Does Quantum Entanglement happen between two wavefunctions?

Question

Does the entanglement happen between two particles or two wavefunctions? If it's wavefunction then what happens to the two wavefuntions after getting entangled?

Answer 1

Any physical system has a wavefunction that describes the whole system. So even when we have two particles that are not entangled the two of them are described by some single wavefunction $\Psi$. However when we say the particles are not entangled we mean the total wavefunction can be separated into different parts for each particle. So we can write:

$$ \Psi = \psi_a \psi_b \tag{1} $$

where $\psi_a$ describes just particle $A$ and $\psi_b$ describes just particle $B$. For convenience we often refer to $\psi_a$ as the wavefunction of particle $A$, and likewise for $B$, but strictly speaking $\psi_a$ and $\psi_b$ are just parts of the wavefunction for the system.

When the two particles become entangled the means the total wavefunction $\Psi$ can no longer be separated into an $A$ part and a $B$ part, so we can no longer write down an equation like equation (1). In effect the two particles have become mixed up and can no longer be distinguished from each other.

Answer 2

When two particles have quantum entanglement, they have a common wavefunction.

The wavefunction is the probability distribution of the particles' position in all of space.

When the two particles are entangled, they common wavefunction describes both of their properties.

That is why as long as the entanglement is there, they can be described by one single wavefunction, and if you check the properties of one particle, you will have checked the common wavefunction, what is the wavefunction of the other particle too. that is why you have affected the properties of the other particle too.

It is a common misconception that at the moment of checking the common wavefunction, you collapse it. Nothing is collapsing. You just affect the common wavefunction itself by checking it, so it will have an effect on the other particle's properties too.