Understanding Wave Packets in Quantum Physics

[Jimmy87]

Hi, I'm fairly new to quantum physics so go easy on me! I've just come across the uncertainty principle and have some confusion over wave packets. If we superimpose multiple waves then as I understand it we get a wave packet with a finite length which lowers the uncertainty in position. If your talking about electrons say, then where do these multiple wave come from? If an electron is a wave then how can you superimpose multiple waves wouldn't that be superimposing many electrons which doesn't make sense to me? Thank you for any help offered.

 

[UltrafastPED]

Think in terms of photons ... then if you have sufficient bandwidth (lots of frequencies close together) you can obtain very short pulses of light by means of interference effects.

There is a very simple Fourier analysis theorem which describes the time-bandwidth relationship - and the HUP is a QM version of this.

For the case of electrons there are additional constraints: Coulomb repulsion, Pauli exclusion principle.

 

[Naty1]

Hi Jimmy: Quantum mechanics is tricky business, so proceed expecting to find unusual results! And be prepared to reread stuff multiple times to understand what is being written...

Particles, any particles, are quanta of fields, local effects we can detect. The theoretical fields are the fields of quantum field theory in the standard model of particle physics not actual particles that can be detected. Particles are not waves. Particles are what we detect in space.

A conventional interpretation of the wave function is associated not with individual particles but rather with the probability for finding a particle at a particular position. In this interpretation, the object always is a particle, not a wave, and the wave aspect is a mathematical abstraction used in the model to make probability calculations.
The relationship between a system's Schrodinger wave function and the observable properties of the system is non-deterministic. Fourier decomposition will always decompose a wave function into an infinite series of equivalent functions.

Example:
The wave function describes not a single scattering particle but an ensemble of similarly perpared particles. Quantum theory predicts the statistical frequencies of the various angles through which a particle may be scattered.

Example: [an example I liked from a discussion in these forums]

Bouncing a photon off an atom tells us nothing about any [Heisenberg] uncertainties. We must bounce many ‘identically’ prepared photons off like atoms in order to get the statistical distributions of atomic position measurements and atomic momentum measurements. What we call "uncertainty" is a property of a statistical distribution.

A simple way to 'decompose' is via trig functions like Sin2x = 2SinXCosX...and you can make such equivalents endlessly...Even better, if you know a little about Fourier transforms, there are more complete illustrations [pictures] showing such decomposition here:
http://en.wikipedia.org/wiki/Fourier_decomposition.

Search 'particle' in these forums or 'Heisenberg Uncertainty' and you'll have dozens of pages of explanations...

 

[Jimmy87]

Think in terms of photons ... then if you have sufficient bandwidth (lots of frequencies close together) you can obtain very short pulses of light by means of interference effects.

There is a very simple Fourier analysis theorem which describes the time-bandwidth relationship - and the HUP is a QM version of this.

For the case of electrons there are additional constraints: Coulomb repulsion, Pauli exclusion principle.

Thanks for your reply. So, are all electrons within an atom (which has a finite space) wave packets (i.e. each electron is composed of a series of multiple waves)?

 

[UltrafastPED]

Thanks for your reply. So, are all electrons within an atom (which has a finite space) wave packets (i.e. each electron is composed of a series of multiple waves)?

Well, it is best to think of them as waves ... they are certainly not localized when they are part of an atom.


See http://en.wikipedia.org/wiki/File:Electron_orbitals.svg
and http://en.wikipedia.org/wiki/Electron_configuration

 

[Naty1]

So, are all electrons within an atom (which has a finite space) wave packets (i.e. each electron is composed of a series of multiple waves)?

Those are not the electrons themselves but rather the probability distribution their location. Those take the form of standing waves, analogous to the resonant waves of a violin string. [edit: Such local distribution of electrons is sometimes called the electron cloud.]

Electrons in any any structure, an atom or a lattice like a metal, are localized, that is confined but not to a point. The wave behavior makes them behave as if they have different vibrational size. But again, all we detect are point particles.

 

[Jimmy87]

Those are not the electrons themselves but rather the probability distribution their location. Those take the form of standing waves, analogous to the resonant waves of a violin string. [edit: Such local distribution of electrons is sometimes called the electron cloud.]

Electrons in any any structure, an atom or a lattice like a metal, are localized, that is confined but not to a point. The wave behavior makes them behave as if they have different vibrational size. But again, all we detect are point particles.

Thanks for all your answers. So this wave function that keeps cropping up when I read, is this just used to calculate the probability of finding an electron within an atom? A description I have read for an electron using QM is that it is like a wave in that you can fit it around a nucleus in such a way that it doesn't cancel itself and this would be an allowed energy level. The amplitudes of the waves are then points where it is most likely to materialise. Is that a good way of thinking about it? Many different people describe it in so many different ways which makes it very confusing for me.

 

[Naty1]

The prior post is ok to get you started...what all this means [how to interpret the math] is still the subject of debate now approaching almost 100 years...so expect to gain new insights the longer you study.

Wikipedia says this:

The Schrödinger equation provides a way to calculate the possible wave functions of a system and how they dynamically change in time. However, the Schrödinger equation does not directly say what, exactly, the wave function is. Interpretations of quantum mechanics address questions such as what the relation is between the wave function, the underlying reality, and the results of experimental measurements.

Remember that

all particles are point like when detected but behave as wavelike otherwise

. QM does not define observables [experimental detections] of wavelike characteristics until detected locally.

If an electron, for example, were truly free, not interacting with any other particles, it's wave would extend across the universe to the cosmological horizon. That puts a boundary on the wave. Think of a limp string just hanging; it can't vibrate and have much in the way of particle characteristics,no energy. But string it tight in a violin,fix the ends firmly, now it resonates when plucked because it can store energy, it has characteristics! That's a view that comes from both string theory and cosmology.

Many different people describe it in so many different ways which makes it very confusing for me.

Yes, that's what the Wikipedia quote implies...interpretations vary! Here are two of my favorites, from other discussions in these forums:

Matter [particles] is that which has localized mass-energy, while spacetime does not.

Particles appear in rare situations, namely when they are registered.