how quantum-mechanical particles react in the potential?

Question

I am reading some materials on quantum mechanics. I am a bit confusing in the chapter on wave-particle duality and following questions arise

In classical mechanics, the force a particle experience is the gradient of the potential. If a particle is in a potential of form $\sin(x)$ where $x$ is the spatial variable, the gradient of the potential is still of sinusoidal form so the particle is experiencing a force back and forward like pendulum, right? So if we consider the particle quantum mechanically, the potential is still of the form $\sin(x)$ so what can we tell about the force that particle experiencing?

In a online material, when talking about the zero-point energy, it said if the temperature is extremely low (close to absolute zero kelvin), all bosonic atoms are staying in the lowest state, so it just like a big atom with wave function in the form of plane wave. If that's true, I am wondering how that "plane wave" will react to the potential of $\sin(x)$, will it see a net force or zero force? Why?

Answer 1

In QM we tend to do away with the concept of "forces" altogether. Instead, a particle is attributed a wavefunction $\Psi (x,t)$. The modulus of the wavefunction $|\Psi|^2$ is a probability distribution which tells you the probability of finding the particle in a particular location. The wavefunction itself behaves according to the Schrodinger Equation:

$$i\hbar\frac{\partial \Psi}{\partial t}= -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + V\Psi$$

where $V=V(x,t)$ is the potential function. For periodic potentials like $V=sinx$ you get a class of wavefunctions called Bloch Waves.

I'm not really sure what you're asking in your second question.