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This may include some quantum physics.

Are there anyways to describe the wave function of Hydrogen's orbital (or that of a Hydrogen-like atom) in terms of the wave function of each orbital? You can write wavefunction as $|\psi\rangle=\sum{c_i|\Phi_i\rangle}$ where $|\Phi_i\rangle$ are each possible states. Suppose that the electron is in a pure state and orbitals are independent states, how can we find coefficient or probability to find the electron in each orbital?

You may consider certain parameters such as temperature in the calculation.

Thank you in advance.

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    $\begingroup$ Orbitals are the pure states. There is nothing to calculate. $\endgroup$ Jun 23 at 7:42
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    $\begingroup$ You can choose the $\Phi_i$’s to be the orbitals, then the probabilities are trivially $|c_i|^2$. $\endgroup$
    – orthocresol
    Jun 23 at 7:44
  • $\begingroup$ Yes, each orbital is the pure state, as well as the electron's superposition between states. But I want to separate the electron's wave function into a superposition of each orbital i.e. solve for $c_i$. $\endgroup$ Jun 23 at 7:46
  • $\begingroup$ @HanchaiNonprasart The inner product of the hydrogen wavefunction with an orbital yields the coefficient, $c_i = \langle \Phi_i|\Psi\rangle$. $\endgroup$
    – Hans Wurst
    Jun 25 at 11:47
  • $\begingroup$ Thanks for your answers and sorry for the confusing question so let make it clear. Can we assume that the electron of hydrogen is always in $1s$ orbital? If not, can we predict the probability of it without knowing its wavefunction beforehand? Or more specifically can we predict coefficient $c_i$ and the wavefunction from the physical property of hydrogen sample such as it's average temperature? $\endgroup$ Jun 26 at 12:27

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