Hamiltonian matrix eigenvalue calculation

[neon.neon]

Homework Statement

A spin system with only 2 possible states

H = ($$^{E1}_{0}$$ $$^{0}_{E2}$$)

with eigenstates
$$\vec{\varphi_{1}}$$ = ($$^{1}_{0}$$) and $$\vec{\varphi_{2}}$$ = ($$^{0}_{1}$$)

and eigenvalues E1 and E2.

Verify this & how do these eigenstates evolve in time?

Homework Equations

I'm even not sure how to start...

Maybe someone could help me out, I'd come up with an attempt if I'd know how. Thanks!

 

[diazona]

Do you know what eigenstates and eigenvalues are?

 

[neon.neon]

Of course I read about it, but it's hard to "imagine".

Is there any "low-mathematical" way of an explanation?

 

[diazona]

Think of it this way: a matrix represents some sort of transformation on a vector space. (Rotation would be one example of a transformation.) Now, for most of the vectors in the space, when you transform them, you get a new vector that points in a completely different direction. But for each transformation, there are some special vectors that, when transformed, continue to point in the same direction, although their length may change. Those vectors are the eigenvectors, and the factors by which their lengths change are the eigenvalues.

Now consider what that means in mathematical language. The matrix is, of course, the transformation. To transform a vector, you multiply it by the matrix. So if your matrix is H, and your vector is ψ, the transformed vector will be Hψ. In order for that vector to be an eigenvector, as I said above, the transformed vector has to be pointing in the same direction as the original vector. Mathematically, that means that it has to be a multiple of the original vector; that can be written as Eψ, where E is some number. So the eigenvectors need to satisfy
$$H\psi = E\psi$$
where E is a number.

Does that help?

 

[neon.neon]

... and eigenstates are?

Does that mean I just have to calculate $$H\varphi_{1} = E_{1}\varphi_{1}$$ and $$H\varphi_{2} = E_{2}\varphi_{2}$$ ?

And what about the time evolving part in the question?


But thanks a lot for now, I think I understood the eigenvectors/eigenvalues !

 

[diazona]

... and eigenstates are?

Let me start by saying that a state is something that encapsulates all the information you need to know about a physical system. A quantum system has an infinite number of possible states, but they can all be expressed as linear combinations of some number of basis states. For example, if the states are denoted by kets ($\vert\text{something}\rangle$), you might use numbered kets ($\vert 1\rangle$,$\vert 2\rangle$, etc.) to denote the basis states, and then you could write a general ket as
$$\vert\psi\rangle = a_1\vert 1\rangle + a_2\vert 2\rangle + \cdots$$
Or the basis states could be unit directions
$$\hat{x},\hat{y},\hat{z}$$
and then a general state would be
$$\vec{r} = a_x \hat{x} + a_y \hat{y} + a_z \hat{z}$$
Anyway, regardless of what kind of things your states actually are, you can take the coefficients $a_i$ and put them into a vector,
$$\begin{pmatrix}a_1 \\ a_2 \\ \vdots\end{pmatrix}$$
or
$$\begin{pmatrix}a_x \\ a_y \\ a_z\end{pmatrix}$$
If that vector is an eigenvector of some matrix, then the state that corresponds to the vector is an eigenstate of the operator that corresponds to the matrix.

In your case, the basis states are actually basis vectors themselves - that is, the vectors
$$\begin{pmatrix}1 \\ 0\end{pmatrix},\begin{pmatrix}0 \\ 1\end{pmatrix}$$
actually are the basis states. And the states, instead of being kets or unit directions, are just 2-component vectors. Because of that, in this particular problem, you can consider "eigenvector" and "eigenstate" to be synonymous.

Does that mean I just have to calculate $$H\varphi_{1} = E_{1}\varphi_{1}$$ and $$H\varphi_{2} = E_{2}\varphi_{2}$$ ?

Yep, that's what you need to show.

And what about the time evolving part in the question?

Well, what do you know about how quantum states evolve over time?