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Forgive me for this dumb question but what are matter waves of particles? are they particles being spread out in a space like waves or the particles are still "particles" but matter waves are probability waves? and if the particle is actually spread out in space like a wave then why does this page from wikipedia about string theory says that strings replace "point-like particles"? https://en.wikipedia.org/wiki/String_theory

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closed as too broad by Aaron Stevens, Qmechanic Oct 19 at 1:16

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Quantum mechanics is applicable in the regime of extremely small (when length scale ~ $h/p$). So lets talk about small particles. For such particles, something strange happens:

  1. when they are detected they behave as if they are particles in the conventional sense: that is they have an exact value for their energy and momenta. Imagine it this way: when the particle's position is detected say at a screen of some detector, its hits it at just one tiny spot.
  2. strangely, if the detection experiment were to be repeated, the energy momentum values are found to be different. They are nevertheless exact this time too.

(by repeated, we mean detection performed by an ensemble of identically prepared systems)

Therefore, there is a probability distribution associated with the energy-momentum of the particle. This is true for any observable. When the observable is position, the associated probability density is called a matter wave.

The matter wave is unlike any notion of wave that you may have. It is not

  1. matter of the particle smeared out in space in the form a standing or propagating wave
  2. a wave in the probability distribution of the observable
  3. a wave in the form of which the original particle travels.

All a particle's associated matter wave represents is the probability of detection of that particle at different points in space. At every detection, the particle exhibits particulate properties. This is the particle nature. However since it doesn't seem to be able decide upon one value of its position, its as if its spread out. This is the wave nature. This is wave-matter duality.

Why can't we say the particle is actually spread out like a wave i.e it travels like a wave? Because it is not experimentally possible to detect the form in which a particle travels-to do so requires detection and at the moment of detection, each particle appears particulate in nature.

What is the wave in the "matter wave"? I am not completely sure. The particle's wave functions (whose mod square gives he probability density) is in general imaginary though it does sometimes appear as if its spreading in space or moving in space with time.. like a pulse expanding. But its nothing like what you associate waves with--sound, water eaves, EM fields etc.

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  • $\begingroup$ -1: "[W]hen they are detected they behave as if they are particles in the conventional sense: that is they have an exact value for their energy and momenta. Imagine it this way: when the particle is detected say at a screen of some detector, its hits at just one tiny spot." This is wrong in multiple ways: 1. Whether a particle can be measured to have an exact value for their energy and momenta depends on the Hamiltonian. If the Hamiltonian doesn't commute with the momentum operators, you can't measure both energy and momenta at the same time. [...] $\endgroup$ – Dvij Mankad Oct 19 at 2:26
  • $\begingroup$ [...] 2. Whether a particle will have a precise momentum or not when you "detect" the particle depends entirely on your mechanism of detecting. If you are measuring the momentum then it would have a precise momentum, if you are measuring the position, it would have a precise position. Both are legitimate ways of detection. 3. A way to think about a particle measured to have a precise momentum is not to imagine it having a localized position like hitting at just one tiny spot. It is the opposite. A particle with a definite momentum would not have a precise position at all. $\endgroup$ – Dvij Mankad Oct 19 at 2:29
  • $\begingroup$ Probability density is a real, not imaginary, quantity. Matter waves are complex waves of probability amplitude whose modulus squared is the probability density. $\endgroup$ – G. Smith Oct 19 at 3:13
  • $\begingroup$ @DvijMankad For your point 3, doesn't applying that reasoning only apply for the distribution of measurements of similarly prepared quantum systems? i.e. if we obtain a smaller spread in momentum measurements we would expect a larger spread in position measurements? $\endgroup$ – Aaron Stevens Oct 19 at 4:24
  • $\begingroup$ @G Smith. Thank you. I have corrected the answer. $\endgroup$ – lineage Oct 19 at 13:18

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