1
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ 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. $\endgroup$
    – Kurt G.
    Mar 10, 2022 at 9:48

1 Answer 1

1
$\begingroup$

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.

$\endgroup$
5
  • 2
    $\begingroup$ This is a very partial answer to the classification question that was asked. But it is an interesting proof that the complete answer is interesting and nontrivial. $\endgroup$ Mar 10, 2022 at 20:13
  • $\begingroup$ @EmilioPisanty the question doesn't read to me like a classification problem. This is indeed a simple example and of course in general separable there's no reason to believe that states will evolve to separable states: in fact in the usual "object/apparatus" framework, the time evolution entangles the initially separable state of the object and the apparatus. $\endgroup$ Mar 10, 2022 at 21:05
  • 2
    $\begingroup$ @. Zero the title definitely reads like a classification problem to me. But I guess YMMV. No biggie. $\endgroup$ Mar 10, 2022 at 21:21
  • $\begingroup$ @EmilioPisanty I guess it’s a case where the body of the question reads a bit differently from the title. As you say: no big deal. $\endgroup$ Mar 10, 2022 at 22:31
  • 1
    $\begingroup$ I agree with @EmilioPisanty, it says "Which Hamiltonians" not e.g. "give an example". But I think part of ZeroTheHero's comment already partially answeres the question. Namely, when you consider an object and an apparatus. There you start with an disentangled wave function of the object and the apparatus. The total wave function then evolves according to the Schrödinger equation and at the end of the experiment you have some entangled wave function. This happens if the Schrödinger equation has interaction terms. So I would say the answer is: Hamiltonians with interaction terms. $\endgroup$
    – Tera
    Mar 14, 2022 at 11:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.