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According to the bootstrap philosophy, not only is a hydrogen atom a bound state of a proton and an electron, but a proton is a bound state of a hydrogen atom and an electron, and an electron is a bound state of a hydrogen atom and a proton.
- Could someone explain how this makes any sense? —Keenan Pepper 22:48, 10 December 2005 (UTC)
- Um.. yea.. it doesn't. Fresheneesz 02:47, 22 May 2006 (UTC)
Nope, seems like a bit of horseshit to me.
Multiple problems in the introduction
There are a number of statements in the introduction to this article that are problematic, to say the least.
If someone gets around to fixing this article, please keep in mind the following: the concept of a bound state in quantum mechanics makes sense even for a single particle in an external potential---say, for the quantum mechanical harmonic oscillator. Let us indeed talk about a particle in a potential, say in 1D for simplicity. Then the particle is said to be in a bound state if it is never found “too far away from any finite position”---in other words, if the probability that it is going to be found somewhere in the interval or goes to zero as goes to infinity.
In other words, it is in a bound state if goes to zero whenever .
In other words, it is in a bound state if is finite whenever .
In other words, a state is a bound state if and only if it is normalizable. The argument here was for a single particle in 1D, but it generalizes.
So, a bound state just IS a state in the Hilbert space, period. Scattering states are not normalizable, and so are technically NOT elements of the Hilbert space. To really treat scattering states within functional analysis, one must introduce the rigged Hilbert spaces.
That's one thing. Another thing: remember that the states of the harmonic oscillator are all bound states, with positive energies. In other words, the oft-repeated statement that “bound states are negative energy states” is only true for a certain class of potentials---those which go to zero at plus and minus infinity (the proof for that is implicit in, for example, the discussion in Messiah's textbook). The harmonic potential blows up at infinities, and so the bound state energies are not obliged to be negative, and indeed they are not.
Moreover, in quantum mechanics it as true as in classical mechanics that what we call zero potential is an arbitrary choice that should have no physical consequences. I may shift what I call zero energy up and down, and the physics should remain the same. But whether a state is bound or not is not conventional---boundedness has real physical content---and so it could not be equivalent to some statement that is conventional, such as whether the energy is positive or negative (the latter is conventional, again, because I can define zero potential energy to be whatever I want).
Having said that, the class of potentials that vanishes at infinity is an important class, and for this class, the link between boundedness (i.e. normalizability) on the one hand, and of the negativity of the energy on the other, does hold. This should be emphasized.
With the above remarks in mind, here come a list of problematic statements in the introduction as it is:
1. “a bound state is a composite of two or more building blocks (particles or bodies) that behaves as a single object.” There is some truth to this at the intuitive level, but it seems to preclude bound states of single particles in external potentials. Also, in what sense does it “behave as a single object”?
2. “a bound state is a state in the Hilbert space that corresponds to two or more particles...” (what about a particle in an external potential?) “...whose interaction energy is negative, and therefore these particles cannot be separated unless energy is spent.” Again: what about the harmonic oscillator? The interaction energy is positive definite; it's just that the trapping potential blows up at infinities.
3. “The energy spectrum of a bound state is discrete...” This simply makes no sense. States do not have energy spectra; Hamiltonians do. The closest correct statement is something like “the energy spectrum corresponding to bound states is discrete,” but probably the whole thing should be rewritten.
4. “In general, a stable bound state is said to exist in a given potential of some dimension if stationary wavefunctions exist (normalized in the range of the potential).” Right idea (as best as I can tell), but awkwardly put. “The energies of these wavefunctions are negative.” Again the negativity of the energy fallacy. Reuqr (talk) 04:17, 29 March 2011 (UTC)
Problems with Chapter: In mathematical quantum physics
This chapter makes no sense at all. It should be removed. On the one hand it does not help any expert, since it just states some definition and on the other hand is is not an appropriate definition for the following reason
It might be right for a particle moving to the right (as stated in the example), but it is clearly wrong for a particle moving to the left, since the definition only took the limit R->inf and x>R —Preceding unsigned comment added by 22.214.171.124 (talk) 14:55, 16 May 2011 (UTC)
Article has serious problems
(A) I agree with Reuqr: the introduction should be rewritten to include the case where a single particle interacts with a potential. An expert can "read through" this, a beginner not.
Also it should be rewritten to allow for potentials that do not go to a constant at infinity.
Most or all of his points are good.
(B ) I agree with 126.96.36.199 that the section called In mathematical quantum physics has serious problems.
It is stated very technically, very hard to read, too formal, but it is also incorrect.
For example, why is ρ(t) a "statistical operator" (whatever that is) and not a state?
No connection is made between U(t) and the thing μ(A,ρ(t)) we are measuring.
The statement in the example about an expanding support is wrong. If the support is infinite (because of infinite propagation speed, as for Schroedinger), or just continually expands, but the probability remains uniformly near zero at large distances, then it's still bound.
(I am assuming that the definition is that the probability distributions
(C) The section called In mathematical quantum physics is very general -- perhaps too general -- in the following ways, some of which seem to contain further errors:
(1) It is stated for any 1-parameter subgroup of a Hilbert space instead of for a concrete physical example, such as the solution operator of the Schroedinger equation.
(2) It is stated for any evolution, not just for stationary states. Actually, this is a good choice. If it were correct.
(3) It is stated for any observable A, not just for position. Is there really such a thing as "bound state" for a general observable A?
(4) μ(A,ρ(t)) is not defined or referenced and I can only guess that it has some definition that somehow generalizes the usual probability density μ=|φ|^2, to something else, depending in A. But since the usual probability density μ=|φ|^2 does not reference the position operator, I don't see how it can be generalized to an A.
(5) Despite all the previous over-generality, the spatial domain is arbitrarily restricted to 1-dimensional, when 2- or 3-dimensional are quite realistic.
This conflicts with the earlier Hilbert space, which was fully general and was not even assumed to be a function space. Can you even define a bound state in an abstract Hilbert space?
(6) The Borel algebra on R is invoked, this is overkill.
In other words, I think this section may have been written by a pure mathematician, while drinking heavily, so that the math isn't right or even coherent.
In the intro, it says the interaction energy should be less than total separated energy. But the bound energy is different than the interaction energy so this is wrong.