Angular Momentum Quantum Numbers

What is the meaning of the six quantum numbers $L$, $S$, $J$, $M_L$, $M_S$, and $M_J$?

The ``old quantum'', Bohrish-Sommerfeldian notion of quantum numbers ran like this:

Certain classical quantities are constrained, for a stationary quantum state, to take on only integer values. These values are called the quantum numbers of the state.

Generalizing this just a bit, we allow any discrete set of real numbers in place of the integers (e.g., the set $\{\sqrt{j(j+1)}\}$ as $j$ ranges over the integers; or more significantly, as $j$ ranges over the set of integers and half-integers).

Dressing this up in mathematical language, we have a space (actually a differential manifold) $\Sigma$ that represents the set of all classical states. Classical quantities are continuous real-valued functions on $\Sigma$. Bohr and Sommerfeld assumed that nature really permits only a subset $\Sigma_0$ to occur as stationary states. They assumed further, for certain classical $f:\Sigma \rightarrow {\bf R}$, that the image $f(\Sigma_0)$ was a discrete subset of R.

The ``new quantum'' notion of quantum numbers (à la Heisenberg, Schrödinger, and Dirac) runs like this:

Classical quantities correspond to Hermitian operators on a complex Hilbert space. Quantum states are specified by elements of the space. If a Hermitian operator $Q$ has a discrete spectrum, and $v$ is an eigenvector of $Q$ with eigenvalue $q$, then we say that $q$ is a quantum number of the state specified by $v$.

I will start with the ``old quantum'' explanation of our six quantum numbers. (The ``new quantum'' version will have to wait till we discuss the Paschen-Back effect.) Consider first the hydrogen atom. The electron possesses orbital angular momentum, given by a vector⁸ l. Since the electron is spinning, it has also spin angular momentum, given by a vector s. The total angular momentum of the system⁹ is then given by the vector ${\bf j} = {\bf l} + {\bf s}$.

For a multi-electron atom, we let L be the sum of the l's over all the electrons; S is likewise the sum of all the s's; and ${\bf J} = {\bf L} + {\bf S}$.

We can now say what $J$ is: it's the quantum number associated with the magnitude of J (likewise for $L$ and $S$). If the term ``associated'' seems too vague, here is the exact relationship: the magnitude of J is $\sqrt{J(J+1)}$.

We pick an arbitrary direction in space and call it the $z$-axis. $M_J$ is the quantum number for the component of J along the $z$-axis. In other words, if ${\bf J} = (J_x,J_y,J_z)$ in some coordinate system, then $M_J$ is the quantum number for $J_z$. Likewise for $M_L$ and $M_S$. Note that $M_J=M_L+M_S$, but we have no such simple relation for $J$, $L$, and $S$; the magnitude of J depends of course on the angle between L and S as well as their magnitudes.

With our six angular momentum quantum numbers, we have enough ammunition to explain the Zeeman splitting (as we will see later). You may be wondering what happened to the quantum number $n$. Our six quantum numbers are not a complete set-- that is, they are insufficient to fully specify the quantum state of the atom (or label the set of basis vectors, in modern terminology). But they are sufficient to explain the Zeeman effect.

Sommerfeld (and Bohr) knew nothing of spin, of course-- that had to wait for Goudsmit and Uhlenbeck. Even so, Sommerfeld had concluded that he needed to augment his arsenal of quantum numbers, just from poring over spectra. Bohr and Sommerfeld supposed that in multi-electron atoms, the ``inner core'' of electrons interacted in some complex way with the outer electrons. That justified the introduction of more quantum numbers. (For historical accuracy, I note that Sommerfeld's set of quantum numbers differed from the ``modern'' set ($J$, $S$, etc.). The ``modern'' set is equivalent to Sommerfeld's, but conceptually clearer.)

Sommerfeld turned the problem of the anomalous Zeeman effect over to young Heisenberg, a student in his seminar. Heisenberg, struggling with it, introduced half-integer quantum numbers into the old quantum theory for the first time. Sommerfeld was shocked, and urged Heisenberg not to publish¹⁰. ``If we know one thing about quantum numbers, we know they are integers!'' Bohr, too, expressed displeasure. Pauli warned that today we allow half-integers, tomorrow it's quarter-integers, then eighths, sixteenths, and before you know it the quantum conditions have eroded away.

© 2001 Michael Weiss

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