Can you cite experiments where, in some excited states of a hydrogen atom, magnetic moment significantly differs from Bohr's magneton was detected? Correction for magnetic moment of nucleus is insignificant. Only experimental data, not theoretical forecasts. Starting from the experiments of Stern and Gerlach, it seems that only moment of one magneton was detected, I could not find other information. But maybe I'm wrong and didn't search well? For single-electron ions, I would also like to get acquainted with the data of experiments, for example, for He+. There is a lot of information on nuclei on the Internet, but somehow there is no information on hydrogen-like ions.

The magnetic moment of hydrogen depends on the quantum orbital and spin states of the electron. The relevant formula looks something like this: \(\mu = \mu_B (m_l + 2m_s)\) where \(m_l\) and \(m_s\) are the magnetic and spin quantum numbers. In the ground state of hydrogen, \(m_l=0\) and \(m_s\) can be either 1/2 or -1/2. In excited states, there are other possibilities. This ignores the contribution of the nucleus which, as you said, is much smaller. Hydrogen-like ions (i.e. atoms with only one electron) have similar magnetic moments, neglecting the different nuclear contributions of course.

We can find formulas in every serious book. But I never saw results of experimental confirmation. And I suspect "orbital" state has no influence, because electron moves chaotically, for example, both in state 2s and 2p, not like planets move around the Sun. Only in Rydberg states electron cloud can move as a whole unit. So only own electron spin affects in lower states.

The Zeeman effect, showing the splitting of spectral emission lines according to the differing magnetic moments, is very well documented. Look it up.

Only in external magnetic fields. And this effect can be also related to electron's own spin distributed statistically over cloud of definite shape. Wherever the practical use of the magnetic moments of atoms is carried out, only the electrons own spins appear. For example, Wikipedia gives the following rule for calculating the moments of transition metals with a large number of unpaired electrons. Many transition metal complexes are magnetic. The spin-only formula is a good first approximation for high-spin complexes of first-row transition metals. Number of unpaired electrons Spin-only moment (μB) 1 1.73 2 2.83 3 3.87 4 4.90 5 5.92 The relationship is almost linear, although it is obvious that electrons occupy d-orbitals with different "magnetic numbers" M at the same L and N. The type of electron cloud does not affect magnetic phenomena, at least at relatively large distances from the atom. It seems that the images of electrons spinning around a nucleus in books for schoolchildren and students are fiction and are of purely historical interest. Except for "Rydberg atoms," where an entire electron cloud can make coordinated movements. Which is not surprising, since the solutions of the Schrödinger or Pauli equations give the probabilities of finding an electron, respectively, the distribution of charge density and proper magnetic moment (spin), but do not indicate the prevailing direction of velocity at that point. Consequently, the movements are either completely chaotic, with equal probability in either direction, or mutually compensated so that no resulting magnetic moment is formed. For example, if the prevailing direction of velocity coincides with the gradient of the wave function or its square, and since the vector potential would be directed so, and the magnetic field represents its curl, and the curl of any gradient is zero.