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The first is about the general solution of the hydrogen atom schrödinger-equation: We learned in our quantum mechanics class that the general solution of every quantum system is a linear combination of all eigenstates (or eqivelant sets of basis functions). So a set of basis functions of the hydrogen atom are the hydrogen wave functions and the general solution of the hydrogen problem should therefore be a linear combination of this states. But every time I get in contact with atomic physics there is stated that the electrons are in the 1s, 2s, 2p,... state corresponding to the basis set of the hydrogen wave functions. But why do the electrons occupy the ‚atomic orbitals‘ and not a state, wich is a linear combination of all atomic orbitals. If an electron occupies a state wich is a linear combination, would this also mean that the energy of this electron is arbitrary (because I can add toghether atomic orbitals with different energys) and how would quantum numbers be defined in such a case (so that no two electrons have the same)?

The second question is related to the expectation value of energy: If we take the stationary schrödinger-equation

H|Φ>=E|Φ>

we can calculate the energy E of an eigenstate |Φ> by<Φ|H|Φ>=<Φ|E|Φ>=E

where on the left is a expectation value <Φ|H|Φ>. Does this mean that if we measure the ground state energy of e.g. hydrogen, we do only on average get the typical ionisation energy of 13.6 eV?I appreciate every help, thank you!