Assuming the electrons to revolve round the nucleus in circular orbits and using the fact from classical electromagnetism that a current loop behaves like a magnetic dipole of dipole moment $\vec{\mu}_L=\int I \vec{dA}$ where $\int \vec{dA}$ is the area of the loop and $I$ is the current through the loop one can relate the magnetic moment $\vec{\mu}_L$ with the classical orbital angular momentum $\vec{L}=\vec{r}\times\vec{p}$ as $$\hspace{5.5cm}\vec{\mu}_L=\frac{e}{2m_e}\vec{L}\hspace{5.5cm}(1)$$ where the symbols have their obvious meaning. What I find uncomfortable is that this derivation is entirely classical. This is because the electrons don't really move in trajectories but are described by wavefunctions and also, in quantum mechanics $\vec{\mu}_L$ and $\vec{L}$ are both operators. We promptly use Eq.1 as a valid operator relation in quantum mechanics: $$\hspace{5.5cm}\hat{\vec{\mu}}_L=\frac{e\hbar}{2m_e}\frac{\hat{\vec{L}}}{\hbar}\hspace{5.5cm}(2)$$

Q1. But why? Why do we expect the quantum mechanical operators $\hat{\vec{\mu}}_L$ and $\hat{\vec{L}}$ to be related? Is Eq.2 a postulate new or is there something simple I am missing?

Q2. Can we take any classical relation, promote the dynamical variables to operators and demand that the relation will hold quantum mechanically?

Q3. Is there a way of deriving/reproducing Eq.2 by rigorous means from the formalism of quantum mechanics?

  • $\begingroup$ en.wikipedia.org/wiki/Correspondence_principle $\endgroup$ – G. Smith Feb 25 '19 at 6:27
  • $\begingroup$ This is no new postulate. You always do this. The postulate of QM is that you're supposed to replace the dynamical observables by self-adjoint operators. This is what you do, for example, when you translate $\textbf{L}=\textbf{r}\times\textbf{p}$ to $\hat{\textbf{L}}=\hat{\textbf{r}}\times\hat{\textbf{p}}$. @mithusengupta123 $\endgroup$ – SRS Feb 25 '19 at 6:34
  • $\begingroup$ can we take any classical relation between dynamical variables, promote them to operators and such a relation will be valid in quantum mechanics at the operator level? $\endgroup$ – mithusengupta123 Feb 25 '19 at 6:46
  • $\begingroup$ Yes. Unless there is a commutativity problem. Since $L_i=i\hbar \epsilon_{ijk}x_jp_k$ and wo different components of $x_j$ and $p_k$ that appear in the definition of $L_i$ commute, there is no ordering ambiguity! Such an ordering ambiguity exists in defining the quantum mechanical counterpart of the Runge-Lenz vector or radial component of liner momentum $p_r$. In these cases, the quantum operator definitions is somewhat different from classical definitions. $\endgroup$ – SRS Feb 25 '19 at 7:08
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    $\begingroup$ Does physics.stackexchange.com/a/138400/37496 help assuage your concerns? $\endgroup$ – d_b Feb 25 '19 at 9:17

Quantum operators have to be related in the same way that classical quantities are, in order for the Correspondence Principle to hold. Otherwise, expectation values of the operators would not obey classical physics in the limit of large quantum numbers.


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