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Certainly a simple question but I am lost; can the Hamilton operator of a particle be a matrix [H] ? If yes, must this matrix be self adjoint [H = h(i,j)] = [H = h(j,i)]*? And if yes: why or because of why ? Thanks
Blackforest said:OK; thanks. Concerning the fact that the Hamiltonian operator under its form of a matrix who should be self adjoint to give us real eigenvalues: does it mean -a contrario- that a non selfadjoint matrix doesn't give real eigenvalues and more generaly doesn't give eigenvalues with physical sense and reality? Or with other words: in developping a theory if I find an Hamiltonian operator whose matrix representation is not self adjoint: did I make somewhere a mistake?
*****************selfAdjoint said:Basically yes. Sometimes quantum physicists use Hermitian instead of self adjoint. The differences are subtle, but the upshot is that self adjointness is better. As you say, for the eigenvalues to be relevant to our world, they have to be real, and the way to guarantee that is to insist on self adjointness..
reilly said:*****************
I'm very puzzled. A standard definition of a Hermitian operator/matrix H, is that H is Hermitian if and only if H is self adjoint, that is it is equal to the complex conjugate of it's transpose. This is commonly accepted by physicists and mathematicians, and has been for a century or so -- or so i thought.
I would be most greatful for a reference or discussion of the difference between a self-adjoint and a hermitian operator/matrix.
Thank you,
Reilly Atkinson
Blackforest said:What is exactly this "LSZ" formalism?
When the Hilbert space is infinite-dimensional, a difficulty arises in that many linear operators cannot be defined on the entire space. For example, think of the position operator Q for a particle moving in one dimension. Since Q takes the function f(q) to qf(q), there will be many square-integrable functions f(q) such that qf(q) is not square-integrable. This means that Q cannot be defined on the entire Hilbert space. On account of this sort of phenomenon, the mathematician is forced to take into consideration the domain (a linear subspace of the Hilbert space) on which a linear operator is defined.rick1138 said:What is the difference between SA and Hermitian operators? I have heard it mentioned a few times, but have never heard anyone state it explicitly.
And Robphy found someone else who, in an example, (just how?) also gets it backwards:turin said:I thought that Hermiticity was the more strict condition: being self-adjoint-ness with the additional requirement on the boundary.
_______________________robphy said:http://mathworld.wolfram.com/Self-Adjoint.html
Near the end of this description, there is a comment confirming what turin said above.
I haven't checked von Neumann's book ... but you are right, there do exist sources which define "hermitian" as you say. In that case, any time I mentioned the word "hermitian" above, you should replace it with the word "symmetric" to denote the corresponding property.reilly said:I'm very puzzled. A standard definition of a Hermitian operator/matrix H, is that H is Hermitian if and only if H is self adjoint ... [and then in a later post] ... In fact, von Neumann in his Mathematical Foundations of Quantum Mechanics gives the definition Hermitian if and only if self adjoint, in Hilbert Space.
On that page, the Hilbert space is finite-dimensional, for which the above mentioned "domain problems" do not exist. At the outset one can talk about linear operators acting on the entire space, and none of the above distinctions need to be considered.reilly said:I checked out the Wolfram page ... If you go to the Conjugate Transpose page via the given link, you'll find my definition. So I'll stick with von Neumann's definition.
What (I think) he wishes to say is:slyboy said:Just to complicate the discussion further, an operator does not have to be self-adjoint in order to have real eigenvalues. It need only be normal, i.e. commute with its adjoint.
Yes. And that is an example of how a physicist's intuition can create something beautiful, useful, and perfectly consistent.reilly said:Historically most physicists have often used math in a non-rigorous fashion. This has been particularly true in quantum physics -- a very well known example is Dirac's delta function.
Yes, often a specification of the domain of a linear operator acting in a Hilbert space of functions amounts to a specification what boundary conditions the functions must satisfy.reilly said:You'll note in the papers cited by robphy, the subtle problems of Hermitean and selfadjoint operators typically involve infinite intervals and boundary conditions ...
Yes, I agree. (... On the other hand, if we are talking about technical definitions, then there are two distinct types of mathematical objects, one described as "self-adjoint" and another described as "symmetric". Some sources equate "hermitian" with "symmetric", while others (apparently, the older ones) equate it with "self-adjoint". The main point is, as far as technical definitions go, "self-adjoint" implies "symmetric", and not the other way around ... and, in a finite-dimensional Hilbert space, such a distinction becomes trivial (in the sense that any linear operator whose domain of definition is not all of H has an extension which acts on all of H.))reilly said:With my old fashioned ways, hermitian and self adjoint are the same thing -- if not, physical intuition will guide you to the appropriate resolution.
The Hamilton Operator, also known as the Hamiltonian, is a mathematical operator used in quantum mechanics to describe the total energy of a system. It is important because it allows us to make predictions about the behavior and properties of quantum systems.
The Hamilton Operator is a mathematical operator, while the Hamiltonian is the corresponding physical quantity. In other words, the Hamiltonian is the observable quantity that is associated with the Hamilton Operator.
The Hamilton Operator can be represented in matrix form by using the position and momentum operators. These operators are represented by matrices, and when combined, they form the matrix representation of the Hamilton Operator.
The eigenvalues of the Hamilton Operator represent the possible energy states of a quantum system. They are the solutions to the Schrödinger equation, which describes the time evolution of a quantum system.
The Hamilton Operator is used in solving quantum mechanical problems by providing a way to calculate the energy of a system and make predictions about its behavior. It is also used in the Schrödinger equation, which is the fundamental equation of quantum mechanics.