Polynomial wavefunction - basic math qu.

[keen23]

Hello all!

I am at the QM-basics, and now a little bit confused, but maybe someone can easily clarify.

A QM-system can be described by a state that lives in a hilbertspace, this was introduced because superposition is essential. In the solutions of different problems polynomial functions turn up like Laguerre-p., Legendre-pol etc. Do these polynomials live in a vectorspace?

On the other hand alway the term wavefunction is used, which I associate exlusivliy with Cos and Sin-functions. Refers this nomination to the timeevolution part, or is this term more the result of the experimental beginings of qm (doble slit experiment), and superpostion is the only relevant point?

Or is this question nonsense, because it doesn't matter which set of orthonormal functions I choose for my series expansion?

As I said, I am confused, so I'm not sure how to put this question. But maybe you get my point, and can light this up for me.

Thank you!

 

[xepma]

Hello all!

I am at the QM-basics, and now a little bit confused, but maybe someone can easily clarify.

A QM-system can be described by a state that lives in a hilbertspace, this was introduced because superposition is essential. In the solutions of different problems polynomial functions turn up like Laguerre-p., Legendre-pol etc. Do these polynomials live in a vectorspace?

Yes, they form in fact the basis of the Hilbert space (which is a special type of vector space). As in any vector space we can take linear combinations of these basisvectors to form new vectors. This is, in some sense, the superposition principle you are talking about

On the other hand alway the term wavefunction is used, which I associate exlusivliy with Cos and Sin-functions. Refers this nomination to the timeevolution part, or is this term more the result of the experimental beginings of qm (doble slit experiment), and superpostion is the only relevant point?

Don't let the 'wave' in 'wavefunction' confuse you too much on this point. The wavefunction is certainly not limited to only Cos and Sin functions. The Laguerre and Legendre polynomials are examples, but later on you will find that you can even have wavefunctions which are not even expressible in terms of 'normal' functions. That will come across when you start treating spin.

Or is this question nonsense, because it doesn't matter which set of orthonormal functions I choose for my series expansion?

Yes and no. Yes, you can choose any set of orthornormal functions for your series expansion. Hell, you can probably get rid of the orthonormality as long as you have a complete basis.

But the thing is, it is not always that useful to just choose some random basis. What you are in fact after is a basis which is 'compatible' with you Hamiltonian. In all the examples of sets of basis functions that you will find in your book, you will notice that they are always eigenvectors of the corresponding Hamiltonian. That means they have a well-defined energy. And states with a well defined energy have a very simple time-evolution, namely a simple factor $$e^{iEt/\hbar}$$. Starting with some random basis, and trying to figure out how wavefunctions behave is somewhat impossible.


As I said, I am confused, so I'm not sure how to put this question. But maybe you get my point, and can light this up for me.

Thank you![/QUOTE]

 

[keen23]

Thank you a lot!
Now things got clear :)