# Wave–particle duality theory question

Note this is more of a coursework understanding question rather than a specific homework question.

## Homework Statement

I have been reading about matter waves and de Broglie relations which suggest that electrons can act as waves. From wikipedia (http://en.wikipedia.org/wiki/Matter_wave) it suggests that the following is true.
$$\lambda = \frac{h}{p}$$ and $$f = \frac{E}{h}$$
In that article it does not suggest the significance of the amplitude in the matter wave. From the wave–particle duality article (http://en.wikipedia.org/wiki/Wave–particle_duality) it suggests that "upon measuring the location of the particle, the wave-function will randomly "collapse" to a sharply peaked function at some location, with the likelihood of any particular location equal to the squared amplitude of the wave-function there."

## The Attempt at a Solution

If I understand this correctly, the probability of finding an ejected electron along a linear trajectory is not equal along its path, but rather the probability is sinusoidal. At one part of the trajectory, I might have a 100% chance of finding an electron when the amplitude is max and $$\lambda$$/4 away there is a 0% chance of finding an electron when the amplitude is zero. This is counterintuitive so I want to check if my reasoning is correct.

PS: For a numerical example, solving the equations for an electron in a 10 kV scanning electron microscope yields a wavelength of 12.3 x 10^-12 m (12.3 pm) as from http://en.wikipedia.org/wiki/Electron_diffraction. If I understand this correctly, as an electron travels in this setup, the probability of finding the electron varies from 0% to 100% sinusoidally with a wavelength of 12.3 x 10^-12 m (12.3 pm).

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So I came across another post in which it describes the photon wave function.

In that post, arcnets, stated that "I mean it in the following sense: "It's not possible to define any photon wavefunction from which a probability amplitude for spatial localization can be calculated". (Landau-Lifschitz IV, chapter 1, §4.)"

If I understand this correctly, the probability of a photon at any position cannot be determined. However, the probability of an electron at a position can be determined by Schrodinger's equation. Is this correct?

So on a related note I have delved deeper into Schrodinger's equations and came across Heisenberg's uncertainty principle. It seems that Schrodinger's equation applys only to electrons and not photons whereas Heisenberg's uncertainty principle apply to photons and electrons. Are these correct limitations to these theories?

So I came across another post in which it describes the photon wave function.

In that post, arcnets, stated that "I mean it in the following sense: "It's not possible to define any photon wavefunction from which a probability amplitude for spatial localization can be calculated". (Landau-Lifschitz IV, chapter 1, §4.)"

If I understand this correctly, the probability of a photon at any position cannot be determined. However, the probability of an electron at a position can be determined by Schrodinger's equation. Is this correct?

ZapperZ
Staff Emeritus
So on a related note I have delved deeper into Schrodinger's equations and came across Heisenberg's uncertainty principle. It seems that Schrodinger's equation applys only to electrons and not photons whereas Heisenberg's uncertainty principle apply to photons and electrons. Are these correct limitations to these theories?

This is totally false. Read the Marcella paper (i.e. don't just read Wikipedia)

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

Zz.

Thanks for your quick response. I am a new student to physics, and I am still in my introductory physics class. I am trying to understand the paper and your statement, "this is totally false."

From the paper's statement, "because position and momentum are non-commuting observables, a particle passing through slits always has an uncertainty in its y-component of momentum," I am guessing that my statement, "Heisenberg's uncertainty principle applies to photons and electrons," is correct. Or is it false?

I am guessing that the "totally false" statement has to deal with my other statement on Schrodinger's equation. If I understand the paper correctly, then Marcella was able to determine the "calculation for the probability amplitude and its corresponding probability function" using Schrodinger's equations. I am guessing this would mean that Schrodinger's equations applies to both photons and electrons. Correct me if I am wrong here.

Also is this quote by Arcnet citing Landau-Lifschitz posted earlier false? Arcnets, stated that "I mean it in the following sense: "It's not possible to define any photon wavefunction from which a probability amplitude for spatial localization can be calculated". (Landau-Lifschitz IV, chapter 1, §4.)"