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Amplitudes or probability amplitudes are the complex coefficients of the linear combination of states which represent other quantum physical states. The amplitude of a wave can be interpreted as a "measure" of the height of the wave. There are different ways to measure it.

AFAIK, quantum mechanics can be formulated from a linear algebra or "wave" perspective.

Hence, the consequent questions of a person who knows almost nothing about quantum mechanics and, in general, physics are the following.

Is there any relationship between probability amplitudes and amplitudes of a wave? If yes, how exactly are these concepts related in the context of quantum mechanics?

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Integrating the probability amplitude over some region tells you the probability of finding the particle in that region. The actual amplitude of the wavefunction does not matter because has to be normalized anyways in order to interpret the wavefunction in terms of probability.

The relative amplitude might be important for interference or answering questions like "is the particle more likely to be here or there?".

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Is there any relationship between probability amplitudes and amplitudes of a wave? If yes, how exactly are these concepts related in the context of quantum mechanics?

Wave equations are differential equations whose solutions are sine and cosine functions.

They are called wave equations because sine and cosine functions model simple waves as observed in strings and water, and other wave equations fit well acoustic data and light/electromagnetic data.

The above waves have an amplitude, which is modeled on the variations of energy transfer in the medium. It was expected that light also traveled on a medium, called luminiferous aether, but experiments showed that the energy it carried traveled in vacuum, due to the properties of electric and magnetic fields. Classical wave amplitude is directly connected to energy transfer.

The wave equations used for describing quantum mechanical data, do not have the amlitude connected with energy transfer. The quantum mechanical amplitude gives the probability of finding the system in a given state.

See the hydrogen atom solutions here.

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