User:Patrick0Moran/Quantum notes

What Heisenberg probably had in mind, is calculating the classical Poynting vector from the expressions for the electric and magnetic fields on p. 880 of the article. However, he restricted himself in the article to one-dimensional examples, so these expressions are not applicable as they stand. The only thing he took from it, is that the fields will depend on the position $\scriptstyle x(t)$ of the electron, this variable being considered as a classical quantity, restricted by quantum conditions of the Old quantum theory. He used for the harmonic oscillator the Fourier expansion

$x(t)=\sum _{n}c_{n}e^{in\omega t},$

in which the quantum condition $\scriptstyle E_{n}=n\omega$ is taken into account. The crucial difference between classical and quantum is that classically the frequency of the field is equal to the rotation frequency of the electron, whereas in quantum theory the Bohr rule is satisfied, equating the frequency of the field with the difference of the rotation frequencies.

Now in the classical theory the amplitude of the field would be proportional to $\scriptstyle x(t)$ and its intensity proportional to $\scriptstyle x(t)^{2}$. Then

$x(t)^{2}=\sum _{nm}c_{n}c_{m}e^{i(n+m)\omega t}.$

For some fixed N in this expression the contribution at frequency $\scriptstyle N\omega$ is

$\sum _{n}c_{n}c_{N-n}e^{iN\omega t}.$

According to Heisenberg this does not work properly, as it does not take into account the Bohr rule (the quantumtheoretical expressions on p. 881) given here as

$\omega (n,m)+\omega (m,n')=\omega (n,n').$

For the harmonic oscillator we have $\scriptstyle \omega (n,m)=(n-m)\omega .$ Heisenberg replaced in the Fourier series the frequencies $\scriptstyle n\omega$ by the Bohr frequencies $\scriptstyle \omega (n,m)$ corresponding to a transition between orbits n and m. Then eq. (7) is directly obtained as the component of $\scriptstyle x(t)^{2}$ at the difference frequency $\scriptstyle \omega (n,m)$:

$\sum _{m'}c_{nm'}c_{m'm}e^{i[(n-m')+(m'-m)]\omega t}=\sum _{m'}c_{nm'}c_{m'm}e^{i(n-m)\omega t}.$

Here matrix multiplication is evident since the component of $\scriptstyle x(t)^{2}$ at frequency $\scriptstyle (n-m)\omega$ (in this one-dimensional example proportional to the intensity of the field at frequency $\scriptstyle (n-m)\omega$) is proportional to $\scriptstyle \sum _{m'}c_{nm'}c_{m'm}.$

Note that this is not quantum mechanics but Old quantum theory. Here no quantum mechanical transition probabilities are calculated, but just field intensities. Note also that eq. (7) is not derived, but guessed by varying the classical theory so as to comply with the Bohr rule.