Exploring the Schrodinger Equation: How Does It Handle Split Wave-Functions?

[San K]

Hamiltonian operator, which characterizes the total energy of any given wavefunction and takes different forms depending on the situation.

Also the frequency of matter waves, as deduced by de Broglie, is directly proportional to the particle's total energy, i.e. the sum of particle's Kinetic energy and rest energy.

when a wave-function splits into two paths (such as in a double slit or a mach-zehnder) how does the Schrödinger equation deal with it?

when one of the path is blocked (before/after the slits or anytime prior to reaching the detector) by an opaque obstruction, how is the energy for the wave-function of that path dealt with?

 

[Simon Bridge]

The wave function flows through the system in a manner analogous to a classical wave.
It is usually more helpful to take the phase representation of the wave functions and sum over the available paths.

See the lecture series starting with:


When one path gets blocked, then any particle attempting that path is blocked - it's energy and momentum are absorbed by the blocking material...
Note: the wavefuction does not carry energy, the particle does. The wavefuction is a representation of probability amplitudes of detecting a particle with particular properties in a particular place.

 

[San K]

Note: the wavefuction does not carry energy, the particle does.

yet it (the energy-less wave-function) can change the path/behavior of the photon...(at least in extrapolation of some of the interpretations)

 

[Simon Bridge]

...for example?

(How would a probability distribulation carry energy?)

 

[San K]

...for example?

(How would a probability distribulation carry energy?)

if both slits are open -- the photon(s), even if sent one by one, land up on a different parts of the screen than if only one slit was open.

its as if (at least in extrapolation of some of the interpretations) the wave-function from the other slit is "pushing" the photon around...as if its changing the photon's path...

similar is with the mach-zehnder...http://en.wikipedia.org/wiki/Mach%E2%80%93Zehnder_interferometer

we may not have the answers yet Simon...

 

[Simon Bridge]

That interpretation is incorrect. The change in pattern is not consistent with the idea of a wave from "the other slit" pushing the photon around. The wkipedia article you link to does not make that interpretation/extrapolation.

Are you picturing the particle as like a cork being pushed along by a water wave?

 

[ZapperZ]

when a wave-function splits into two paths (such as in a double slit or a mach-zehnder) how does the Schrödinger equation deal with it?

Read this:

http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

Zz.

 

[San K]

That interpretation is incorrect. The change in pattern is not consistent with the idea of a wave from "the other slit" pushing the photon around.

yes, most interpretations don't support the above idea. we can discard it.

would you like to share the correct interpretation(s)? that you have in mind

The wkipedia article you link to does not make that interpretation/extrapolation.

the link was to give information about mach-zehnder (for those not familiar with it) and not any interpretation.

Are you picturing the particle as like a cork being pushed along by a water wave?

I have discarded that picture. let's hear what you have in mind.

 

[Simon Bridge]

I make no claims to have any "correct" explanations of anything.

The paper ZapperZ has shown you (link above) has a nice discussion of the common misconceptions about "wave particle duality" which is the broad umbrella-term for what you are wrestling with.

My personal favorite description comes from Richard Feynman in the lecture series he gave in Auckland NZ ... quite old now but still relevant.

QM waves are not classical waves: they are not waves of anything in the way water waves are waves of water molecules. They are tools for predicting probabilities. They get called "waves" because the math has the same structure as the class of equations in mathematics called "wave equations".

When you get introduced to them, authors like to spell out how weird they are by making comparisons with classical mechanics. This is just why we need quantum mechanics to replace the classical.

So - when you have both slits open - the wave-function in the "other slit" is zero ... because we know the photon went through "this" slit right? Thus the probability that the wave went through the other one is zero.

If we abandon knowledge of where the photon went, i.e. if we only care about the screen, then we work out the probability of detecting it some distance from the slits by working out the amplitude and phase that it would have if it had come from one slit, and the amplitude and phase if it had come from the other slit, and add the two amplitudes together (making a vector sum). The square of this resulting amplitude is the probability of finding a photon at that position.

Here's the Feynman lectures. Watch all of them.
http://vega.org.uk/video/subseries/8/
... and don't worry - it takes everyone a while to wrap their minds around this stuff.

 

[San K]

thanks Zapper and Simon

 

[Simon Bridge]

No worries - these questions seem to have been raised by you before now. I know some concepts can be persistent. But really watch those videos.