# Beginner, wondering about wave functions

• imjustaguy
In summary, the conversation revolved around the concept of wave functions and their usage in quantum mechanics. The participant, a filmmaker, was seeking an explanation in English as opposed to math language. They offered a metaphor of a fly flying in figure eights at immense speed to explain wave functions, but were advised against using physical analogies as they can be misleading. The conversation also discussed the relationship between wave functions and physical reality, with one perspective considering it as a mathematical symbol with no direct connection to reality, while another seeing it as a calculus to predict particle behavior. The final point made was that in some interpretations of quantum mechanics, there are no particles at all and they are just properties of a universal wave function.
imjustaguy
I am a filmmaker, and I have some questions about wave functions. I have read as much in English as possible (as opposed to math language) but unfortunately, I haven't dedicated my life to mathematics and understand math languages, so I suppose I'm asking for an explanation in English.

I had a Physics teacher in school tell me about particles acting like waves, or waves being made up of particles. The only way I can explain it in my own words is through the only way I know how, metaphor. In my understanding, one way of thinking of a particular wave function is like imagining a fly buzzing around in figure eights at immense speed. It's moving so fast that it creates a blurring effect from our perspective, an effect similar to a propeller looking like a translucent circle while in full motion. At any given instant, the fly appears to be in one position, but given the normal human understanding of time, we see a solid looking eight. In the metaphor the fly could be flying around in any possible pattern as well, but I chose figure eight because it seems to be a readily understandable flight pattern. Also, if these figure eights are what really have an effect on other elements in nature, is the relative position the instant it makes contact with another element what we use to define it as one particular attribute or another? Would the specific direction and momentum the particles strikes the element with cause different reactions in the element?

How incorrect is this metaphor?

I am very interested in getting as up to date understanding as possible, so thanks in advance!

Please, don't try to find any physical analogies for the wave function. All these analogies are more misleading than helpful.

In the beginning of the last century physicists understood one important fact about nature. The fact is that microscopic bodies and particles do not behave predictably. Their dynamics is sort of random. It is impossible to predict where a given electron will go. The best one can do is to calculate probabilities of different outcomes. Quantum mechanics is the theory which calculates these probabilities. This theory operates with certain mathematical symbols. Wave function is just one of those symbols. As a mathematical symbol, it has no direct relationship to physical reality. Don't try to find wave functions in nature.

Let me give you this analogy. Suppose you hold in your hand a gaming die with 6 faces. You can mentally assign to each face the number 1/6. This number tells you what is the probability of this face being up if you throw the die on the table. These numbers "1/6" do not actually exist in nature. They exist in our mind only. They are just imaginary mathematical quantities, which are, nevertheless, helpful for describing the behavior of the die. These quantities are the best analogs for wave functions that one can find in real life. Even this analogy is not perfect, because wave functions normally have complex values, while the probabilities (1/6) are real numbers.

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http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/wvfun.html

I just based my metaphor off of this more than anything. When predicting the behavior of a particle, it is understood that our particular perception sees a blurred bloom of possible spots where the particle is. That was my point is using the fly buzzing around in a figure eight, at the very least, we have a model of where the most likely spots various particles might be. Why would it be inaccurate to assume that these particles move fast enough to create these shapes in essence, in the physical world? If we are predicting them to be in these spots anyway with accuracy, what stands in the way of assuming that these particles don't actually travel in these patterns?

So it seems now I might have to consider one perspective where the wave function is not real at all, and another in which it is. Isn't the wave function a calculus to predict, at least, where a particle will be at any given moment? Could it be possible that our perception is inaccurate because we don't perceive particles that move that fast in an accurate way? If it's only a mathematical model, then why is it that the particle will actually be where we predicted sometimes? Is it impossible that the particle travels the path that we predicted it to travel to get into the position it's in? This is all very fascinating, thank you very much!

imjustaguy said:
1
Why would it be inaccurate to assume that these particles move fast enough to create these shapes in essence, in the physical world?

2
If we are predicting them to be in these spots anyway with accuracy, what stands in the way of assuming that these particles don't actually travel in these patterns?

3
Isn't the wave function a calculus to predict, at least, where a particle will be at any given moment? Could it be possible that our perception is inaccurate because we don't perceive particles that move that fast in an accurate way? If it's only a mathematical model, then why is it that the particle will actually be where we predicted sometimes?

1 Because particles do not have a trajectory.

2 You can make 1 measurement and locate a particle in some position. However, fixing it position you make the momentum unknown. So, if you measure a position of an elector inside an atom you ionize that atom - electons flies away after such meaurement!
You can decrease the precision so electorn won't fly away, but then the potential precesion of the measurement will be less then size of an atom

3 Again, 'what is actually a wavefunction?' depends on the interpretation.
BTW, in some interpretations, MWI for example, there are no particles at all - they are just an illusion.

Try reading the first section here: http://en.wikipedia.org/wiki/Wave_function

Also, regarding the above post about dice, notice as well that the value of the next roll is never know with precision until it is actually rolled (measured). "What is the value of the dice before the roll?" "Where is the location a particle before I measure?" just have no absolute answers.

and as noted the "meaning" of the wave function is still a subject of debate after a hundred years...try wikipedia here: http://en.wikipedia.org/wiki/Copenhagen_interpretation

I think it goes: the wavefunction describes a probability of finding the mass of an electron as position/momentum, when these have a non-commutating phase when mixed with a proton's.
Free electrons can divide their wavefunctions and create a phase between two halves, described by two probability amplitudes.

The wave function (actually the wave function squared) is just a mathematical function that can tell you the probability of finding a particle in a particular location. If particles had exact positions (like they have in classical mechanics for example), the wave function would simply be a spike at exactly that location (let's call it x) and zero everywhere else: the probability of finding the particle when you measure it is 100% at the location x, and 0% everywhere else.

In quantum mechanics however, it has been verified that particles simply do not have an exact location. You cannot say that the particle is 'at location x' anymore. What you can do is examine the wave function, and determine that the particle is somewhere 'near location x', with some uncertainty.
Let's imagine that the wavefunction looks like a cone for example (just an arbitrary example which probably never occurs in reality, but I suppose you are familiar with the shape of a cone, but not with the shape of complicated mathematical function ;) ).
The probability of finding the particle at the position corresponding to the 'peak' of the cone (where it is at its highest point) is relatively high, while the probability of finding the particle at the edges of the cone is nearly zero.

One common misunderstanding is that the uncertainty I spoke about is actually due to the non-perfection of our measuring devices. We cannot measure anything to infinite precision, but even if we could, particles would still have an uncertainty in their position. It is just the wierdness of quantum mechanics...

Most of QM is often very hard to interpret, and you cannot often think of any useful analogy, because there is simply nothing in our direct world that behaves like a single particle does in QM.
Let's take a ball for example. If you see a ball you will say that it has an exact position, obviously exactly where you see it is. But even the ball has a very very very small uncertainty in its position. All the particles that make up the ball have this uncertainty, so the ball has it too. You could theoretically calculate the wave function of the entire ball, but it would mean you had to do that for each individual particle (around 10^23 I think, or a 1 with 23 zeros) in the ball. What you will probably find is that the ball as a whole has a very small chance of actually being very far away, like on the moon! However, the probability of finding the ball on the moon is soooo astronomically small that you will never observe that happening.

imjustaguy said:
Why would it be inaccurate to assume that these particles move fast enough to create these shapes in essence, in the physical world? If we are predicting them to be in these spots anyway with accuracy, what stands in the way of assuming that these particles don't actually travel in these patterns?

The uncertainty in the measurement is not a result of the particle moving too fast for us to see it, it's something weirder. If that's all it was, then as our measuring devices increased in precision, there would be less and less uncertainty. But quantum mechanics says there is a fundamentally necessary amount of uncertainty, and in fact, we've been able to get a level of precision in our experiments where we can see this for almost a century now.

There is a more accurate picture of quantum mechanics that is superficially similar to what you described, called the sum over histories. Here we say that the particle doesn't take one particular path between the two points we measure it, but it takes all possible paths simultaneously. We assign a number to each path (roughly speaking, the paths that are harder for the particle to take get smaller numbers), and take an average of these numbers over all the paths connecting two points to find the probability of the particle ending up at the final point.

## 1. What is a wave function?

A wave function is a mathematical function that describes the behavior of a quantum system, such as the position and momentum of a particle. It contains all the information about the system and can be used to calculate the probability of finding a particle at a certain position or with a certain momentum.

## 2. How is a wave function related to quantum mechanics?

In quantum mechanics, particles behave both as particles and waves. The wave function is used to describe the probabilistic nature of a particle's behavior, as it can exist in multiple states at once. This concept is known as wave-particle duality and is a fundamental aspect of quantum mechanics.

## 3. Can a wave function be observed or measured?

No, a wave function is a mathematical representation and cannot be directly observed or measured. However, the effects of a wave function can be observed through experiments and calculations, such as the probability of finding a particle in a certain state.

## 4. What is the Schrödinger equation and how does it relate to wave functions?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how a wave function evolves over time. It relates the energy of a particle to its wave function, allowing for the prediction of its behavior and properties.

## 5. Are all particles described by a wave function?

Yes, all particles, including atoms, molecules, and subatomic particles, can be described by a wave function. This is because all particles exhibit wave-like behavior at the quantum level.

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