How to distinguish two different systems which have the same Hamiltonian in the Schrodinger equation?

Suppose we have 4 hydrogen atoms and 2 oxygen atoms. If we write the Hamiltonian containing all the possible interactions for the Schrodinger equation, how can we distinguish the system is two interacting water molecules or 2 hydrogen and an oxygen molecule?

• You seem to be asking two different questions in your title and in the body of your question. – probably_someone Oct 20 '18 at 13:19
• I changed the title. – MOON Oct 20 '18 at 13:24

One way to do so would be by looking at spatial correlations for each pair of atoms. If the positions of hydrogen and oxygen atoms are often correlated, then you likely have water; in contrast, if there is little or no correlation between hydrogen and oxygen atoms, then you either have separate molecules of hydrogen and oxygen or you have a bunch of isolated atoms. To distinguish between the latter two cases, look at correlations between the positions of oxygen atoms and the positions of other oxygen atoms (and likewise for hydrogen atoms). If a high degree of correlation exists in the positions of particular pairs of oxygen atoms, it's likely that those oxygen atoms are in an oxygen molecule (and likewise for hydrogen atoms). If not, then that particular pair have not formed a molecule.

• So at the end the Hamiltonian of the two systems I mentioned is different, right? One of the Hamiltonians include the correlation between the position of Hydrogen and Oxygen atoms while the other does not include such correlations. – MOON Oct 20 '18 at 14:47
• @MOON No, the Hamiltonian will be the same. What's different is the actual state (wavefunction) of the system. The various correlations are numbers extracted from the wavefunction, not the Hamiltonian. – probably_someone Oct 20 '18 at 14:49
• So there would be extra constraints. For example if the hydrogen's position is not correlated with that of oxygen, the we have the constraint that the expectation value of <r_h*r_o>-<r_h><r_o> should be zero during solving the equations numerically. – MOON Oct 20 '18 at 14:58
• @MOON Constraints on what? Looking at correlations will give you an idea of whether the state you currently have contains molecules, and, if it does, whether they are hydrogen/oxygen or water molecules. – probably_someone Oct 20 '18 at 15:14

I would argue that a quantum mechanical system is defined by its Hamiltonian and the underlying Hilbert space. Therefore, two systems which exist in the same Hilbert space and which have the same Hamiltonian are, by definition, the same system.

Case in point, your example. The configurations you speak of (two H$$_2$$O molecules vs. two $$H_2$$ molecules and one $$O_2$$ molecule) are in fact two separate states$$^\dagger$$ lying in the same (extremely complicated) Hilbert space.

$$^\dagger$$Of course these are not individual states, but large collections of similar states which are identified with each other by virtue of the bond structure which exists between different atoms. For example, two atoms can be considered to be bonded together if one or more of their valence electrons exist in bonding orbitals - see here for more.