# Hermitian operators = values of variables

1. Jun 4, 2012

### arlesterc

I have read in different places something like the following:
Hermitian operators have real eigenvalues
Hermitian operators/their eigenvalues are the observables in Quantum Mechanics e.g energy
I am not sure what this means physically.

Let us say I have a Hermitian operator operating on a Ket like:
H|A> = λ|A>
H|B> = β|B>

etc. for however many eigenvalues there may be - all real numbers.

And let us say we are talking about the 'energy' operator.

Also for sake of argument the real numbers of the eigenvalues are 1, 2, 3, 4, etc.

What exactly is the energy operator doing to the Ket vectors in physical terms - if there is a physical meaning to H operating on A? Lamda operating on A? How does energy 'operate' on anything?

What is the physical meaning of the ket vector - if there is one?

Similarly what are the physical meanings, if any, of the eigenvalues i.e. 1, 2, 3, 4 ?

What does it physically mean that the eigenvalue 2 as opposed to 1 is multiplying ket vector A versus Ket Vector B?

What type of experiment/s - again in the energy area - would I do to get the eigenvalue 'observables'?

I appreciate in advance any time spent on clarifying the above for me.

2. Jun 4, 2012

### Matterwave

The state vector (kets) mathematically represents the state of the particle or system in question (for simplicity, let's just talk about one particle). This ket represents everything we know (and everything we could know) about the particle.

However, we need a way to extract the information that we need. That is where operators come in.

One should always remember that these objects are mathematical representations of the physical thing and as such they obey certain rules that we impose on them. We must always know what question we are asking corresponds to what mathematical operation.

So, for example, "What is the expectation value for the energy of this particle?" would be answered by <E>=<A|H|A> given that the particle is in state |A>. If |A> happens to be an eigenstate of H, then we know that <A|H|A>=<A|λ|A>=λ<A|A>=λ (assuming your state is normalized). Thus, eigenstates of H have definite energies.

So, to answer any quantum mechanics question, we must first turn the English question into a mathematical question (just like for any word problem) and then find the solution given the rules of quantum mechanics.