thank you very much for your explanation, i think my second question is not so clear, i mean if we take the Fourier transform of a wave packet (for an electron as an example) , which component is the frequency of the electron? is it the principle frequency of the Fourier transform?

And what will be the unit of time?

thank u for your kind replay, but in electron diffraction for example, electrons must satisfy Bragg's condition (2dsin(theta)=nlambda) lambda is the electron's wavelength, if electron does not have well defined wavelength it cannot satisfy Bragg's condition for diffraction, hence it must have certain frequency according to speed=frequency*lamda. Is it the case?

welcome Mr. Oscar Chavoya-Aceves i did not notice your question when i wrote my note, i think its no matter what is the unit of time only to be in range that gives frequencies (energies) in the range of some ev i.e the electrons energies.

When you say about sines and cosines the wave function, being a superposition of them, represents a plane wave with a definite momentum and energy (frequency) that is the most commonly used representation. Unfortunately, the simple thing is that there are no plane wave in Nature. At all! We have learned this well already from the Maxwell electrodynamics.

The real particles are the wave-packets whose wave functions have finite transverse extent and longitudinal extent (duration). These wave packets are not superpositions of sines and cosines, since there are not plane waves any longer. So they have no definite momentum and energy but rather some distribution over some set of values with some averages. In other words, such real particles as waves are not monochromatic that is in accordance with the Heisenberg uncertainty principle. What could be a bit confusing here is that these real wave packets are used in practical calculations rather rarely.

I am afraid either my question is not clear or there is a confusion in understanding my question..well i am just talking about how can we interpret the wave particle duality by a wave packet physically...

Dear Charles,

I do understand how the Fourier transform works. And I did not say that a wave packet, which is not a plane wave, could not be represented as a superposition of plane waves either. In fact, it could be of course.

Gentlemen,

I believe you have a foundational problem here.

Mixing mathematical techniques with representational assumptions produces a dangerous cocktail.

Quantum mechanics is about quantum states. These states are elements of a Hilbert Space. This is an abstract space.

From this point is difficult to get at a representational model. Particles and waves are classical physical elements. Quantum states associated to a given "materiality" is still today a challenge; not solved.

It is not strange that the question serving as a base for these exchanges produce confusion.

If nothing is well defined, it will be very difficult to obtain something meaningful.

I apologize for this rather negative intervention.

@Orlando Tapia

I do not see the negativity of your intervention. There is nothing negative in a clarification of concepts.

@Abdulghefar Faiq

I appreciate your clarification, however the question was not for you: it was for Charles, because he related Planck's constant to the unit of energy and it was motivated by the uncertainty relation \delta E \delta t ~ \hbar. As to electron diffraction, those experiments take a lot of time to complete, so the physical situation is modelled using a wave function with a very small \delta \omega (almost monochromatic), so that Bragg's analysis applies well. That's at least the justification to use non-normalizable wave functions to study electron diffraction.

thanks for all contributors , does Hilbert space deal with continuum?

@Charles

I understand that, we can define a unit of energy by means of the relation E = \hbar \omega and for \omega we can take any angular frequency we can consider well defined. Defining a unit of energy will give us a unit of mass from the relation E = mc^2. I take your answer in the sense that we will continue to use the properties of Cesium for a while, to define the unit of time.

Q. M. postulates that for a particle in bound states, energy states is quantized (in contrast to the free particle's energy which is continuous), which means that we have an infinite number of discrete states, this is an important key point that differentiate Q. M. from Classic mechanics ( converting continuum to discrete). While we see that Hilbert space defined an infinite number of states that all continuum (there are no gaps between them). is it an odd state? how can we interpret this case?

i mean that Hilbert space is look like classical space.... is it the case? can we compare?

I am answering the original question. How a localized function

can be split in terms of periodic functions. This question

arise because the wave packet is representing a particle hence

it has to be non-zero in a small region of space. But the packet

as a whole can move. Whereas the basis in which the wave packet

is being decomposed is made out of infinite number of sines

and cosines, which are periodic functions.

The answer to this question is that we can adjust wavelengths,

phases and amplitudes of component sinusoidal waves.

By this adjustment constructive interference happens in a

small region of space whereas destructive interference

occurs everywhere else in space.

y(x,t)= int A(k) sin(kx-wt) dk

limits of integration -infinity < k < infinity

Here, k=2 pi/lambda. Thus when k is varying continuously

so is lambda. To localize a wave packet in a small region

we need a wide range of values of k, which in turn means

that we require a wide range of values of lambda.

For a pictorial explanation of what I just wrote above

please see the link.

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/wpack.html

Dear Biswajoy Brahmachari 

thank you very much for your kind explanation I think that you understood what i am asking about !

thank you very much 

Dear Abdulghefar,

You seem pretty convinced that wave functions represent the physical system.

The question you raise shows that such is not the case. Representationalism is a classical physics ideology. I would suggest you to have a look to one of my latest paper to be found in my site.

Yours sincerely