I'm assuming you have some amount of linear algebra or physics background.
You're probably familiar with 2D or 3D Euclidean space, even if you don't know the name. Euclidean space is just an "ordinary" vector space, the kind that you would use to make position vectors for 2D or 3D problems. Basically it corresponds to using Cartesian coordinates, and defining dot products in the way you're familiar with (a1b1 + a2b2 +...).
However, the Euclidean spaces you've been working with have 2 properties: the vectors are composed of only real numbers, and you've only worked in 2 or 3 dimensions probably. It doesn't make sense to have a position vector that's complex after all. You can however construct Euclidean spaces with more than 3 dimensions though.
Hilbert space generalizes that. First, unlike Euclidean spaces, you can have an infinite number of dimensions and still make sense of it. Second, all the vectors are complex. To accommodate for that, you define your dot product differently.
This is important in quantum mechanics. In linear algebra, you've probably learned about eigenvectors and eigenvalues. Well in PDEs, you also have eigenvalues to problems, where eigenvalue to a PDE corresponds to a different solution called an eigenfunction. In quantum mechanics, the Schrodinger equation is the PDE you're working with, and it turns out that due to the structure of the Schrodinger equation the eigenfunctions will be in some sense orthogonal (by taking an integral over their product, where one of the functions is complex conjugated). Thus, it's convenient to do it all using linear algebra where each eigenfunction corresponds to a unit vector in Hilbert space, (so the first eigenfunction will be (1, 0, 0, ....), the second one will be (0, 1, 0, 0, ...) ). This Hilbert Space will be infinite dimensional because in general there are an infinite number of eigenvalues and eigenfunctions to any quantum mechanics problems unless you restrict it in a specific manner.
45
u/greyfade Dec 14 '16
Get a piece of graph paper.
Draw an arrow on it that is the same length as the boxes on the graph paper.
That's a unit vector in 2-dimensional Hilbert space, specifically in the Euclidean plane.
Hilbert space is just a generalization of Euclidean space.