Wavefunctions and states

chill_factor

this reminds me of a question that I'd like to ask.

as a chemist, my view of quantum mechanics is that it is a useful tool to give the right answer for spectroscopy calculations and to make a model of complicated molecules so that we can pin down some parameters with instruments, then get the whole molecule's structure.

however, i keep reading on some book reviews at Amazon that the "core of quantum mechanics is not the wavefunction, but the state". What is that supposed to mean? I never paid attention to the "philosophical" meanings of quantum mechanics. All that mattered was knowing how to calculate the wavefunction for various boundary conditions and knowing how to use them.

that was until I took a look at the first few pages of Desai's book and didn't actually see many wavefunction symbols in there. this is important news to me because i'm entering a MS in physics soon and don't want to slam into a brick wall due to "learning QM wrong".

Jorriss

Check out bra-ket notation. It's called bra-ket notation but it's really a more abstract formalism for quantum mechanics. When you say the wave function, you are implicitly saying the state of the system projected onto either the position or momentum basis (or maybe energy basis too, as in the fourier coefficients, but that's not too important for what I'm talking about). The most abstract state of a system though, is just some vector in a space without regard to a particular basis.

chill_factor

i learned Dirac's braket notation in quantum chemistry, in the book they just said it was a shorthand way of writing out integrals between wavefunctions and operators. like for example they had an example <1|N|2> is just Integrate[psi1* x operator N(psi2), dAllspace]. Had some examples where they make us calculate expected values for momentum, kinetic energy, etc. in Dirac notation.

i still can't understand the Wiki page.

Jorriss

You learned quantum from McQuarrie?

M Quack

Wave functions are just one way to represent a quantum mechanical state.

In many cases you can derive the physics of a simple system by just knowing the Eigenvalues of an operator (or a few operators, for example angular momentum or spin), you don't have to determine the complete wave function.

The states then can be written in vector form, and the operators as matrices. (that is why expectatoin values are often called matrix elements) We can then do all the fun physics using simple matrices and leave the derivation of the exact wave functions to the chemists :-) The Bra-Ket notation is particularly useful for that.

chill_factor

Atkins Quanta Matter and Change.

i think i will have to cut my vacation this summer short and spend a month at academic boot camp relearning quantum.

Jorriss

I wouldn't put it that way, it's not like you really learned stuff wrong, just not the full story :). But it would be a good idea to get a different text book like Shankar or Sakurai and get a taste of more abstract quantum.

nonequilibrium

But what can a "molecule's structure" mean then? I mean, if you regard QM simply as a calculational tool, and not something that gives ontology, then what do you mean/imagine when you're saying "molecule's structure"? Perhaps a Newtonian image of some spheres hanging about? But then again, you know Newton is/Newtonian concepts are heavily outdated...

My intention is not to go off-topic or something, but I'm just wondering if you realize that the naive approach you're painting (or am I misinterpreting?) doesn't seem to create a coherent picture. Just trying to get you to think about it :) not trying to belittle/act superior...

cgk

To OP: In the common understanding in quantum chemistry and all parts of solid state physics I am aware of, a 'state' and a 'wave function' are the different names for the exactly same thing (and for one-particle wave functions, a further synonym is 'orbital', but unlike orbitals, states and wave functions also come in N-body varieties). Maybe some physicists like to emphasize different aspects of wave functions with the two words, but there is little reason to go with them. You have not learned quantum mechanics "wrong".

EDIT: Okay, maybe one note: In statistical mechanics, some there also are "non-pure states", described by thermodynamic (ensemble-) density matrices (projectors onto wave functions). Those density matrices represent ensembles of wave functions (not single wave functions) and can and often are regarded as "states" themselves, as they are the maximal description of statistical ensembles. However, in some sense they are not as fundamental as wave functions, because statistical quantum mechanics can be derived from standard quantum mechanics alone and a few statistical assumptions (ergodicity principle etc). Maybe the author was referring to that. But unless you are working with statistical mechanics, there is not need to regard such density matrices as fundamental 'states'.

Oudeis Eimi

However, it should be noted that a wave function is actually a set of expansion coefficients of a ket (abstract state vector) in a certain base. In elementary QM this base is usually chosen as the position base, but IMHO if one is going to bother with different bases at all, it more than pays off going all the way and learning the more general Dirac formulation.

I have to disagree completely. Some books do introduce density matrices as statistical ensembles, but that's not the whole story. Indeed, density matrices are the general states, and it is only a subset of them (those who are also projector operators) which can be associated with a ray of state vectors. The rest cannot be represented with kets, and this is so regardless of how much information we possess about the system. The typical example of this is an isolated system composed of two interacting particles; in principle we can assume that the system can be prepared in a pure state (with no statistical uncertainty), but the state of each component particle will in general only be representable by a strictly impure (Tr ρ² strictly lesser than Tr ρ) density matrix, for which a coresponding vector ray doesn't exist. It just happens that the machinery of density matrices can also represent statistical ensembles together with (and in addition to) 'objective' impure states, using the same formulation.

cgk

No one says that wave functions have to be represented in a position basis. In fact, in most cases they are not. But if you re-express 〈x₁x₂x₃...|ψ〉as 〈q₁q₂q₃...|ψ〉it is still a wave function.

I did not claim otherwise. But both open-system treatments with density matrices and statistical treatments with density matrices can be *derived* from quantum mechanics on pure states (in the former case from the QM on the total system, of which the density matrix describes a partial trace over some degrees of freedom, in the latter case by the entropy principle and pure states). That means: You get equations referring to density matrices alone, but what you are doing is still solidly founded in wave function theory, even if the wave functions do not explicitly occur. But you *could* describe the same system with wave functions if you wanted (in your open-system case by treating the full quantum system with a wave function instead of the partial system with a density matrix).

Thus, there is no reason to regard density matrices as the fundamental objects instead of wave functions, because both formalisms can be derived from each other. And I personally find it hard to regard density matrices as anything but calculation tools to use when lacking some information about about a system, but this is only a matter of taste, nothing more (in both ways!).

chill_factor

i think my problem may be weak linear algebra and just not getting dirac notation.

i never thought about this actually. depending on problem, its sometimes more useful to think of a molecule as Newtonian sticks and balls, and other times in quantum terms.