Talk:Hamiltonian (quantum mechanics)

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[edit] Some questions

  1. "In quantum mechanics, the Hamiltonian is the observable state of a system corresponding to the total energy of that system."
    • Is there anything wrong with phrasing it this way?
  2. "The eigenkets" — link to Bra-ket notation? Explain the relationship with Bra-ket notation?
  3. "Depending on the Hilbert space of the system" How do Hilbert spaces differ? How do they relate to the modelling of physical systems? Are the Hilbert spaces used here of infinite dimensions?
    • The relationship between the Hilbert space containing all possible states of the system (?) and the equation given is not clear. Where is the vector in this space?

In response to 3, it depends. Say your system consists of one particle with two internal states (say, "spin up" and "spin down") and some finite number n of possible positions. Then the particle has a total of 2n possible states, and your Hilbert space is 2n dimensional. However, if your particle can be anywhere in some region of space (i.e., it's not constrained to a discrete set of positions) then your Hilbert space will be infinite dimensional. Your Hilbert space also depends on how many particles are in your system, how many internal states those particles have, whether they're distinguishable or indistinguishable particles, etc.

If this is unclear, it should probably be clarified somewhere, but I'm not sure if it should be here or in the article on Hilbert spaces. The latter seems much more focused on the mathematical definition of a Hilbert space... although I think that's probably appropriate. Perhaps we could create a separate article (or subsection of that article) entitled "Uses of Hilbert spaces in physics" -- Tim314

I've not found the information I was looking for (what "Cauchy propagators" are). I think this is the wrong branch of QM; I need one of the time dependent formulations (?) such as Feynman's. Interesting, though, and worth reviewing before I try to understand that.

Mr. Jones 21:07, 1 Oct 2004 (UTC)

[edit] What is the hamiltonian?

This page (like other pages that just jump into bra-ket notation) doesn't explain what the article's subject really is. Hyperphysics seems to suggest that the hamiltonian can be defined like this:

H = - \frac{\hbar^2}{2m} \frac{\partial^2}{(\partial x)^2} + U(x)

where U is potential energy.

Is this correct? Incorrect? Close? I have no idea how to read bra-ket notation (and the article on it doesn't help) - so I'm completely lost on this page, as I would guess that *most* people are who've come here. Fresheneesz 06:37, 30 April 2006 (UTC)

I agree, this page does jump into rather technical definitions right away. People don't really need to understand Hilbert spaces to get an idea of what the Hamiltonian is. The hyperphysics definition is an appropriate one for a 1-dimensional system. In general, it's a measure of the total energy of the system - the first term above is the kinetic energy (simply p2 / 2m, see the article on momentum), and the second term is potential. KristinLee 00:28, 13 May 2006 (UTC)

[edit] spectrum conflicts with set

The text says that the spectrum ... is the set...., but the Wiki article for spectrum says that it represents a continuum, whereas, although not linked, the Wiki definition of set refers to "distinct things".

My guess is that spectrum is referring to some quantum mechanics definition, which is not presently in Wikipedia, so probably needs to be set (quantum mechanics) (broken at time of writing). I.E. it looks like an over enthusiastic wikification to me.

David Woolley 19:28, 13 November 2006 (UTC)

[edit] "Free States" article has nothing to do with quantum states.

I just noticed that the link to "Free States" takes you to a political page about governments and other such nonsense unrelated to the free states of particles in quantum physics. Don't know how to fix it, but there you have it.71.125.60.42 05:44, 10 December 2006 (UTC)

thanks for pointing it out. that link has been removed. Mct mht 15:27, 10 December 2006 (UTC)
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