Interpretation of QM as discrete derivative

Question

In a couple of letters dated 1923, Pauli writes to Sommerfeld and Lande about Zeeman effect, and he describes (a piece of) quantisation in a peculiar way: the substitution of the "Differentialquotient" $$ {d \over dj} {1 \over j} $$ ($j$ is called a impulsquantenzahlen? is it an angular momentum, isn't it?) by the "Differenzen quotient" $$\frac 1j - \frac 1{j-1}$$ (see page 3 of letter to Sommerfeld).

The later being argued as plausible from the integration of $\int {dj \over j^2}$ (footnote in page 4 of letter to Landé). I wonder, did the concept of "Differenzen quotient" survive in the literature, as a quantisation method or even as some fundational concept of quantum mechanics?

Answer 1

the "difference quotient" (click) which is the straightforward translation of "Differenzen quotient" is simply the ratio of differences whose limit defines the derivative: $$\frac{\Delta F}{\Delta X} = \frac{F(X+\Delta X)-F(X)}{\Delta X}$$ This simple ratio, as all simple mathematical expressions, obviously wasn't invented by Wolfgang Pauli and it appears at many places in physics (although fewer places than derivatives - because fundamental equations of physics depend on continuous variables). In your example, $X\equiv j$ and $\Delta j\equiv 1$.

On the other hand, a speculative partially hand-written letter from 1923, years before quantum mechanics was understood, is no longer relevant to understand quantum mechanics. Bohr's old model of the hydrogen atom has been superseded by quantum mechanics and this letter is even more vague, imprecise, and ephemeral than Bohr's old model.

Of course that if $j$ is discrete, one cannot differentiate with respect to $j$ - at least not without an analytical continuation. The difference quotient would be the closest thing to differentiation one could find. On the other hand, there is usually no good reason to differentiate with respect to $j$, so the previous two sentences are largely inconsequential.

Best wishes Luboš