebra in which you had to show that the set of complex numbers endowed with 2 internal operations (addition & multiplication) is a field.

 

[AJ Bentley]

I struggled with this concept for a while.
I guess you can say it's because the math works.

It's a bit like fitting a curve to some experimental points. If you notice they form a sort-of-straight-line you can draw a line and say that's it.
Initial QM experiments (on spin) gave results that didn't fit any obvious rules until someone noticed that the behaviour appeared to parallel vector maths with a few 'i's thrown in..

Using vector calculations on the phenomena at least gives the right answer even if we can't find a real vector to justify it.

 

[bhobba]

Why QM uses complex vector spaces is a deep issue, but without going into the details its got to do with the necessity of infinitesimal transformations from, for example, an infinitesimal displacement of the measurement apparatus, which can be shown to require complex rather than real vector spaces, if you want, for simplicity, such transformations to be linear. Another reason is some really nice theorems such as Wigners theorem only work in complex spaces.

But over and above that in applied math the eigenvalue problem often arises (eg in Markov Chains) and generally, even if you start with real numbers, complex numbers tend to creep in because, in general, eigenvalues and eigenvectors are complex.

The elements of a complex vector space are just as much vectors as those in a real vector space - its by definition.

Thanks
Bill

 

[AJ Bentley]

It's interesting that the apparently artificial mathematical concept of the square root of a negative number should turn out to be a 'solid' part of the real world.

I find the fact that we can invent such a crazy idea and then find that it fits into the physical world somewhat disturbing.

Most mathematical ideas are based on some conceptual model of a 'real' process. But 'i' ? We just dreamed that up for a laugh surely?

 

[CrazyNeutrino]

Thanks Guys... I am kinda getting an intuition on why complex vector spaces are used instead of real ones. But i think its really weird that operations that you would do with classical vectors or pointers work with elements of a set.

 

[AJ Bentley]

Thanks Guys... I am kinda getting an intuition on why complex vector spaces are used instead of real ones. But i think its really weird that operations that you would do with classical vectors or pointers work with elements of a set.

Not really - after all, you defined it as a set of vectors.

 

[waveandmatter]

I think the point is, that a vector space is defined as a set and some operations defined on the set, which in the end obey the same rules as the "traditional" vectors. Actually i had to prove/disprove the vector space axioms for lots of sets and associated operators in school and again at the university.
Especially in the case of "complex vectors" like in QM, which are simply vectors where the elements are complex numbers, i think that iss easy to show and understand. (Just find the required axioms for a vector space on wikipedia and plug in the definition of your hilbert space vectors and the operations on them and see that the axioms work out).
If now you go from vectors with finite dimensions (=="elements") to vectors with infinitely many dimensions, you get the usual wave-functions, which also obey the vector space axioms, though it might be slightly harder to show.