Complete basis for quantum oscillator Hilbert space

[Anonym]

I am trying to study the question in the simplest possible environment, that's why I started with HO and went to quantization of scalar field.

Cosmology, althought quite speculative, is full of quantization of scalar fields.

The quantization of other fields starts with the same assumtion about unique vacuum state and complete basis only the details are different. If you don't like the question for scalar field, then answer it for other fields with all the possible extra complications.

Sure, I’ll try. My remark intended to attract your attention that it is probably impossible to do for the scalar fields. The possible extra complications of spin 1 fields may turn out to be the simplifications.

Regards, Dany.

 

[vanesch]

Let consider this thread as a model. Smallphi started with the simple and clearly defined question. There exists the clear and unique answer to that question (Hans de Vries, post #4). So, what we study from that?

I was taking on a mathematical approach, because the question of proof was posed, and we started from different axiomatic bases. If you talk about the "one-dimensional mechanical harmonic oscillator", that means that you already have introduced a lot of postulates. One is for instance that the system is a classical system with a classical configuration space that follows the usual quantization rules. In other words, you have a classical configuration space, which consists in this case of the real line, and which we describe, say, with the variable "x" which we call "position". We then go on introducing the canonically conjugate variable p, and use the standard rules of quantization of a classical system to obtain a quantum system. You express your hamiltonian inspired by the classical hamiltonian and so on. As such, the quantum system is entirely defined starting from the classical system and the "quantization rules". The question of whether there is a unique ground state has not much to do with the fact that it is a harmonic oscillator.

However, if you talk about the "quantum-mechanical oscillator" you don't have a particular model in mind, you just have an abstract system in mind with a number operator and ladder operators. This is the system I thought (and still think) the OP was talking about. It doesn't need to have any physical interpretation, it is just an axiomatic system. Now, as I said previously, this is like distilling the axiomatic structure of an algebraic field out of usual real-number algebra (you know, you have a set with 2 operations, each operation gives you a commutative group (when you exclude the unit element of the first group from the second), and you have distributivity).
With only this axiomatic structure of a "harmonic oscillator" you cannot prove that there is a unique ground state. The best illustration is a counter example, that was my spin-1/2 particle in a mechanical oscillator.
If you have an example structure which satisfies all the postulated axioms, and it doesn't satisfy the conjectured theorem, then that theorem cannot be deduced from the axiomatic structure.
It is not because in another example, the theorem is right, that this means that it is derivable from the axiom set.

So the point is: *purely* from the axiomatic structure of the abstract harmonic oscillator, one cannot derive that the ground state is unique. This is my point.

Now, by going to a specific case, or by adding axioms (that's in fact the same!), you CAN of course arrive at cases where you can demonstrate that the ground state is unique. An example is indeed the case of the 1-dimensional mechanical oscillator of a point particle. But that was not the question. The question was: can one PROVE, starting from the axioms of an ABSTRACT harmonic oscillator, that a unique ground state exists. The answer is: no one cannot prove that. And the proof of that statement is a counter example.

We go now to the more complicated case. Let say, no one know the answer. So, what is expected? One way: since in the previous case I found the unique answer, it is reasonable to try finding the unique and clear answer also in more complicated case (QED, QCD, QG, etc). Now, you came with the opposite attitude. Using all possible arguments, legal and illegal, legitimate and not legitimate you try to prove only one statement: the previous case was also not unique and clear. It is illusion that Smallphi learn something, Smallphi do not know anything, and will know nothing. It is called agnosticism and you consistently present that POV in every post you wrote.

I wasn't agnostic about the claim! I clearly stated that from the axioms of an abstract harmonic oscillator, one cannot derive that the ground state is unique, and if one wants a unique ground state, one will have to postulate something else. Or one postulates directly that the ground state is unique, or one uses another model in which one can embed a harmonic oscillator. If in that other model, one can derive that the ground state is unique, well then so be it. But I have the *freedom* not to do so, and still use an abstract harmonic oscillator. Whether that is a physically useful thing to do or not doesn't matter here. What matters is that the harmonic oscillator structure is not incompatible with degenerate ground states.

The only questions I have: why you choose science in general and physics in particular to be your profession? And why you consider that it is your duty to convince a newcomer that the study physics is meaningless?

It is always good to know what are the logical relationships between the different postulates and results one obtains. If not, one locks oneself up into a kind of religion where absolute truths have been hammered in stone. A typical example is non-Euclidean geometry. It is not because everybody "knew" that the 5th postulate was "true" that it was meaningless to question it and to explore its logical relations to other statements.

The study of physics, or the study of mathematics, or any study is the study of relationships between statements. It is not because in summer the days are longer, and hot things usually expand, that it is because the days are hot that they expanded. It is not because we think at a certain moment that two statements are true that you can necessarily derive one from the other. And you always have to keep in mind that "absolute truth" is a fata morgana

But it is not because we cannot know anything with absolute truth that we must give up trying to think about it. It is not because you don't know what tomorrow will bring that you must not live tomorrow.

P.S. Vanesch, perhaps you intuitively looking for the justification of the statistical interpretation of QM on the level of the single particle system. You try to catch fata morgana, illusion. There is no such animal in nature.

How do you know with such certainty ?
But this was not the discussion here. It was a purely mathematical discussion: what properties, within an axiomatic system, can be derived from what axiom set.

 

[Anonym]

I consider your post beautiful. I consider your duty to write your posts always at the same level which now has the empirical justification. However I consider also your duty to emphasize that neither mathematics nor mathematical physics and nor theoretical physics is the area of your professional competence, that you are the experimentalist. And I know only three examples of the physicist that had the professional level of competence in both: G.Galilei, I. Newton and E. Fermi.

I was taking on a mathematical approach, because the question of proof was posed, and we started from different axiomatic bases. If you talk about the "one-dimensional mechanical harmonic oscillator", that means that you already have introduced a lot of postulates. One is for instance that the system is a classical system with a classical configuration space that follows the usual quantization rules. In other words, you have a classical configuration space, which consists in this case of the real line, and which we describe, say, with the variable "x" which we call "position". We then go on introducing the canonically conjugate variable p, and use the standard rules of quantization of a classical system to obtain a quantum system. You express your hamiltonian inspired by the classical hamiltonian and so on. As such, the quantum system is entirely defined starting from the classical system and the "quantization rules". The question of whether there is a unique ground state has not much to do with the fact that it is a harmonic oscillator.

However, if you talk about the "quantum-mechanical oscillator" you don't have a particular model in mind, you just have an abstract system in mind with a number operator and ladder operators. This is the system I thought (and still think) the OP was talking about. It doesn't need to have any physical interpretation, it is just an axiomatic system. Now, as I said previously, this is like distilling the axiomatic structure of an algebraic field out of usual real-number algebra (you know, you have a set with 2 operations, each operation gives you a commutative group (when you exclude the unit element of the first group from the second), and you have distributivity).
With only this axiomatic structure of a "harmonic oscillator" you cannot prove that there is a unique ground state. The best illustration is a counter example, that was my spin-1/2 particle in a mechanical oscillator.
If you have an example structure which satisfies all the postulated axioms, and it doesn't satisfy the conjectured theorem, then that theorem cannot be deduced from the axiomatic structure.
It is not because in another example, the theorem is right, that this means that it is derivable from the axiom set.

So the point is: *purely* from the axiomatic structure of the abstract harmonic oscillator, one cannot derive that the ground state is unique. This is my point.

“one cannot derive that the ground state is unique.” You should write:” I cannot derive that the ground state is unique.”

I did not understand even one single statement and certainly this is not what I have in my mind. But it is not a matter. You present your POV and I presented mine above. There are others. You cannot convince me and I do not try to convince you. I am only sensitive to the stupid propaganda. The only statement that I agree with you is:” "absolute truth" is a fata morgana By the way, the line is described by x in mathematics, in physics it is described by the acceleration.

A typical example is non-Euclidean geometry. It is not because everybody "knew" that the 5th postulate was "true" that it was meaningless to question it and to explore its logical relations to other statements.

I do not understand what you say and what you want to say. Please provide your formulation of the 5th postulate of the Geometry.

Regards, Dany.

P.S. I would like to attract your attention to the discussion in “HUP and Particle Accelerators”. I would appreciate to read your comment. It is strictly your area of competence and I need your help.

 

[Careful]

It is always good to know what are the logical relationships between the different postulates and results one obtains. If not, one locks oneself up into a kind of religion where absolute truths have been hammered in stone. A typical example is non-Euclidean geometry. It is not because everybody "knew" that the 5th postulate was "true" that it was meaningless to question it and to explore its logical relations to other statements.

The study of physics, or the study of mathematics, or any study is the study of relationships between statements. It is not because in summer the days are longer, and hot things usually expand, that it is because the days are hot that they expanded. It is not because we think at a certain moment that two statements are true that you can necessarily derive one from the other. And you always have to keep in mind that "absolute truth" is a fata morgana

But it is not because we cannot know anything with absolute truth that we must give up trying to think about it. It is not because you don't know what tomorrow will bring that you must not live tomorrow.



How do you know with such certainty ?
But this was not the discussion here. It was a purely mathematical discussion: what properties, within an axiomatic system, can be derived from what axiom set.

Hi Patrick, if you consider logical positivism to be the statement : ``all true things are measured through experiment'' then you (actually Bertrand Russel was the first to notice it) arrive at a contradiction because the statement itself isn't measurable. That is why Bohr and Heisenberg had to define the ``subjective truth'', some can live with it, others can't.

Cheers,

Careful

 

[Anonym]

You cannot discuss physics without taking into account special relativity, that is like going to restaurant and eat with bare hands (which might still be a habit in some parts of the world).

Hi, Careful

I think I made some progress in my understanding of the relativistic QM. I hope invite you to our restaurant in the near future. However as you may see above I remained poor mind.

By the way, the statement itself is easily measurable, the content is not measurable(C.E.Shannon).

Regards, Dany.

 

[Careful]

Hi, Careful
By the way, the statement itself is easily measurable, the content is not measurable(C.E.Shannon).

Regards, Dany.

Ok, fair enough

 

[vanesch]

“one cannot derive that the ground state is unique.” You should write:” I cannot derive that the ground state is unique.”

This is starting to become ridiculous. The statement "one cannot derive that the ground state is unique" is proven in my posts. As such it is not a statement of incompetence, but a statement of impossibility.

(btw, the proof of that statement is by counter example: I set up a simple example which satisfies the proposed axioms for the harmonic oscillator, and which has a degenerate ground state. As such, it is IMPOSSIBLE to find a proof that such an example cannot exist! (which is exactly the claim that "unique ground state" follows from the axioms of the harmonic oscillator))

I did not understand even one single statement and certainly this is not what I have in my mind. But it is not a matter.

It does. If you cannot follow the simple mathematical reasoning I presented in previous posts, then I do not think you are qualified to criticize what I'm qualified to do. Although I am indeed an experimentalist (but not only, I have other degrees too, and I'm preparing others), even an experimentalist knows some mathematics, and the level needed here is what every physicist should know.

To repeat, the starting axioms were those of the "harmonic oscillator", which are:

You have a hilbert space.
1) In that hilbert space, there is a hermitean operator N, which has only whole-number eigenvalues.
2) That operator N = (a+) a, where a+ and a are two operators defined over the space, which are each other's hermitean conjugate and which have the property that:
3a)If | n > is an eigenvector of N with eigenvalue n (integer), then a+ | n > is an eigenvector of N with eigenvalue n+1.
3b)If | n > is an eigenvector of N with eigenvalue n (integer), then a | n > is an eigenvector of N with eigenvalue n - 1 (or is the 0 vector)

(an alternative to the axioms 3a and 3b is to state the commutation relations between N, a and a+)

That's it. These are the axioms one starts with. They are the axioms of the quantum-mechanical harmonic oscillator.

theorem 1:
One can easily prove that n is limited to the positive integers:
Indeed, n = < n | N | n > = (< n | a+) (a | n > ) = | a |n> |^2. So n is the norm of a vector in hilbert space, and must hence be a positive number. It was already an integer (axiom 1), so it must be a positive integer QED.

theorem 2:
All positive integers appear as eigenvalues of N (and not only some).
The proof is again based upon the relation n = ( < n | a+ ) (a | n >), from which it follows that
a |n> = (phase) sqrt(n) |n - 1>
and from which it also follows that
a+ | n > = (phase) sqrt(n+1) |n+1>.

Imagine that the positive integer m doesn't appear in the list of eigenvalues, and that the number k does. Assume k < m for starters. So there exists a vector | k > with eigenvalue k under N. Apply a+, we find a non-zero vector |k+1>. Apply a+ again. We find an eigenvector k+2... Do this m-k times. We find a non-zero vector |k + m - k > with eigenvalue m. So m was, after all, an eigenvalue of N.
Same argument if k > m: we apply a (k - m) times, and we again arrive at a non-zero eigenvector of N with eigenvalue m. So no positive integer m exists which is not an eigenvalue of N. QED.

definition:
we call a *ground state* an eigenvector of N with eigenvalue 0.

theorem 3:
There exists a ground state.
Apply theorem 2 up to eigenvalue 1. There exists hence an eigenvector |1> of N, with eigenvalue 1. Apply a. a | 1 > = 1 |0>, a non-zero vector. Hence there exists a vector |0> with eigenvalue 0 of N. QED.

FALSE theorem 4:
the ground state is necessarily unique.

This cannot be derived from the above axioms. Proof by counter example.

I do not understand what you say and what you want to say. Please provide your formulation of the 5th postulate of the Geometry.

Someone who doesn't know what is the 5th postulate of Euclidean geometry, the 2000 years of looking for a proof of it from the 4 others, and the final realisation that it is an independent axiom, and can hence be altered, which gave rise to the whole class of non-Euclidean geometries, tells me that I have no qualifications to do some maths ?
For your information, the 5th postulate is simply:
"given a line and a point not on the line, there is one and only one line that can be constructed which contains said point, and which is parallel with the given line"
(or equivalent formulations)

 

[olgranpappy]

This is starting to become ridiculous.

Yeah. These threads seem to have a way of dragging on...
Cheers.

 

[Anonym]

If you talk about the "one-dimensional mechanical harmonic oscillator", that means that you already have introduced a lot of postulates. One is for instance that the system is a classical system with a classical configuration space that follows the usual quantization rules. In other words, you have a classical configuration space, which consists in this case of the real line, and which we describe, say, with the variable "x" which we call "position". We then go on introducing the canonically conjugate variable p, and use the standard rules of quantization of a classical system to obtain a quantum system. You express your hamiltonian inspired by the classical hamiltonian and so on. As such, the quantum system is entirely defined starting from the classical system and the "quantization rules".

You are not making distinction between the mathematics and the physics. If you define trajectory as set of positions, you already know everything and you don’t need momentum, you do not need the notion of the canonically conjugated variables and so on. You do not need the analytical mechanics. Equally well you may define the real line through the 10th derivative of x. Then you need additive constants to define it uniquely. You may choose (postulate using your language) the funny subset of the Fibonacci numbers and call it the “physical model”. However the physics is only what is defined by Newton/Hamilton equations. They followed from the principle of the Least Action which is the Principal Physical Postulate.

This is starting to become ridiculous.

Yes. Your “axiomatic” approach I consider ridiculous (not only in this session).

For your information, the 5th postulate is simply:
"given a line and a point not on the line, there is one and only one line that can be constructed which contains said point, and which is parallel with the given line"

The 5th postulate is the definition of the parallel. You provide the classical example of the circular logic.In addition, if so, how you call that:

Consider (using auxiliary definitions introduced previously) line and a point outside that line. There exists at least one line which contains that point and do not contains any point that belong to the original line.

As I mentioned above, this session is about Foundations of Quantum Theory. I suggest reading the book written by top expert in the Foundations of Physics where he discusses the questions that you like very much to consider:

J.M.Jauch, Are Quanta Real?, Indiana University Press, (1973).

After that I hope you will be qualified to criticize what you are qualified to do.

Regards, Dany.

P.S. I identify that you in the martial mood. There is no Sticky: Forum Rules in Beyond the Standard Model sub-forum. I consider that it must be there.