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I started reading Quantum Mechanics and I'm being told that matter and light have both a particle and wave aspect to them. It is easy to understand the particle aspect of light: experiments show that light can push an electron out of it's orbit, so one imagines that light has to behave to some extent "as if" it was a particle that can collide with an electron.

Furthermore, when we say that "light" behaves as a wave I understand this as simply meaning that light can be viewed as a specific solution of Maxwell's equation of the form $f(x - c t)$ and in particular it can be expanded into a superposition of "waves" of the form $\exp(2\pi i u (x - c t))$.

However, I am confused as to what is being meant when people say that matter behaves like a wave. In particular I am confused as to the meaning of the de Broglie relation between the momentum of a particle and it's wavelength. What is the physical meaning of this wavelength associated with matter? And how is it associated to matter?

My best guess is as follows: we can conceptually view a beam of electrons of a certain energy directed in a certain direction as behaving like a "wave". This allows us to explain certain experiments in a consistent way, by maintaining the principle of superposition: for instance when two beams collide we can view the resulting beam as a superposition of the original ones. In this way coupled with the statistical interpretation of the wave, we manage to obtain physical information on the collisions of electrons having different energy levels (within the limits of precision that are enforced by the uncertainty principle).

blaber
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According to QED, light consists of photons. How do we know that? Because in any experiment with dim light, We always found that light comes in a discrete packet of energy. The phenomena of light can be understood by introducing what is call probability amplitude. With the help of this probability amplitude, We can find out the probability for some event to occur. More on this here.

The wave nature of the particle is not in a sense of a classical wave but It means to associate with particle a wave function $\psi(x)$. The interpretation for this is that the square of the wave function (called Probability amplitude) is probability density for the particle. So when de-Broglie gave the wavelength for the particle, that's mean the wavelength of probability amplitude associated with the particle. See here.

Young Kindaichi
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  • Can you please elaborate on "we always found that light comes in a discrete packet of energy"? I thought that QM allows for a wave packet of light consisting of all energies in an interval (a,b), say. – blaber Nov 15 '20 at 06:21
  • The device we use to detect the light is what we call photomultiplier. When we use a dim light, it's signals that the light is actually coming in a discrete way. More on this here: https://en.wikipedia.org/wiki/Photomultiplier – Young Kindaichi Nov 15 '20 at 07:00
  • Quantum mechanics doesn't deal with light as far I know. – Young Kindaichi Nov 15 '20 at 07:00
  • Quantum mechanics doesn't deal with light as far I know. – Young Kindaichi Nov 15 '20 at 07:00
  • How can quantum mechanics not deal with photons (and thus light) when the Compton effect was one of the foundational experiments that led to the discovery of quantum mechanics? – blaber Nov 15 '20 at 22:41
  • I do not think that the statement that the photons come in "discrete packets of energy" is correct. In general I think the energy of the photon can be anything you like, it is not bound to a discrete set of values (in fact photons correspond to a scattering state with a continuous spectrum of energies). Of course you will not be able to measure consistently the energy to an accuracy smaller than the Planck scale, but that is another story. – blaber Nov 15 '20 at 22:45