Wave functions, i don't get em

[3m0k177y]

OK, so i understand that wave functions have something to do with something...i'm clueless, please help on this one.

First, what ARE wave functions? i haven't an answer that i can understand, explain this to me dumbed down please.

Second, what do they do exactly in the double slit experiment?
If you don't know what that is, its where particles are shot at a sheet with two slits in it, and a sheet behind that catches the particles. when there is an observer, the sheet that catches the particles has two rows where the particles built up, the rows correspond to the slits of course. when the observer isn't present, the particles scatter all over the sheet that catches them, this is because the particles are in quantum state therefor they are everywhere they can possibly be and the individual particles interfere with themselves.

 

[bhobba]

I will presume you know a smattering of linear algebra - if you don't then its going to be a bit hard. But basically its an element of a complex vector space. If you expand an element of unit length in some orthonormal basis then that basis is considered the possible outcomes of some experiment. The square of the absolute value corresponding to a particular basis vector (ie the complex quality multiplying the particular basis vector in the expansion) gives the probability of getting that particular outcome.

In the case of of the double slit experiment the basis is considered to be positions in space and the wave-function (when the absolute value is taken and squared) gives the probability of finding the particle in that position. The best way to analyse the double slit is using Feynmans sum over history approach. This relies on the fact by rewriting the equations determining wave-functions in a certain way you can express it as the sum of values it has over all the possible paths it has to get to that point from its starting point. Because those paths are limited by the fact it must go through the two holes you end up with an interference type pattern for the wave-function at the screen so that when you take the square to get the probability you get an interference pattern.

Check out Feynmans QED lectures:
http://vega.org.uk/video/subseries/8

Thanks
Bill

 

[Agerhell]

OK, so i understand that wave functions have something to do with something...i'm clueless, please help on this one.

First, what ARE wave functions? i haven't an answer that i can understand, explain this to me dumbed down please.

Second, what do they do exactly in the double slit experiment?
If you don't know what that is, its where particles are shot at a sheet with two slits in it, and a sheet behind that catches the particles. when there is an observer, the sheet that catches the particles has two rows where the particles built up, the rows correspond to the slits of course. when the observer isn't present, the particles scatter all over the sheet that catches them, this is because the particles are in quantum state therefor they are everywhere they can possibly be and the individual particles interfere with themselves.

Historically I think it began with the "Schrödinger equation". By using that equation, which is a wave equation, Schrödinger was able to compute the energy levels of the hydrogen atom.

Apparently, as bhubba points out, the wave function concept can also be use to explain the outcome of the double slit experiment. Can someone inform me on other ways in which the wave-function concept is used to explain things?

 

[genericusrnme]

Historically I think it began with the "Schrödinger equation". By using that equation, which is a wave equation, Schrödinger was able to compute the energy levels of the hydrogen atom.

Apparently, as bhubba points out, the wave function concept can also be use to explain the outcome of the double slit experiment. Can someone inform me on other ways in which the wave-function concept is used to explain things?

The uncertainty principle with position and momentum
Since position and momentum are Fourier transforms of each other, the more 'delta like' you make one wavefunction, the more spread out other one gets.
If you don't know what a Fourier transform is (I'll assume you don't) then it can be thought of like this;
Take a wave that extends, with a constant frequency from -inf to +int, you can tell what it's frequency is, but you can't tell WHERE it is.
Next, take a wave that is 0 almost everywhere except for a small area. In this case, you can tell where it is but not what it's frequency is.
Now mathematically, it turns out that the wave that goes out to infinity is a linear combination of every position and the wave that is 0 everywhere except a small area is a linear combination of every frequency but for our intents and purposes it serves fine to say that we don't know where or what frequency the wave has.
Now, replace frequency with momentum and you have the uncertainty principle without the maths!The wavefunction is a hard thing to understand regardless of your background the first time you approach it.
If it helps, you can think of particles as waves when you're not measuring their position, the wavefunction describes that wave. When the wave then interacts with something the wavefunction 'collapses' onto whatever point you measured it on. The amplitude of the wave (well, the absolute value squared of the wave technically) at a point gives you the probability that wavefunction will collapse on that point.
When the wavefunction collapses to a point, all the previous waves have gone away and new waves start propogating from the point where it collapsed.

Using this framework, when we look at the double split experiment;
1. No measurment at the slits
-Wave function is collapsed at the projector
-Waves propogate through the two slits without collapsing
-Waves make interferance pattern
-Wavefunction collapses mostly at the peaks in the interferance pattern because that's where the highest probability is
-Physicists see interferance pattern on their detector
2. Detector at the slits
-Wave function is collapsed at the projector
-Waves propogate to the two slits
-The wave function collapses at one of the slits
-The wave function starts propogating from the slit it collapsed at
-The wavefunction collapses mainly directly across from the slit since that's where the highest probability is.
-Physicists see no interferance pattern

Now, I know this framework of thinking isn't perfect but it's better than a lot of the other ones I've seen floating about the place and it's served me well enough in my studies. Ofcourse, it is a completely mathematical construct and an understanding of linear algebra will serve you far better but without this should do well enough.

 

[mal4mac]

Ouch bhobba! Surely we can dumb that down a bit more?

In one sentence: A wave function is just a particular kind of mathematical function.

In the context of a real simple quantum mechanical system:

Imagine an electron in a box. Using quantum mechanics, you can't say exactly where the electron is in the box. The possible positions of the electron are given by the wave function for that electron. When you look for the electron, you find it in a particular position. The probability of finding it at that position is given by the wave function.

Move the electron to the double slit experiment:

The probability of finding the electron at any position on the screen is given by the wave function. The form of the wave function is determined by the set-up of the experiment (one slit or two...)

 

[mal4mac]

... double slit experiment?
If you don't know what that is, its where particles are shot at a sheet with two slits in it, and a sheet behind that catches the particles. when there is an observer, the sheet that catches the particles has two rows where the particles built up, the rows correspond to the slits of course. when the observer isn't present, the particles scatter all over the sheet that catches them...

This is wrong! Where do you get this sheet/observer stuff from?

The original double-slit experiment was a demonstration that light can display characteristics of waves and particles. A light source illuminates the two slits. The light waves passing through the two slits interfere, producing bright and dark bands on the screen. However, at the screen, the light is absorbed as discrete photons. For instance, you can send the photons through one at a time, and the screen absorbs them one at a time until you see the wave pattern emerging.

 

[gianamar]

Hi can anyone help me? I am trying to calculate the evolution of ψ(x,t) (a value which I have calculated numerically for my specific problem), and I would need to find another state; phi(x,t), which has changed over time. Can someone explain to me how to do this please. Do I use the Schrodinger equation? I have most numerical parameters, I just don't know how to find the new wavefunction. In addition, I would then need to calculate the probability of wavefunction collapse.
Thanks

 

[Naty1]

[do not ascribe any importance to the order of the following...stuff is in a rather random order...]

You should leave this discussion with the understanding that different 'experts' ascribe different meanings to the wave mathematics...

Some say a wavefunction, like the Schrodinger wave equation, is nothing but a probability density function representing the probability of finding a particle at some particular location. Others think it represents something like a field, say like the electromagnetic field...maybe even something 'physical' or 'real'.

Regardless of what the wave function 'really' means, it has an amplitude and phase. [The amplitude [height] is the intensity of the wave, the phase, like sine and cosine being 90 degrees offset in time from an origin...]

Here is one abbreviated interpretation...http://www.rochester.edu/college/faculty/alyssaney/research/papers/Ney_ReductionWaveFunction.pdf

Wave function realists have put this point this way:

No one can understand Bohmian mechanics until he is willing to think of the wave function as a real objective field… Even though it propagates not in 3-space but in 3N-space. There is nothing in this theory but the wavefunction. It is in the wavefunction that we must find an image of the physical world, and in particular of the arrangement of things in ordinary three-dimensional space. But the wavefunction as a whole lives in a much bigger space, of 3N dimensions.

The sorts of physical objects that wave functions are… are (plainly) fields –
which is to say that they are the sorts of objects whose states one specifies by
specifying the values of some set of numbers at every point in the space where
they live, the sorts of objects whose states one specified by the values of two numbers (an amplitude, a phase) at every point in the universe’s so-called configuration space. The wave function is a field in the sense that it is spread out completely over the space it inhabits, possessing values, amplitudes in particular, at each point in this space.

The high-dimensional space in which the wave function exists is what physicists
refer to as ‘configuration space’. Traditionally, ‘configuration space’ refers to an abstract
space that is used to represent possible configurations of particles in three-dimensional
space. ...

For wave function realists, this is the fundamental space of the universe.
It is the space over which the wave function is spread. The proper way to understand the dimensionality of configuration space is in terms of the number of degrees of freedom needed to accurately capture the quantum state of the universe.

Other views:

When we say a particle "behaves like a wave," we are talking about a wave function that gives the probability of finding the pointlike particle at a particular location.

What is a 'particle: In classicla QFT, its a pointlike object; but in relativistic QFT theory, like
QED, or relativistic quantum field theory in general is not based on the notion of ''point particles''...Electron wavefunctions tend to take the size of their container. If you place an electron in a quantum well, it will tend to spread out to fill the well. If you bind it to an atom, it takes generally ends up a different size..that of the electron “cloud” which reflects local interactions, degrees of freedom, with other electrons and nucleons..

PArticles and virtual particles:

Wikipedia sez:

There is not a definite line differentiating virtual particles from real particles — the [wave] equations of physics just describe particles (which includes both equally). The amplitude indicating that a virtual particle exists interferes with the amplitude for its non-existence; whereas for a real particle the cases of existence and non-existence cease to be coherent with each other and do not interfere any more. In the quantum field theory view, "real particles" are viewed as being detectable excitations of underlying quantum fields. As such, virtual particles are also excitations of the underlying fields, but are detectable only as forces but not particles. They are "temporary" in the sense that they appear in calculations, but are not detected as single particles. Thus, in mathematical terms, they never appear as indices to the scattering matrix, which is to say, they never appear as the observable inputs and outputs of the physical process being modeled. In this sense, virtual particles are an artifact of perturbation theory, and do not appear in a non-perturbative treatment.

For the double slit 'meaning': try reading here as a start:

http://en.wikipedia.org/wiki/Double-slit_experiment#Interpretations_of_the_experimentFROM THE ROAD TO REALITY. Roger Penrose, age 528- 530:

The parenthesis [] are my clarifications of terminology.

“The Schrodinger wave equation [state vector] is a deterministic equation: the time evolution is completely fixed once the state in known at anyone time….and provides for the evolution of a quantum particle in a very precise way- until some measurement is performed on the system.

This may come as a surprise to some people, who may well have heard of quantum uncertainty, and of the fact that quantum systems behave in non deterministic ways. [Non deterministic means limited to ‘statistical’ result observational measurements.]

Generally a measurement would correspond to an operator of some sort [ a mathematical component] and the effect of a measurement on the state [wave function] would be to make it jump into some eigenstate….which eigenstate is a matter of chance! [This means a measurement forces the wave equation to take on some value, the one that is observed, but that exact value cannot be predicted in advance because it turns out a statistical distribution of results occurs, not a particular value.]

This jumping of the quantum state to a specific eigenstate [a specific value] that is referred to a 'state vector reduction' or 'collapse of the wave function'.[These are equivalent terms] It is one of quantum theory’s most puzzling features…I believe most quantum physicists would not regard state vector reduction as a real action of the physical world, but it reflects the fact that we should not regard the state vector as describing an ‘actual’ quantum-level physical reality. ..” Note this last description conflicts with the 'realist' above.

And now you can understand most of what Penrose told a group of famous physicsts celebrating Stephen Hawkings' birthday [Cambridge England, 1993] :

"...Either we do physics on a large scale, in which case we use classical level physics; the equations of Newton, Maxwell or Einstein and these equations are deterministic, time symmetric and local. Or we may do quantum theory, if we are looking at small things; then we tend to use a different framework where time evolution is described... by what is called unitary evolution...which in one of the most familiar descriptions is the evolution according to the Schrodinger equation: deterministic, time symmetric and local. These are exactly the same words I used to describe classical physics.

However this is not the entire story... In addition we require what is called the "reduction of the state vector" or "collapse" of the wave function to describe the procedure that is adopted when an effect is magnified from the quantum to the classical level...quantum state reduction is non deterministic, time-asymmetric and non local...The way we do quantum mechanics is to adopt a strange procedure which always seems to work...the superposition of alternative probabilities involving w, z, complex numbers...an essential ingredient of the Schrodinger equation. When you magnify to the classical level you take the squared modulii (of w, z) and these do give you the alternative probabilities of the two alternatives to happen...it is a completely different process from the quantum (realm) where the complex numbers w and z remain as constants "just sitting there"...in fact the key to keeping them sitting there is quantum linearity..."

Read the above at least five times and you'll be on your way! If you are a 'dummy' like me, go for six to eight times...then copy and start your file of notes.]

 

[Naty1]

Some good perspectives and complementary descriptions here,,,,

http://en.wikipedia.org/wiki/Wavefunction

If you are interested, see EXAMPLES and SEE ALSO at the bottom of the artiicle and follow any links of interesed within the article...

 

[batabek]

I'm just wondering,
from the above descriptions can we think of the wave function as a random variable ?