Wave-particle duality?

Question

What does it mean for an electron to behave as a wave? I can visualize electrons or other subatomic things as particles. But what do we mean when say it's all a wave. What is waving actually? Waves are just a way by which energy can get transferred right with the oscillation of particles. But in this case the particle is an electron and is it oscillating to produce waves? And then we say that normal day to day objects also have wave properties but not really significant. How can I visualize it?

Answer 1

What behaves like a wave is the quantum wave function $$\Psi(\vec x_1, \dots, \vec x_n)$$ defined on configuration space. It evolves according to the Schrödinger equation \begin{align} \label{eq:S_eq} i \hbar \frac{\partial}{\partial t} \Psi(t, \vec x_1, \dots, \vec x_n) = - \sum_{i= 1}^n \frac{\hbar ^2}{2 m_i} \Delta_i \Psi(t,\vec x_1, \dots, \vec x_n) + V(\vec x_1, \dots, \vec x_n) \Psi(t, \vec x_1, \dots, \vec x_n)\,. \end{align}

It must be stressed that this is not a classical wave like and EM-waves, as it is not defined on physical space. This creates a lot of confusion.

According to quantum mechanics, the electrons' positions are random with probability density

$$ \rho(\vec x_1, \dots, \vec x_1) = |\Psi(\vec x_1, \dots, \vec x_n)|^2. $$ The waves are not produced by the particle.

As an example, consider the double-slit experiment with a single particle. In this case, it follows from the Schrödinger equation that $\rho(x)$ has the same interference pattern as a classical wave passing through the slit. By doing many double-slit experiments in a row, we can sample the probability density $\rho$, i.e., the interference pattern builds up on the screen.

The similarity between this statistical pattern and the classical wave interference can be regarded as the wavelike behaviour of the particle. On the other hand, the flashes on the screen in each individual run are sharply localized. This part of the phenomenology is rather similar to the behaviour of classical particles. Thus, we have also particle-like behaviour.

However, both the classical wave picture and the classical particle picture are wrong. Instead, we have a new theory called quantum mechanics that describes what is going on.

While the classical wave picture and the classical particle picture are incompatible, there is no such logical inconsistency with the quantum mechanics.