From S.M. Blinder's Introduction to QM book, p. 116:
In seeking an approximation to the ground state, we might first work out the solution in the absence of the $1/r_{12}$ term. In the Schrodinger equation thus simplified, we can separate the variables $r_{1}$ and $r_{2}$ to reduce the equation to two independent hydrogen like problems. The ground state wavefunction (not normalized) for this hypothetical helium atom would be:
$$\psi(r_1, r_2) = \psi_{1s}(r_1)\psi_{1s}(r_2) = e^{−Z(r_1 + r_2)}$$
Why is it only the product and not some linear combination of the two wavefunctions? I heard somewhere that it has something to do with "tensor product". Can someone provide a detailed explanation about this?
Reference:
- Blinder, S. M. Introduction to Quantum Mechanics: in Chemistry, Materials Science, and Biology; Elsevier, 2012,. ISBN 978-0-08-048928-5.
NB: This question has also been asked on Chemistry Stack Exchange: Why is the electronic wavefunction of helium a product of the two 1s wavefunctions when ignoring electron-electron repulsion?
Best Answer
QMechanic gave a hint to the answer in the comments section. It is usually taken as a separate axiom of QM that if a quantum system A has the identifiable subsystems B and C (chosen two for simplicity, the argument can be easily extended to an arbitrary but finite number of subsystems), then the Hilbert space of A is the tensor product of the Hilbert spaces of B and C. This entails that:
$$\Psi_A = \psi_B \otimes \psi_C$$ at the level of normalizable (pure) quantum states and
$$H_A = H_B \otimes \hat{1}_C + \hat{1}_B \otimes H_C $$ at the level of Hamiltonians.
This description is consistent and leads to experimentally verifiable predictions for any multiparticle system (the simplest would be a Hydrogen atom). This axiom is amended for subsystems made of identical elements (for example the two electrons in the three-particle Helium atom) case in which the states and operators are multiplied or acted on by symmetrization or antisymmetrization operators.
The answer by Emilio Pisanty should be read after mine.