AlfieD said:
Could you please explain matrix elements?
That's what I thought :)
At "A-level"? That would take a while. Students normally meet these concepts in their second year of University. I might be able to give you an idea - at the risk of giving you the wrong one.
Hopefully you've met matrixes in math already?
And you know about writing a vector as a column of numbers - and you've seen matrixes transform vectors when you multiply them?
You can make make a vector out of a function by expanding it in a basis ... for example, if:
$$f(x)=\sum_{n=0}^N c_nx^n$$ ... then I can represent f(x) in the "polynomial basis" as an (N+1)-dimensional column vector of all the ##c_n##'s. i.e. if ##f(x)=5x^5+3x^2## ... then that would be: ##f=(0,0,3,0,0,5)^t## as a 6D vector.
I don't have to use the polynomial basis - there are lots of them, some more useful than others.
The main advantage is that transformations of the function come from pre-multiplying the vector by a matrix... this can make all kinds of math involving multiple integrations by parts etc much easier.
In fact - if we define "vector" in terms of what it does rather than how we write it down, then a function actually
is a vector and all we've done is changed the way it is written down.
If quantum mechanics the functions are normally complex-valued, so there's extra tricks for handling them. But in particular, if you know any quantum mechanics at all, you can use the eigenfunctions of an operator as a basis.
Have you met complex numbers?
If I have a set of complex numbers ##\{z_n\}## then I can multiply any two of them like this ##a_{i,j}=z_i^\star z_j\; 0\leq i,j \leq N## then all the ##a_{i,j}## will be real, which is handy, and I can represent them as the elements of an NxN matrix A.
If you examine A it has some handy properties. i.e. an element from the diagonal has the property: ##a_{i,i}=|z_i|^2##.
I want to stop there - take a breath - it's a lot to take in. These concepts are usually introduced slowly with exercises at each step and I've glossed over a lot of stuff. The point here is to give you a glimpse, not to provide a complete picture, so be cautious about drawing conclusions.
Should give you an idea of what you are in for :D
In the meantime, treat "matrix element" as a special jargon.
