# Which Hamiltonians turn disentangled states into entangled ones?

I have been solving example oral-exam questions about quantum mechanics, and I am stuck on the last question:

"Describe the spin space for $$N$$ particles of spin 1/2. Give an example of an entangled spin state. Which Hamiltonians turn disentangled states into entangled ones?"

My guess is: If the potential is of the Form $$V(x,y)\neq V_1(x)V_2(y)$$, where x and y are variables of the each state, then the corresponding Hamiltonian would suffice to entangle these two states. Is it right? How can I prove it?

• Doesn't look right because spin states don't depend on $x$ in physical space. Do you know what is the Hilbert space describing spin up/spin down states ? Hint: look up the Pauli matrices and/or what is nowadays called a qubit. Mar 10, 2022 at 9:48

Consider for instance the separable two-spin state $$\vert +\rangle\vert +\rangle$$. Then the Hamiltonian $$H= \omega L_x=\omega (L_++L_- )$$ will take $$\vert +\rangle\vert +\rangle$$ to $$H\vert +\rangle\vert +\rangle = \vert -\rangle \vert +\rangle + \vert +\rangle\vert -\rangle$$ which is not separable.