Dear Michael Dada,

Gert Van der Zwan pointed out the actual reason. Here, I am giving a more transparent way of looking at the same which may be helpful to you.

Thanking you and best regards....NG.

No:

indeed, the eigenfunctions psi_n of the eigenvalues E_n=n+1/2

form a COMPLETE ortonormal basis.

@ Michel Dada

I do not understand your question.

Try to tick one , which you have in mind.

1. Eo= a

En= a+ n

2.Eo=a

En= a+b_{n}

In above a,b_{n} are not integers .

B.Rath

Dear Gert Van der Zwan,

the completeness condition of my firs answer makes impossible energy

values different from n+1/2, because the resolution of the identity of the

energy operator is a step function with discontinuities just in the points

n+1/2. Only by modifying the hamiltonian, for instance with pertubations,

other energy values can be measured.

No I don't think so. The x and p operators are the quantization and there's no place for fractional quantization in the HO, in any one of three methods of derivation: spectral by Hermite ploynomials power series that uses integers in its solution: ladder operator, because the raising and lowering operators can operate on 1 particle at a time ; other method because the natural length scale is fixed in units of (h/mw)^1/2~ .09 A

I've come up with another thought that would be very difficult and might give any satisfactory results but..you always can learn something. Try solving the Shroedinger Equation with fractional differentiation. That way each time a derivative is taken a fraction will show up instead of a whole number, that's the simple part.

@Michael Dada

😂

Probly you did not see my earlier answer.

In short answer to your question is YES .

1. If you take n=1.45 Energy levels are

1.45,3.90,5,90,7.90 ,••••••••••••••••••;

2.For n=1.65 Energy levels are

1.65,4.30,6.30,5.90,7.90 ••••••••••••••••

I am cross checking my results . If find different , let you know.

You can do your calculation to verify my results.

Good Luck

B.Rath

@Michael Dada

Correction ;

n=1.65 the energy levels are

1.65,4.30,6.30,8.30 •••••••

Sorry for typing mistakes of earlier answers.

B.Rath

Dear Michael Dada,

As a result of the harmonic oscillator potential being continuous everywhere, the respective energy eigenfunctions must be C1, i.e. they must have continuous first derivatives.

This is easily proved by integrating the energy eigenvalue equation of the harmonic oscillator (the known time-independent Schrödinger equation) on an ε-neighborhood of an arbitrary point x and then taking the limit ε->0, taking into account that the potential is continuous, and thus it is also bounded in the given neighborhood.

Now, the restriction that n must be an integer results solely from the previous condition, i.e. from the fact that the energy eigenfunctions of the harmonic oscillator must be C1 functions, because if n is not an integer, the first derivatives of the resulted eigenfunctions are not continuous at the origin (x=0).

Otherwise, you can find square-integrable – and thus normalizable – solutions to the energy eigenvalue equation of the harmonic oscillator even for non-integer values of n.

This is nicely and easily explained in the paper

Article Solution of the quantum harmonic oscillator plus a delta-fun...

(pages 3 and 4), where the authors solve the energy eigenvalue equation of the harmonic oscillator with the addition of a delta function at the origin.

Hello all: This does not depend on mathematics, or parabolic cylinder functions. It is not observed. There are basic set theory and topological reasons for this non-observation, but the determining factor is experimental -- physics is defined by nature, not by mathematics.

Although multi-photon absorpton exists, due to the heat bath --with many energies and bands in-between, not just one energy or multiples thereof. How to explain it, mathematically? Fundamentally, the set of positive rational numbers and the set of positive integers are of the same size, there is a one-to-one correspondence.

So, if we see a set of what seems to be rational levels of energy for a molecule, are they fractional quantum levels? No, we can break them down into their components, including ro-vibrational levels, and ionized parts, to reach positive integers in a global respective series, which is now in a one-to-one correspondence to the set of positive integers. This represents the Cantor theorem, in cardinality. The observations that cannot be so re-ordered would be for the reals, which have higher cardinality (size) as the rationals. However, a real number, with its infinite precision, is not measurable, although they may exist in nature. Therefore, all measurable numbers can be quantum, in a one-to-one correspondence to the set of positive integers.

This is one basic argument, but there is, perhaps, a more fundamental reason.

Linking also to underlying topological properties of homotopy, as even more basic than QM and the Cantor theorem given above. The number of topological holes is an integer number, you cannot transform topologically, a very general transformation, a 1-hole into a 2-hole; a cup with a loop-handle into a doughnut is OK, they both have 1-hole.

The correspondence with the quantum representation seems more general, than using the Cantor argument. There is also no such thing as a topological half-hole, or 1/3-hole, and so on -- only integers.

Cheers, Ed Gerck

Hello Gert an all: Schrödinger's equation is not valid, is not relativistic. Topology is intuitive, there are no half-holes, or partial. Either a hole exists, or not. Thank you for citing my paper, where instead of adapting a series to a problem, I first adapted the problem, with its physics, to the solution in the given boundary conditions, as a series.

In the case of the wave function, we can follow a similar path, I suggest. In the question, one can adapt the mathematics to experimental results, and find no fractional values for n.

But one could use mathematics further, to demonstrate the impossibility of fractional n, which was attempted, to demonstrate under Cantor and topological evidence. We did not attempt to calculate, but to prove that it is impossible to calculate a fractional or real value.

Hence, n must be an integer, all other possibilities being excluded.

Hello Gert and all, Thank you. Topology is more fundamental than even metric spaces, so topology sets the stage for physics. Please do not doubt this, it is well-known, also in physics.

In RG level, in my previous response, I proved that anything measuarable in physics, including continuum levels, CAN be quantized without any prejudice in precision. Anything a computer calculates, btw, is quantized for the same reason.

However, I pointed that these results show what this thread did not say until then. All this does not matter at all in physics, quantum levels are integers because they are so verified experimentally. This is not due to mathematics, whatever it may fancy. The reason is in nature, billions of years old evidence..

Hello Gert: Your posting is off-topic and false, please consider deleting it. You already wrote, in respose to my comment, "I agree that it is an experimental fact that the spectrum is discreet." It does not depend on mathematics, it was true billions of years ago...

Careful that sarcasm not be used in academic discussions, as it would reveal a lack of arguments and an infantile desire to "win a discussion", whereas we should be working for the benefit of all.

Newton and Einstein also would agree with you, things fall when one drops them: it is experimentally verified, and the reason is in billion years old nature.

So, let us not cross the river to misleading anyone: Things happen because they can be experimentally verified. It does not matter who said what, nor how much mathematics is used.

Hello Gert, Thank you for your support. The question is not decidable by mathematics, even though it may look very beautiful, logical, or simple. The center of the Solar system may be my fingertip, and all physics works!

The question is settled by experimentation (I did not write observation, that is reserved for astronomy). In chemistry, it was actually once settled personally by a dream, but Kekulé had to show to others by experimentation.

This connects to a current discussion in physics, what is the demarcation between science and non-science? Today, it is direct experimental confirmation, tomorrow that may still be experimental, but indirect may also be acceptable.

RG has gone back working after updates, after changes and hard work it seems. I am writing a draft on RG, on this "new physics". I am close to a new version, at https://www.researchgate.net/publication/326398997_The_next_physics

Hello all,

In principle, the quantization of energy in a bound quantum system can be considered as an axiom, since it is actually a consequence of the separability of the (Hilbert) state space of the system.

The separability of the state space of a bound quantum system is a postulate in von Neumann’s formulation of quantum mechanics (see, for instance, https://plato.stanford.edu/entries/qt-nvd/#2.1), a formulation establishing the equivalence – the isomorphism – between matrix mechanics and wave mechanics.

A separable Hilbert space has a countable orthonormal basis, which is actually a formal and abstract way of saying that the energy spectrum is discrete.

However, in elementary quantum systems, like the infinite well, the particle on a ring, etc, the quantization can be mathematically verified, as it results merely and solely from the boundary conditions imposed on the energy eigenvalue equation of the system (the known time-independent Schrödinger equation).

In the case of harmonic oscillator, the fact that the n of the energy spectrum is a natural number results from the square integrability of the energy eigenfunctions and the continuity of their first derivatives at the origin, two conditions that, in turn, result from the properties of the harmonic oscillator potential, namely from the fact that it is confining and analytic.

The proof is demonstrated in pages 3 and 4 of the article

Article Solution of the quantum harmonic oscillator plus a delta-fun...

Dear Prof. Gert Van der Zwan,

Thank you for your very kind and educating responses. They have been quite enlightening. I wish to also appreciate you for making your book chapter available to us. The chapter has been a blessing for us here. It provided applications and solutions to some challenges we have had with Abramowitz and Stegun Book. Thank you once again!

Dear Prof. N. Gurappa,

Thank you so much for your contribution. I guess the problem has to do with the variable to be measured and if normalization is always necessary (eg wave packet in free space).

Dear Prof. Ed Gerck,

Thank you for your response and contributions. We shall look into the issues you have pointed out. Thanks!

Dear Prof. Giuseppe Antonio Nisticò.

Thank you so much for your contribution. Points noted!

@ Drs Edgar Korteweg, Spiros Konstantogiannis and Dann Passoja,

Thank you very much for your contributions.

I really appreciate the scholarship!

Dear Prof. Biswanath Rath,

Thank you for your calculations and I am sorry for my late response.

They have been quite useful!