Exploring the Destructive Interference of Probability Waves in Quantum Mechanics

[thebiggerbang]

I'd like to clarify my concepts about how a electron probability wave works. For a light wave, we can say that there is a destructive interference produced as and when the trough coincides with the crest of suitable amplitude. But since there can't be negative probabilities, how can there be a destructive interference pattern be observed in the experiment? What is the physical meaning of a trough and crest in a probability wave? What opposite character do they posses? Thus, how does the destructive interference take place?

I've read a little bit of Feynman; and could make out that it is related to complex numbers. I know the basics, so can make some sense outa why probability =|$\Psi$|$^{2}$

If possible, please try to make an explanation at my level, although I have a good idea about the basic principles of QM.

 

[vanhees71]

This is the very fact that makes quantum theory quantum vs. classical.

A nice example is the double-slit experiment. If you direct a particle beam (take electrons as an example) towards a plate with two slits in it and observe the distribution of particles hitting a detector behind the two slits, you'll find a pattern which cannot be explained by classical physics.

In classical physics you'd expect to simply see a pattern, which is given by the sum of the particle densities from each slit, i.e., you add the probability s of a particle to go through slit 1 and the probability to go through slit 2 since each particle can go only through one slit.

Contrary to that, quantum theory predicts a probability distribution, which looks like the interference pattern of a wave going through two slits, i.e., you must not add the probabilities for the particle to go through the one or the other slit but you add the probability amplitudes first and then squaring it to get the total probability. The amplitudes can have either sign, postive or negative (or as you correctly say in general one needs complex amplitudes in quantum theory) and thus you can have destructive interference.

BTW: This is only true, if there is no way to say for each individual particle through which slit it has gone. As soon as you give each particle a marker through which slit it came, the interference pattern vanishes and you are back to the classical pattern from adding the probabilities rather than the probability amplitudes.

Don't ask, why nature behaves like this. Physics only describes as good as one can, how nature behaves, but it cannot answer the question why the fundamental laws of physics are as they are.

 

[quietrain]

This is the very fact that makes quantum theory quantum vs. classical.

A nice example is the double-slit experiment. If you direct a particle beam (take electrons as an example) towards a plate with two slits in it and observe the distribution of particles hitting a detector behind the two slits, you'll find a pattern which cannot be explained by classical physics.

In classical physics you'd expect to simply see a pattern, which is given by the sum of the particle densities from each slit, i.e., you add the probability s of a particle to go through slit 1 and the probability to go through slit 2 since each particle can go only through one slit.

Contrary to that, quantum theory predicts a probability distribution, which looks like the interference pattern of a wave going through two slits, i.e., you must not add the probabilities for the particle to go through the one or the other slit but you add the probability amplitudes first and then squaring it to get the total probability. The amplitudes can have either sign, postive or negative (or as you correctly say in general one needs complex amplitudes in quantum theory) and thus you can have destructive interference.

BTW: This is only true, if there is no way to say for each individual particle through which slit it has gone. As soon as you give each particle a marker through which slit it came, the interference pattern vanishes and you are back to the classical pattern from adding the probabilities rather than the probability amplitudes.

Don't ask, why nature behaves like this. Physics only describes as good as one can, how nature behaves, but it cannot answer the question why the fundamental laws of physics are as they are.

maybe the whole notion of electrons is wrong ?

i don't know.,.. i saw a lot of "pseudo" science on youtube, like saying anti-gravity , electrons don't exist etc...

i mean i have studied science till uni and seriously, i have never seen an electron or witness electricity at its raw form. everything seems to be taken for granted.

:( like this guy called edward leeskalin? who built the coral castle, he claims that education nowadays are wrong because they are passed down wrongly. :(

its confusing:( like quantum mechanics... it ain't intuitive at all...

 

[thebiggerbang]

Don't ask, why nature behaves like this. Physics only describes as good as one can, how nature behaves, but it cannot answer the question why the fundamental laws of physics are as they are.

Ouch! That hurts :-(

So is it like that's the way nature behaves, and we just make use of complex amplitudes to simplify the concept and get it down to a level that we can comprehend?

 

[vanhees71]

Right! Physics is, as any natural science, an empirical science, i.e., everything we know about it comes after all from observations of nature. Of course, one has to get observations under clearly defined simple conditions, and that's why phsycists do experiments and measure quantitatively certain properties of such setups of experiments.

In the history of physics it has turned out that one finds certain basic patterns of the behavior of such setups, and that these basic patterns can be formed into a mathematical description, of such common aspects of the behavior of nature. These form the theories (or models), which lead in turn to prediction about the behavior of nature under other setups than investigated before, and these predictions can be tested to check the theory against reality.

At the end of the 19th century it turned out that the classical models of physics (at the time these were basically Newtonian mechanics and Maxwellian electromagnetics) cannot adequately describe the behavior of matter and the interaction of electromagnetic fields with matter, if it comes to the description on the level of atomic (and lateron also subatomic) scales. In an amazingly short time quantum theory has been developed to take care of this somewhat "strange" behavior of microscopic entities of matter. Already in 1925 the first formulation of quantum theory in the modern sense has been formed by Heisenberg, Born, Jordan, Dirac, and Pauli. Only little later, Schrödinger found another formulation of the same theory in terms of his wave function.

Then, the physicists were in a somewhat strange situation with this new theory of matter and interaction of matter with radiation: They had a nice mathematical description, but it has not been clear from the outset what the quantities involved in this theory mean in terms of physics. This is a unique case in the history of sciences!

The basic conceptual problem was, what the meaning of the wave function might be, and it turned out that only Born's interpretation as a probability amplitude is compatible with all observations.

Now, even today, not all physicists are satisfied with this rather radical step: The basic laws of nature do not describe the behavior of matter in a deterministic way, and the theory tells us that instead we have to use a probabilistic description. Even if we prepare the system of our investigation in an experiments in a perfectly possible way, we cannot predict with certainty the outcome of measurements of other observable quantities than that we have determined by our preparation. The theory also tells us that we never can prepare a system such that all possible observables have well-defined values simultaneously. E.g., it says that one cannot simultaneously determine the position and the momentum of a particle exactly, but if one determines the position of a particle very precisely, one necessarily has a very uncertain momentum (an vice versa). Thus, there's no way of a deterministic description as in classical physics.

This implication has not been appreciated by some famous physicists like Einstein or even Schrödinger, who invented one formulation of quantum theory himself! However, all the implications of quantum theory have been tested very carefully and always have been found to describe nature very accurately.

As I said before, quantum theory cannot tell us, why nature behaves in such a way, but it only describes this behavior in a very accurate way.

One should also mention that not all physicists follow this socalled "minimal statistical interpretation" but have other interpretations of quantum theory. However, in my opinion, all those "non-minimal" interpretations do not help to comprehend quantum theory but make it even worse to do so. That's why I prefer the "minimal interpretation", sometimes called "shutup-and-calculate interpretation" (Feynman).

 

[thebiggerbang]

One should also mention that not all physicists follow this socalled "minimal statistical interpretation" but have other interpretations of quantum theory. However, in my opinion, all those "non-minimal" interpretations do not help to comprehend quantum theory but make it even worse to do so. That's why I prefer the "minimal interpretation", sometimes called "shutup-and-calculate interpretation" (Feynman).

Ouch! That hurts!
But if nature doesn't want to reveal itself, we are no-one to change that! QM is totally counter intuitive, maybe the very same unpredictable nature fascinates me! Everytime the world thinks we are at the edge of decoding the universe, the comfort bubble is burst! I hope by the time I graduate, the stage is set for such an upheaval!
Well would it help to have another such revolution in physics, can you sense any?

 

[Fredrik]

The phrase "shut up and calculate" originated with David Mermin, not Richard Feynman.

http://fisica.ciencias.uchile.cl/~emenendez/uploads/Cursos/callate-y-calcula.pdf

 

[Drakkith]

Ouch! That hurts!
But if nature doesn't want to reveal itself, we are no-one to change that! QM is totally counter intuitive, maybe the very same unpredictable nature fascinates me! Everytime the world thinks we are at the edge of decoding the universe, the comfort bubble is burst! I hope by the time I graduate, the stage is set for such an upheaval!
Well would it help to have another such revolution in physics, can you sense any?

Since when has the world ever thought we were about to "decode" the universe? Science is constantly finding new things! Are you saying that you hope we get rid of quantum mechanics and find the "real" way things happen?

Also, I don't really see Quantum Mechanics as a "Revolution" really. It just looks like another step in the process to me.

 

[Naty1]

So is it like that's the way nature behaves, and we just make use of complex amplitudes to simplify the concept and get it down to a level that we can comprehend?

We find ways to explain what is observed...and oddly, for reasons no one really understands, man's mathematics often fits the universe we observe. Math is good to use because it can be used to make predictions as well as explain observations.

Like, distance = velocity times time, d = vt : that relationship enables you to predict how far an object has traveled at constant velocity at some future time ...


The trick is to pick the mathematics that matches this universe...there is a lot of math that we have not yet found a fit for. Often, the math is sitting there without physical interpretation, pure mathematics, and along comes a clever physicst and realizes it fits something being studied in the physical, observable world.

Einstein was perhaps the pre eminent genius who could fit real world observations and unanswered questions to largely existing math. And that's how string theory got "accidently" started...

 

[BruceW]

In answer to the OP, $\Psi$ is sometimes called a probability wave, but this is a very bad name for it. Because it is actually ${| \Psi |}^2$ which represents the probability. (not $\Psi$ itself).
So when you have destructive interference, one $\Psi$ is the negative of the other, so they cancel out.

 

[BruceW]

Since when has the world ever thought we were about to "decode" the universe? Science is constantly finding new things! Are you saying that you hope we get rid of quantum mechanics and find the "real" way things happen?

Also, I don't really see Quantum Mechanics as a "Revolution" really. It just looks like another step in the process to me.

I think it was pretty revolutionary. Before quantum mechanics, they thought reality was deterministic. That's a pretty big philosophical change.

 

[Drakkith]

I think it was pretty revolutionary. Before quantum mechanics, they thought reality was deterministic. That's a pretty big philosophical change.

I guess if we were to discuss the most "Revolutionary" theories, my vote would be Quantum Mechanics yes. I guess I meant that while it was a huge step along the path, it was still just a step.

 

[mathal]

I think it was pretty revolutionary. Before quantum mechanics, they thought reality was deterministic. That's a pretty big philosophical change.

I would add

'Before quantum mechanics they thought reality was "necessarily" deterministic. That's a pretty big philosophical change.'

There are still a few QM models that have a deterministic representation of the universe, just not in a classical determinisic way. I am referring to the 'pilot wave' model and it's cousins and the Block Universe models. Both these models are of course in their own way even bigger philosophical changes from classical modeling.

In answering the OP, no matter which modeling you subscribe to measurements will adheres to the already inherently unassailable tenets that QM rests on. In the double slit experiment, the photon or electron or whatever, cannot be precisely nailed down as passing through this or that slit or there is no interference. You are right to see this is a physical phenomena that doesn't require an observer to occur, so one want's to be able to picture it.
Each modeling offers it's own explanation for a phenomena which is necessarily always true, given the same stringent requirements for the experiment.
So you can take the
1. it's a black box- we cannot determine any more than the probability of any particular circumstance occurring.
2. Inside the 'black box' is a pilot wave which governs the 'actual' path of each particle to it's particular measured destination
3.The 'black box' is the universe and the possible paths of each particle are 'determined' by events the particle 'will' necessarily encounter in each possible 'real' path it 'could' take. Further to this last model you have to picture the particle simultaneously traveling each and every possible path, interfering with itself constructively or destructively for the purpose of determining the actual probability of any particular path being taken.
This last model is the one I prefer but it is a matter of taste. The underlying math is necessarily the same. They are just different pictures of things we cannot see.

mathal

 

[ZealScience]

I don't quite understand your question. But I think as long as it is some magnitude squared how could it be negative?

 

[thebiggerbang]

I don't quite understand your question. But I think as long as it is some magnitude squared how could it be negative?

Don't worry. You are not alone! many have failed to understand my question when I asked to a few people personally.

Squaring a complex number will give you a negative number my boy! This is avoided by dropping in a modulus sign. So even though the amplitude might be a complex number, the probability will never be negative! it fits in perfectly, describing the wave nature of matter!

Clever math, ain't it?

 

[ZealScience]

Don't worry. You are not alone! many have failed to understand my question when I asked to a few people personally.

Squaring a complex number will give you a negative number my boy! This is avoided by dropping in a modulus sign. So even though the amplitude might be a complex number, the probability will never be negative! it fits in perfectly, describing the wave nature of matter!

Clever math, ain't it?

But I meant magnitude in my post which is usually denoted by √<Ψ|Ψ> or Z·Z* in mathematics that I've learnt. It is always calculated using conjugation, right? I think people don't square complex numbers to get magnitudes. Just what I've learnt.

 

[Fredrik]

Squaring a complex number will give you a negative number my boy! This is avoided by dropping in a modulus sign. So even though the amplitude might be a complex number, the probability will never be negative! it fits in perfectly, describing the wave nature of matter!

Squaring a complex number will in general give you another complex number. Squaring an imaginary number on the other hand...

 

[BruceW]

Don't worry. You are not alone! many have failed to understand my question when I asked to a few people personally.

Squaring a complex number will give you a negative number my boy! This is avoided by dropping in a modulus sign. So even though the amplitude might be a complex number, the probability will never be negative! it fits in perfectly, describing the wave nature of matter!

Clever math, ain't it?

Did you know the answer to your question before you posted it?

 

[twofu]

The things you ask are merely consequences of solving Schrodinger's equation and integrating all over space and set the probability equal to one. If you aren't fluent in math, none of this makes sense. Quantum mechanics is a mathematical construct with much much physical proof.

 

[twofu]

Don't worry. You are not alone! many have failed to understand my question when I asked to a few people personally.

Squaring a complex number will give you a negative number my boy! This is avoided by dropping in a modulus sign. So even though the amplitude might be a complex number, the probability will never be negative! it fits in perfectly, describing the wave nature of matter!

Clever math, ain't it?

Squaring an IMAGINARY number will give you a negative. However, if the probability amplitude is complex, we are told to take the absolute square, which is defined as zz* where z* is the conjugate leaving you with a positive quantity defined on the Reals.

 

[Uroboros]

I would like to know more about the individual steps performed in Young's experiment. In particular I would like to know this, if particle detectors are in place at the slits and are activated while a wave interference pattern is being observed, is it possible to watch the wave pattern change into a particle pattern in real time or can these only be viewed separately?

 

[StevieTNZ]

BTW: This is only true, if there is no way to say for each individual particle through which slit it has gone. As soon as you give each particle a marker through which slit it came, the interference pattern vanishes and you are back to the classical pattern from adding the probabilities rather than the probability amplitudes.

It seems even with a marker, you get fringe + anti-fringe interference patterns, which results in an overall scatter pattern which looks like two one-slit patterns added together. If erasure is done on the marker, then the interference patterns would become evident.