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In Shankar's QM book Chapter 10 pg. 274, it was said that quantum mechanically, the separability of the hamiltonian $$H=H_1(x_1, p_1)+H_2(x_2,p_2)$$ leads to the factorization of the wave function: $$\psi(x_1,x_2)=\psi_1(x_1)\psi_2(x_2).$$

How can this be shown to be true?

I know that for a separable Hamiltonian, the energy eigenvalue equation $$(\hat{H_1}+\hat{H_2})\psi_E(x_1,x_2)=E\psi_E(x_1,x_2)$$ gives energy solutions which are separable: $$\psi_E(x_1,x_2)=\psi_{E_1}(x_1)\psi_{E_2}(x_2).$$

The general wavefunction will hence be a superposition of the separable energy eigenfunctions $\psi_E$:

$$\psi(x_1,x_2)=\sum_{E_1+E_2=E}\psi_{E_1}(x_1)\psi_{E_2}(x_2)$$

How do I proceed to show that the above expression can be written as

$$\sum_{E_1+E_2=E}\psi_{E_1}(x_1)\psi_{E_2}(x_2)=\psi_1(x_1)\psi_2(x_2)?$$

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  • $\begingroup$ Does this answer your question? Separable Hamiltonian systems in quantum mechanics $\endgroup$
    – Niall
    Jun 23, 2022 at 9:55
  • $\begingroup$ The answers under that post do not show that the eigenstates are factorizable, only that they are common eigenstates of $H_1(x_1,p_1)$ and $H_2(x_2,p_2)$ $\endgroup$ Jun 23, 2022 at 9:59
  • $\begingroup$ I think Shankar was being vague and what he really meant was that the energy eigenfunctions are separable. In a few lines later, he wrote that symmetrical states like $|a\rangle \otimes |b\rangle + |b\rangle \otimes |a\rangle$ in general fails to be factorized. $\endgroup$
    – TaeNyFan
    Jun 23, 2022 at 11:01

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I think your interpretation of Shankar's statement is too strong and that cannot be true. Shankar's statement is that separable states make up a complete eigenbasis of the Hamiltonian $H$. Your interpretation is that every eigenstate of $H$ is a separable state. The two statements are equivalent only if the spectrum of $H$ is non-degenerate. Otherwise, you are free to add up two separable eigenstates with the same energy to make up an entangled eigenstate, which cannot be separable in any way. (Quantum entanglement is a basis-independent statement. There is no way to separate an entangled state)

Actually a simpler way to see this is to consider the trivial case $H_1=H_2=0$. Then any state, either entangled or unentangled, is an eigenstate. However, it is still possible to find a complete basis of $H=0$ which only consists of unentangled state (that is, the total Hilbert space is a tensor product of the two Hilbert spaces).

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