A useful distinction is:

  • Product state: We can assign each particle its own wavefunction and multiply them to get the joint amplitude.
  • Entangled state: Only the combined system has a complete wavefunction. There is generally no separate pure wavefunction for each particle that can be multiplied to recover it.

explain what it means to multiple the wavefunctions of two separate particles that are interacting in some manner

For two distinguishable particles, “multiplying their wavefunctions” usually means building a joint wavefunction from two independent states:\[ \Psi(x_1,x_2)=\psi_1(x_1)\psi_2(x_2). \]

Here \(x_1\) is particle 1’s position and \(x_2\) is particle 2’s position. The product is an amplitude for the joint possibility that particle 1 is at \(x_1\) and particle 2 is at \(x_2\). Its squared magnitude gives the joint probability density:\[ |\Psi(x_1,x_2)|^2 =|\psi_1(x_1)|^2|\psi_2(x_2)|^2. \]

So this multiplication describes a state in which the particles’ position probabilities are independent. It is often written more precisely as a tensor product, \(|\psi_1\rangle\otimes|\psi_2\rangle\). We are not multiplying two waves at the same position; the full wavefunction has two position arguments.

Interaction changes the picture. Suppose the particles exert forces on each other. Their Hamiltonian contains an interaction term \(V(x_1,x_2)\):\[ H=H_1+H_2+V(x_1,x_2). \]

Even if the state starts as a product, the interaction generally evolves it into a joint wavefunction that cannot be written as one wavefunction for particle 1 times another for particle 2:\[ \Psi(x_1,x_2)\ne\psi_1(x_1)\psi_2(x_2). \]

That is entanglement. The amplitude for finding particle 1 at one position now depends on what position we consider for particle 2. For example, if two particles attract, configurations in which they are near each other may have greater probability than a product of two independent position distributions would predict.

A useful distinction is:

  • Product state: We can assign each particle its own wavefunction and multiply them to get the joint amplitude.
  • Entangled state: Only the combined system has a complete wavefunction. There is generally no separate pure wavefunction for each particle that can be multiplied to recover it.

There is one further wrinkle for identical particles. Quantum mechanics does not let us physically distinguish “particle 1 at \(x_1\)” from “particle 2 at \(x_1\).” Their joint wavefunction must be symmetric for bosons or antisymmetric for fermions. Even without a force between them, a simple labeled product \(\psi_a(x_1)\psi_b(x_2)\) is therefore generally not the complete physical state.

In short, multiplying two wavefunctions is a way to describe independent amplitudes within one joint system. Once an interaction creates correlations, the joint wavefunction usually contains more information than that product can express.