[Physics] Getting Energies and Probabilities from the Hamiltonian

hamiltonianquantum mechanics

So I need to find the possible energies and the probabilities of these using the eigenvalues of a Hamiltonian.

Once I obtain the eigenvalues, are those the energies E_n in and of themselves?
Or do they simply give me the n values, i.e. n = 1, 2, 3, that I would then plug in to the equation

Equation for finding energies

Or is it both? Do they both yield the same answer? (I am still waiting on the installation of the computer program to use to find the eigenvalues)

Finally, I am completely at a loss as to how to go on to find the probabilities of the energies. I am not given a traditional wavefunction to normalize, so how do I find the probability without the normalization constant?

EDIT: I normally don't like to put problem specifics from my homework on here, but I suppose it's hard to understand what I mean by "I am not given a traditional wave function." As such, the exact problem is stated:

Models describing electrons on a crystal lattice are very important to
understanding various phenomena in solids. Here we consider a model in
which an electron lives on a one-dimensional lattice of N sites. The
sites are labeled by i=1,2, ….,N. The system looks like

  o----o----o-- .... --o----o
  1       2       3           N-1       N

The state of the electron is then a vector of dimension N. The
Hamiltonian is given by an N by N matrix whose elements are:

                / - 1,  if i and j are near-neighbors; 
   H_{ij} = |
                 \ 0  ,  otherwise.

Physically, the electron can be thought of as hopping from site to
site through a near-neighbor hopping. As you see, the Hamiltonian
resembles the one we obtained in class when we discretized the problem
of a particle in a box. Suppose we prepare the electron in a state
|a> with equal amplitude for all N sites, i.e., a(1)=a(2)=….=a(N).
To be specific, let's consider N=5.

What is the lowest value we can find if we measure the energy of the electron? With what probability?
List all the possible (i.e., with non-zero probability) energy values that we could find in such a measurement.

So I used Maple to obtain eigenvalues for N=5, and thus have the energies.

So I suppose the root of my question is: if I am only given this information, how do I know what state the electron is in? Is it as simple as the corresponding eigenvector for each eigenvalue? Can I assume anything about the wave function (i.e. follows particle in a box method)?

Thank you.

Best Answer

Just "plug into the equation" is always a bad idea. So here is a short overview:

  • Given a Hamiltonian, the possible energy levels correspond to the eigenvalues of the Hamiltonian (no "plugging in" needed). More precisely, we have $H|\psi\rangle=E|\psi\rangle$ for every eigenvector.
  • Given a normalized eigenvector, you can find the probability by $\langle \psi |H|\psi \rangle$, otherwise you have to normalize (divide by $\langle \psi|\psi \rangle$).
  • In the density matrix formalism, this means that given a state $\rho$ (positive semidefinite matrix with trace one - otherwise normalize the trace) the probability is given by $\operatorname{tr}(\rho H)$ gives you the probability.

So the question is: what do you mean "I am not given a traditional wavefunction to normalize"?

EDIT: To me, it seems that you are given a perfectly reasonable wave function (in matrix formulation, though). An electron that sits just at site $j$ will have corresponding wave function $|e_j\rangle$, where $e_j$ denotes the $j$-th basis vector (i.e. $|e_j\rangle=(0,\ldots, 0,1,0,\ldots 0)^T$ with the $1$ at position j. Following your assignment, this tells you the wave function of your particle looks like: $$ |a\rangle=\sum_{i=1}^N a|e_i\rangle$$ where $a$ is a complex number and $N|a|^2=1$ for normalization.

In order to find the probabilities, you can now either compute the eigenvectors of $H$ and then decompose $|a\rangle$ in terms of these eigenvectors, or you can compute the spectral decomposition of $H$, i.e. the eigenvalues $\lambda_i$ and projectors $P_i$ such that $H=\sum_{i=1}^n \lambda_i P_i$ and compute $\langle a|P_i|a\rangle$ to obtain the probability of measuring $\lambda_i$.

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