# Why do we think that the relation $\vec{\mu}_L=\frac{e}{2m_e}\vec{L}$ will be valid in quantum mechanics?

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?

• en.wikipedia.org/wiki/Correspondence_principle – G. Smith Feb 25 at 6:27
• 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 – SRS Feb 25 at 6:34
• 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? – mithusengupta123 Feb 25 at 6:46
• 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. – SRS Feb 25 at 7:08
• Does physics.stackexchange.com/a/138400/37496 help assuage your concerns? – d_b Feb 25 at 9:17