Talk:Hamiltonian (quantum mechanics)

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Cross terms for kinetic energy?[edit]

Which is it?

\hat{H}=-\frac{\hbar^2}{2}\left( \sum_i \frac{1}{m_i} \nabla^2_{\mathbf{r}_i}\right) + V(\mathbf{r}_1,\ldots ,\mathbf{r}_N,t)

or

\hat{H}=-\frac{\hbar^2}{2}\left( \sum_i \frac{1}{m_i}\nabla^2_{\mathbf{r}_i} + \frac{1}{2}\sum_{i\neq j} \frac{1}{m_i+m_j}\nabla_{\mathbf{r}_i}\cdot \nabla_{\mathbf{r}_j}\right)+ V(\mathbf{r}_1,\ldots ,\mathbf{r}_N,t) ???

(N-particle Hamiltonians)

Under what conditions is the latter more appropriate than the former? Where, mathematically, do these terms come from? This needs expansion. My text doesn't mention anything about cross terms. I would be very grateful for clarification!

24.22.53.72 (talk) 06:32, 10 April 2013 (UTC)

Total Energy?[edit]

I came to this page looking for clarification on what my physics text states, however it falls into the exact same esoteric pattern. Exactly what does "total energy of the system" quantify? Further, the next sentence talks about variability in the total energy measured. Thus, in what sense is it total energy? Is it a probability distribution? An average?

"Total energy" just means kinetic + potential energy. Why don't they just call it energy and be done with it? It's because the motion of bodies is calculated a different way. Normally, we use Newton's law F=ma, and we can say that a body of mass m moved with this trajectory because ot was accelerated by these forces. If we want to calculate the energy total energy = sum of potential and kinetic energies, we can derive it from the speed of the body, and it's movement against the forces. However, you could derive the trajectory of the body a completely different way: use your knowledge of the total energy of the body. If it was a bead moving along a wire, you might know that the total energy is constant. It would be hard to use Newtonian mechanic to calculate what the bead will do. The alternate way of calculating it is called Hamiltonian (classical) mechanics, and the Hamiltonian is just the energy, as I said. But we came at the problem from a different direction, so historically "energy" here has a different name - "hamiltonian".

Similarly, I think it's foolish that no where on the page can the equation H*Phi=E*Phi can be found. I know that equation is a huge simplification, but I think I can say with certainty that 80% of the people who come to the page will be most familiar with that equation given that it's frequently the first equation presented in Quantum classes having reference to the Hamiltonian operator.

Finally, I agree with the previous poster that this page has come down with "properties but not essence" syndrome, wherein the text rambles on about the properties of the subject but not the subject itself. If, as with many things in physics, it's difficult to describe it should be broken down into different conceptions or interpretations.

"total energy of the system" isn't jargon, it states the Hamiltonian of a system is the total amount of energy of the system. Besides lacking a definition of energy which is not approriate here.

Per WP:JARGON terms that will not be understood by the average reader are especially important to define before they are used. I believe this is recursive, and introductory background is often paragraphs in length in similar articles. Listing Port (talk) 23:04, 9 April 2008 (UTC)

Some questions[edit]

  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 modeling 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


  1. The Hamiltonian is the operator, not the system itself, so this change would be incorrect.
  2. Bra-ket notation has nothing to do with the Hamiltonian specifically, but that link will be for people who haven't come across bra-ket notation yet.
Marbini (talk) 23:29, 27 August 2009 (UTC)

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)

What is the hamiltonian?[edit]

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 p^{2}/2m, see the article on momentum), and the second term is potential. KristinLee 00:28, 13 May 2006 (UTC)
The problem here isn't that the article isn't clear, simply that people are demanding you be able to read the text without having any knowlage needed to understant what an Hamiltonian is. Referencing hilbertspaces and dirac notation should be sufficient. The Hamiltonian in the link is one possible Hamiltonian, depending on what forces and interactions you include, the total energy of the given system be defined differently, and thus the Hamiltonian will change.
WP:JARGON has some different advice, which, it says, is particularly important for math articles. Listing Port (talk) 23:12, 9 April 2008 (UTC)

spectrum conflicts with set[edit]

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)

The term spectrum is used in a variety of contexts. Here, the term is probably borrowed from functional analysis but I don't think it should be thought of as a quantum mechanics term of art. In functional analysis, the spectrum of an operator is defined by its set of eigenvalues. The spectrum of an operator is sometimes discrete, sometimes continuous and sometimes a combination. Outside of quantum machinics, the term continues to be used in the same sense. A given operator may have various particular spectrums depending upon boundary conditions. In general fixed boundaries produce descrete spectrums and free boundaries produce continues spectra. (I hope you are still monitoring after the long wait for an answer!) --scanyon 06:14, 6 June 2007 (UTC)


I am not really sure, whether you should give some definition of the hamiltonian without using fundamental concepts like hilbert space and diracs notation. The hyperphysics definiton is just a special case for 1D, 1 Particle etc... for the sake of unversality, one should use the proper notation, maybe with a comment like "If you are not familiar with Dirac-Notation click ->here<-" or something... regards, sascha —Preceding unsigned comment added by 91.11.194.95 (talk) 11:34, 3 June 2008 (UTC)

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

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)

Eigenket?[edit]

Shouldn't it be eigenkeit instead of eigenket? It's spread around the article and even appears in one of the headings. Sakkura 14:55, 30 November 2007 (UTC)

I've always heard eigenket- not that I've covered bra-ket notation in university yet but I read a lot, and have never come across the word eigenkeit- it's not a word I can find a translation for online either. I'm inclined to say it's a typo, as I think Dirac invented the notation and he certainly refers to them as kets.Marbini (talk) 16:14, 6 January 2008 (UTC)

Have a look here. --TraceyR (talk) 11:22, 16 May 2009 (UTC)

Eigenket is correct, since it is referring to Dirac bra-ket notation.

Font[edit]

Shouldn't the Hamiltonian be represented with a curly capital H? <math>\mathcal{H}</math> produces \mathcal{H}, which would appear to be the closest symbol available, and is used in the Hamiltonian mechanics article. Modest Genius talk 03:15, 16 October 2008 (UTC)

I've always seen the Hamiltonian written as a straight H or as \hat H The \mathcal{H} symbol usually represents Hilbert space. Hilbert space is relevant to the Hamiltonian operator, so the symbols should be different to avoid confusion.
Ghazwozza (talk) 23:20, 20 October 2008 (UTC)
The Hamiltonian written as H usually refers to the quantum mechanical Hamiltonian, while a cursive version is reserved for the classical Hamiltonian (not the \mathcal{H}, but an even fancier version which I haven't found in TeX yet). See (for example) Shankar's quantum mechanics text book, which uses this convention. I believe I have seen other texts which use this convention as well, and I have certainly had professors use this convention in most of my undergraduate and graduate physics courses. zipz0p (talk) 04:44, 19 April 2009 (UTC)

While this does not pertain directly to quantum mechanics, Jackson's electrodynamics text uses the script H as a Hamiltonian density (describing the Hamiltonian in terms of continuous fields rather than discrete coordinates), and reserves the typical H for the discrete Hamiltonian of point particles (and charges). 128.101.126.238 (talk) 05:27, 21 April 2009 (UTC)

It turns out this does have relevance to QM, in the context of quantum field theory. Marbini (talk) 23:16, 27 August 2009 (UTC)

Notation[edit]

Shouldn't there be some sort of link explaining the use of the vertical bar preceding and the arrow following the wave-function phi ? The meaning is not at all clear and therefore the article in its entirety is of little use. Could the author not correct this please ? —Preceding unsigned comment added by 92.236.77.63 (talk) 14:46, 21 November 2008 (UTC)

Sir William Rowan Hamilton[edit]

The above Irish mathematician seems to be ignored throughout Wikipedia wherever the term "Hamiltonian" is used. --TraceyR (talk) 11:18, 16 May 2009 (UTC)

Definition at the top[edit]

Regarding this sentence at the beginning: "It is a Hermitian matrix, that, when multiplied by the column vector representing the state of the system, gives a vector representing the total energy of the system."

It gives a vector "representing" the total energy? Are you sure? In what way does this vector represent the total energy? As I understand it, the vector that it produces will be the sum of the total energy eigenstate components of the original vector, each multiplied by its corresponding total energy. Not much of a "representation", if you ask me. I think this statement may be someone's misunderstanding.

Also, a lot of people seem to be complaining that the page uses jargon and is inaccessible to laymen. While I agree with these statements entirely (as a neophyte the subject myself), and in other cases I sympathize, I can't imagine how you can make such a sophisticated and complex concept as a quantum operator accessible to a non-physicist. I mean, even physicists often struggle to completely understand these things at an intuitive level. This stuff is just plain intrinsically hard. What is the solution to such a problem? Should the page just be deleted? 24.174.30.146 (talk) 03:07, 25 June 2009 (UTC)

Re "if you ask me. I think this statement may be someone's misunderstanding.": Yes, that is not correct. The Hamiltonian itself is the energy observable. Mct mht (talk) 17:42, 1 July 2009 (UTC)
Agreed. I'll try and clean the introduction up a bit now. Marbini (talk) 23:33, 27 August 2009 (UTC)

I really don't think the amount of "jargon" this page uses can be substantially further reduced without teaching people quantum mechanics. Should this tag really still be there? Marbini (talk) 00:25, 10 October 2009 (UTC)

I concur. Perhaps some of the words can become links?
bi  — Preceding unsigned comment added by 42.106.85.153 (talk) 15:11, 9 October 2011 (UTC) 

Sign error?[edit]

Under "Schrodinger formalism", shouldn't the kinetic energy be given as below?

\hat{T} = \frac{\bold{\hat{p}}\cdot\bold{\hat{p}}}{2m} = \frac{\hat{p}^2}{2m} = -\frac{\hbar^2}{2m}\nabla^2

Fixed.--Maschen (talk) 17:39, 8 January 2012 (UTC)

Hamiltonian...what?[edit]

I realize that one often sees the the thing discussed in this article referred to as "The Hamiltonian", but names like this, in the obvious form of an adjective, cry out the question "Hamiltonian what?". The absence of that being spelled out propagates foggy confusion.

I gather from the article that the full name of the thing under discussion is "Hamiltonian operator", or "Hamiltonian function" (are these the same thing?), and to say so explicitly would be an improvement, I think.

There are apparently also Hamiltonian spaces, paths, matrices and so on. Any overview info that would help clarify which of these are related would be welcome!

By the way, the above point aside, I applaud the clarity of the first sentence "In quantum mechanics, the Hamiltonian is the operator corresponding to the total energy of the system." One question though: I wonder if the word "corresponding" is less precise than it might be: Are these other bolder sentences still correct:

  • "In quantum mechanics, the Hamiltonian is the operator which specifies the total energy of the system, or
  • "In quantum mechanics, the Hamiltonian is the function which calculates the total energy of the system."

If these aren't correct, then what does "correspond" mean?

Gwideman (talk) 11:09, 19 August 2012 (UTC)


Using "the Hamiltonian" as a noun to refer to the Hamiltonian operator (an operator is a kind of function, so yes they are the same) is a standard shorthand in physics. Another that might trip you up is "Hamiltonian of a system" to mean the value of the Hamiltonian operator for a given system. The other ways "Hamiltonian" is used as an ajective are afaik unrelated. Hope this helps.
Also, I am not totally sure of this, but I believe "corresponding" refers to the Correspondence principle.
Xuanji (talk) 11:12, 20 August 2012 (UTC)

I'm no expert, but the first and second sentences of the Introduction seem to me to say practically the same thing, or almost. Surely this badly needs editing. Seems that way to me anyway. AP — Preceding unsigned comment added by 58.109.71.99 (talk) 04:41, 1 September 2013 (UTC)

Notation question[edit]

What is the significance of the following notation variants in this article:

  • hat vs no hat (on H for example)
  • bold vs not bold on various variables

I appreciate that these variants may signify vector, normalized, array and so on, but it's sometimes not clear which meanings apply on this page, why they appear only sometimes, and indeed whether this page uses a particular convention consistently. Thanks. Gwideman (talk) 23:37, 14 September 2012 (UTC)