# B Double Slit experiment and the path of least Action

#### sqljunkey

Using the principle of least action can you figure out which path the photon took, or which slit it went thru given some initial condition. Or is this not possible and why.

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#### phyzguy

Science Advisor
The path integral formulation of quantum mechanics tells you that the photon takes all paths. So to find the amplitude of finding the particle at the detection screen, you have to sum up the amplitudes of all possible paths for arriving there. As Feynman clearly points out in the Feynman Lectures, Vol 3, Chapter 1, the assumption that the photon goes through one slit or the other leads to a contradiction with experiment. So the answer to your question, "...which slit it went through..." is, "both". Feynman showed that the path integral formulation of QM reduces to the principle of least action in the classical limit, because only along the classical path do the amplitudes add up constructively. Everywhere else they cancel out.

#### sqljunkey

But assuming this principle, there could be only one right path. if the photon had to go through both slits that means one of these trajectories was longer than the other one and would take more time and more action. and we know light takes the path of least action and least time.

Unless we are assuming that the photon, when we aren't looking, breaks apart into this probability distribution wave-like glob and travels through these slits, while interacting with itself and then when it meets the screen collapses into a measurable quantity.

But also having all these possible trajectories traveled, and interacting destructively or constructively with each other violates some other principles, because some of these paths would be going to the end of the universe and back just to get eliminated by destructive interference.

What I am thinking about this principle of least action is that if I know the initial position and the final position of the photon i can calculate it's trajectory. And it's either going to be one of these holes, because I have to take the minima of the action.

Correct me if I'm wrong. :3

#### Nugatory

Mentor
if I know the initial position and the final position of the photon....
You also have to know the times at which the photon is at these positions. It’s not enough to say “there’s a dot on the film here”, you also have to be able to say when that dot appeared. You can do that for the detection event but not the emission event.

#### sqljunkey

so If I measure the photon at the source, way before it travels through those slits, I would still destroy the interference pattern ?

#### zonde

Gold Member
But assuming this principle, there could be only one right path. if the photon had to go through both slits that means one of these trajectories was longer than the other one and would take more time and more action. and we know light takes the path of least action and least time.
Let's say you shine a beam of light trough long cylinder of glass. Let's say that cylinder is so long that the path around cylinder trough the air would take less time for light to travel. Would you expect that light will go around cylinder just because this path takes less time to travel?

#### sqljunkey

zonde I'm not going to make physics up and tell you that the photon is going to knowingly make an effort to bend around that rod of glass to shorten it's trajectory in time. I don't know.

But it's certainly an alternative idea to a shy particle that quickly turns into a distribution wave-like glob when people aren't looking at it and then back into a particle again when people are looking.

#### zonde

Gold Member
But it's certainly an alternative idea to a shy particle that quickly turns into a distribution wave-like glob when people aren't looking at it and then back into a particle again when people are looking.
Idea that quantum particles are somehow switching between particle like behavior and wave like behavior is simply flawed. If this idea is hunting you, you can try to adopt Bohmian interpretation to get rid of it.

#### phyzguy

Science Advisor
... And it's either going to be one of these holes, because I have to take the minima of the action.
Correct me if I'm wrong. :3
You're wrong. As I stated earlier, and Feynman eloquently explains, the assumption that the photon goes through either one slit or the other leads to a contradiction with experiment, so it cannot be true. Again quoting Feynman, "You'll have to accept it. It's the way nature works. If you want to know how nature works, we looked at it, carefully. Looking at it, that's the way it looks. You don't like it? Go somewhere else, to another universe where the rules are simpler, philosophically more pleasing, more psychologically easy."

The principle of least action (or least time) only holds in the classical limit.

#### Nugatory

Mentor
But it's certainly an alternative idea to a shy particle that quickly turns into a distribution wave-like glob when people aren't looking at it and then back into a particle again when people are looking.
The quantum mechanical explanation for the double-slit experiment isn’t anything like that.

#### DrChinese

Science Advisor
Gold Member
I don't know.

But it's certainly an alternative idea ...
The issue here is that you don't know enough about the physics to be coming up with fanciful ideas. There are a lot of variations around the double slit, and without being familiar with those, you will end up with "alternatives" that are contradicted by experiment.

For example: read up on the double slit with polarizers. In this variation, interference appears (or not) depending on the relative setting of polarizers over each slit. This contradicts the notion that a particle goes through one slit or the other fairly convincingly.

#### vanhees71

Science Advisor
Gold Member
Using the principle of least action can you figure out which path the photon took, or which slit it went thru given some initial condition. Or is this not possible and why.
QED tells you that the idea of a "path of a photon" doesn't make sense. That's why the action principle is applied to fields rather than to "trajectories" in the evaluation of QFT path integrals: It's derived from applying the saddle-point approximation to the path integral, which provides a formal expansion in powers of $\hbar$, i.e., is a systematic expansion around the "classical field-theory limit", i.e., in this case Maxwell classical electrodynamics.

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