I know that wave-packet is the super-imposed form of the waves. It is created when some waves have interference and a particle (say electron) can be represented by a wave-packet. I am bit confused actually a single wave is associate with a particle then why in case of wave-packet we associate more then one wave with a particle?
The concept of wave packet has undergone many, more or less subtle, variations during the last 100 years. Sure, interference is important, but it is not all. The common motive is that when a system admits a continuous band of wave solutions (classical or quantum, it does not really matter), its dynamics can exhibit localized-interference features that behave differently from the individual waves. They appear as persistent "objects" with strikingly different properties than those of the waves that compose them (for example, their apparent collective - or "group" - velocities can be quite different). Examples are vortices, maelstroems and tsunamis in hydrodynamics and, why not, particles and "resonances" in subatomic physics. The concept is nice, interesting, and probably contains a lot of truth, though it need not be strictly applicable to all the situations where it was tried. In any case, any kind of wave equation (be it Schroedinger or other) leads to this kind of discussions.
A wave-packet is connected to physical quantities and not to the number of particles. If the quantity is concrete, say it has a known value the probability wave of the particle has a Dirac delta shape in that state (the peak of the delta function is at the value of the quantity). If the quantity is uncertain the probability function of the particle in this state is a "thick Dirac delta", say a wave packet. According to Fourier's theorem a wave packet can be constructed from single sinus waves in which the single values of the quantity occure mostly inside of that area where it is important. If the quantity is very very uncertain its function is a single sinus wave (and its conjugate quantity has then a Dirac delta form).
Consult any good book on introductory quantum mechanics, such as [1] (Sec. 2.4 herein).
[1] DJ Griffiths, Introduction to Quantum Mechanics (Prentice Hall, NJ, 1995).
The concept of wave packet has undergone many, more or less subtle, variations during the last 100 years. Sure, interference is important, but it is not all. The common motive is that when a system admits a continuous band of wave solutions (classical or quantum, it does not really matter), its dynamics can exhibit localized-interference features that behave differently from the individual waves. They appear as persistent "objects" with strikingly different properties than those of the waves that compose them (for example, their apparent collective - or "group" - velocities can be quite different). Examples are vortices, maelstroems and tsunamis in hydrodynamics and, why not, particles and "resonances" in subatomic physics. The concept is nice, interesting, and probably contains a lot of truth, though it need not be strictly applicable to all the situations where it was tried. In any case, any kind of wave equation (be it Schroedinger or other) leads to this kind of discussions.
I think the simplest answer to your original question is that there is a distinction between a particle and a particle state. If by a "single wave" we mean one component in a series expansion for a wave function, and if that series contains terms that have physical interpretations (e.g., the terms in a Fourier series are momentum states), then each single wave represents a very specific particle state. But a particle need not be in any one of these states, in which case a superposition of such states is used to represent the particle state.
The idea of way packet is to at least partially reconcile particle and wave,
weather this achieves the goal is subject to debate. It gives the particle more localization in space while maintaining the wave. It balances the extremes of the uncertainty principle.
In addition to other talents (like being super smart) Richard Feynman also had a gift for explaining things in a way that could be understood by an audience that had no prior education in the topic. A good introduction to the topic of quantum mechanics is in Volume 1, Chapter 37 of that well-known book called "The Feynman Lectures on Physics." You might want to look at that before getting into the more advanced stuff.
Vk Singh
A wave packet is a joke that no one will tell you, even the Feynman could not understood it, so repeated the same joke in his own way of joking.
That is the truth here no one will tell you this fact.
(Actually a wave packet is collection of standing waves in a single setup. it is not a joke.)
You may be having so many questions just ask and you will find no one has clear insight.
Vk Singh
don't read this book Griffiths, Introduction to Quantum Mechanics
author does not have competence in this field just he uses jokes for every thing
Good text on Quantum Mechanics are
1) Kevin F Brennan, the physics of semiconductors...
2) N Zettili. Quantum Mechanics
3) Ajoy and Ghatak Quantum Mechanics
there are few more but just read them
for electrodynamics use as a beginer
Griffiths, Electrodynamics which best for upto 8 chapters , when the wave section starts in that book don't rely on Griffith's wording, instead just solve numerical problems only as after this chapter Griffith's view is not correct for waves and relativity.
There are at least two topics in quantum mechanics. One is, how do you make it work. In other words, what calculation methods produce answers that agree with observations. The other topic is why does it work. Textbooks tell us how to make it work. Richard Feynman has said that nobody understands why it works. That is still true today and finding an answer is still an area of active research in theoretical physics. Lots of textbooks explain how to make it work. Feynman's QED is one of the most successful applications of how to make it work. We still don't understand why it works.
The fourier transform of a wave packet is another packet in k space.
It just means a probability distribution, not a whole bunch of different waves.
The product of widths has to do with the uncertainty principle.
Juan Weisz
please dive deeply in the definition of wave packet, instead of telling the properties, just don't make less educated guess.
According to the statistical interpretation of qm the behavior of a simple particle is exactly the same as that of a bundle of particles. So the shape of the probability function of a single one looks like if it would be built up from many particles with their own sinus waves.
Éva Szőke
you get the correct meaning of it, elaborating as waves are standing waves in one setup so they willy naturally superimpose. but don't forget particles don't have spin it is there associated wave which have this unless there is a torque on the setup. more remarkably, particles do not superimpose. which is the deepest insight drawn from "Paired Statistics".
Dear Mr. Maurya,
I read about 10 years ago this book, but I guess the author makes clear the meaning of the Stern- Gerlach experiment. As I remember he deals with mirrors instead of holes but the meaning is the same. In the experiment are free particles which have then sinus waves and they interfere because they are from the same origin. The experiment can be done with one single particle with the same result.
I have several quantum mechanics books that all say pretty much the same things but none of them mention quantum entanglement. I have read that Einstein was debating that topic in the 1920's (spooky action at a distance) so it is not a new topic but my books don't mention it. Can anyone recommend a book that does include this topic?
Thank you for the suggestion. I think the actual title is "The Emperor's New Mind" but your title came close enough for me to find it. Looking through the table of contents found in a website, it looks like some pretty good stuff. I am going to get a copy.
A wave packet is like a cross between a particle (single particle) and a wave,
describable by a superposition of many sine like waves (this is the fourier transformation or packet in k space) but the many waves are not to be interpreted as many particles, just many states for one particle.
It is true that in some statistical sense the one particle acts as many , this is one or
the most confusing elements of QM. This is the effect of many states for one particle.
Entanglement is interpretable in the sense that one wave function is used for many particles, which share one wave function. (This is the many particle wave function) A single state for many particles.
For Fermions as electrons there can be one particle per state, but a state can be either full or empty. So we should not confuse the two.
Juan Weisz Good View.
electrons there can be one particle per state ---> per pair of states (don't forget the degeneracy of an electron).
What about photons?
Dinesh, yes you are right. There are two spins states, for each
spatial state for electrons.
You can have one particle for each fully specified state, including spin.
Photons are bosons with spin 1, you can have any number
of them in a single state.
regards, juan
Juan Weisz
spatial state for electrons??? what is this?? how many types are there, any guess??
Photons are bosons with spin 1 (how is there any proof or just a joke), you can have any number of them in a single state----> single level. ???
To answer the question of " spatial state for electrons??? what is this?? "
There are a variety of ways to specify a spatial state. One way is to specify three location coordinates (for an electron at a definite location). Another is to specify three linear momentum components (for an electron having a definite momentum). A third is to specify energy, magnitude of total angular momentum, and one component of angular momentum (often done for electrons in atoms).
Hi Dinesh
Yes wave functions are spread out in space. The spin degrees of freedom are normally discussed using angles, and angular momentum.
regards, juan
Juan Weisz thankyou Sir for reply, just dive deeply in our understanding of particles/waves. Just do not go on with Dirac as every work from him is misleading and most probably wrong.
Photons are bosons with spin 1 (how? is there any proof or just a joke), you can have any number of them in a single state----> single level---one wave function---> wave packet
Photons also follow the same state structure.
Would not agree that
Dirac is mostly wrong..The inventor of relativistic QM for electrons. Can make mistakes like everyone else.
Bosons have integer spin. Feyman vol. III has a good description of spin 1 for photons.
Ever heard of a Bose Einstein condensate? Such states are
achievable with a whole bunch of integral spin atoms inside traps.
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