How Do Matrix Representations Relate to Wave Functions in Quantum Mechanics?

[cooev769]

Just trying to get my head around undergraduate quantum mechanics and they throw a lot of stuff at you, so some help is appreciated.

I understand that the wave function is some abstract vector living in an infinite dimensional hilbert space, and that it's a function. But then the textbook I'm reading goes on to represent the state as superpositions of two states:

s=al1>+bl2>

I've actually done quite a lot of QM calculations and am very familiar with the maths involved in ket-bra notation and what they represent. But I'm just confused as to how they tie in with the Hilbert space and infinite dimensional vectors and such. They've now represented the hamiltonian as a 4x4 matrix, the state by 2/2 matrix and I'm just confused as to how the functions tie into this at all. Do these states have anything to do with functions, or the wave function. I mean we were specifically told not to think of the wave function or QM in matrix form and just think of it as an abstract vector function, but now all the math we are doing is using superpositions and using vectors to calculate eigenvectors and eigenfunctions of the hamiltonian and so on. It's just all very confusing and overwhelming haha.

Thanks for any help.

 

[cooev769]

Are these l1> and l2> just arbitrary kets, which can represent any function? Because in this case l1> is a ket vector (1 0) and l2> is a ket vector (0 1) so they're linearly independent. But how do these vectors have anything to do with functions for example if I wanted to imagine this in terms of the wave function.

 

[UltrafastPED]

In abstract form the Shroedinger equation is H |psi> = E |psi>, where H is the Hamiltonian operator, |psi> is a state vector, and E is the energy associated with that state vector.

This is in the form of an eigenvalue equation (from linear algebra) ... since H is a linear operator it also has a matrix representation. The dimensionality of the matrix for H corresponds to the dimensionality of the state space for the eigenvectors (including any degeneracy), and the number of eigenvalues.

Especially for infinite dimensional cases it is useful to think of the vector space as a space of functions ... then Fourier analysis is the tool of choice. The Hilbert space is often viewed as a function space: the vectors are linear combinations of functions.

But most undergraduate QM courses don't discuss it in such abstract terms because ... then the course would be an advanced course in linear algebra and Fourier theory. Instead they just teach you the mechanics of what you need, and show you how to setup and solve problems.

Much like introductory calculus. You don't really find out what is going on unless you are a math major, or a brave soul who stumbles into a course on real analysis.

 

[Matterwave]

I would like to point out that the actual Schroedinger equation is $$\hat{H}\left|\Psi\right>=i\hbar\frac{\partial}{\partial t}\left|\Psi\right>$$

The equation that PED gave is the time-independent Schroedinger equation valid for time-independent Hamiltonians for which one uses separation of variables on the original PDE to turn the Schroedinger equation into a ODE.