Matter wave is the wave associated with a particle having momentum $p$ ; its wavelength being $$\lambda = \dfrac{\mathbf{h}}{p}$$. So, particle moving with high momentum has lower wavelength. Ok, upto it quite conceivable.

Now, it is written that

We shall abandon the classical concept of particles moving on trajectories. From now on, we adot the quantum-mechanical view that a particle is spread through space like a wave. The wave provides direction or pilots ** the particle in the **path. To describe this distribution, we introduce the concept of a wavefunction $\Psi$ in place of the precise path. It is the blurred version of a trajectory. Like a classical wave where either particle, pressure or electric field component oscillate, in matter wave, at a certain point , $\Psi$ oscillates periodically like a wave.

Now, if the particle doesn't move in a precise path and is behaving as a wave, then what does this mean that a particle is spread through space like a wave??

Now, also the Born Interpretation asserts that

the probability of finding the particle in a small region of space is proportional to ${\psi(x,t)}^2 . dv$ around $x$.

What is $\psi$?

It is called the amplitude of matter wave. Now, at a point at time $t$ , $\psi$ can be of any value; so why does it specially saying the amplitude?? In a sound wave, at a certain point , pressure varies periodically and after a time , the pressure becomes maximum. So, at that point, there cannot be always maximum value of wavefunction- it varies with time. So, why does it say that the probability at that point is proportional to the amplitude of the matter wave? Really confused.


The idea that

The wavefunction provides direction or pilots the particle in the path.

is a particular interpretation of quantum mechanics known as Bohmian mechanics, also known as pilot-wave theory.

In contrast, the statement of the Born rule that the probability to find a particle with wavefunction $\psi(x,t)$ at time $t$ in the interval of space $[x_-,x_+]$ is

$$ P([x_-,x_+]) = \int_{x_-}^{x_+}\lvert\psi(x,t)\rvert^2 \mathrm{d}x$$

is independent of the chosen interpretation.

The wavefunction $\psi(x,t)$ is also called the probability amplitude (in the position representation) of the state described by $\psi$, which is a slight misnomer, because it is not the difference between extremal values like an ordinary amplitude, but just a (complex) valued function.

Staying free of any interpretations, the object $\psi(x,t)\in L^2(\mathbb{R})$ is what specifies uniquely the quantum state of a particle confined to one dimension, because, by the Born rule, it tells you exactly how likely it is to find the particle at any given time in any given segment of space. This is all quantum mechanics will ever tell you, and all you need to know.

Whether you choose to think of this in the Bohmian way that the $\psi$ "guides" the particle, or in the usual way that the particle "exists" in a superposition of states with definite position, or not at all, has no impact on the theory. The apocryphal

"Shut up and calculate"

variously attributed to Feynman, Dirac, Gell-Mann or Mermin is supposed to mean exactly that - there is no unique "intuitive picture" of quantum mechanics, just a theory and its predictions (by the Born rule).


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