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There are a lot of papers on optical rotation which cite Rosenfeld's (German) 1928 paper "Quantum mechanical theory of natural optical rotation..." [Quantenmechanische Theorie der naturlichen optischen Aktivitat von Flussigkeiten und Gasen, Zeitschrift für Physik, Volume 52, Issue 3-4, pp. 161-174] as background. Here is a paid link--

Apologies that I cannot link to the paper. I have a paper copy and can't find a paywall-free version.

The symbols below are: $\lambda$--wavelength; $c$ is the speed of light; $\nu$ is frequency; $n$ is the index of refraction; $\rho$ is a quantity that does not change between equations (42) and (43).

The last two equations in that paper (42), (43), are a conclusion about an angle of rotation $\theta:$

$$\Delta n = \frac {2\pi ~\rho}{\lambda~ n} \tag{42}$$

and

$$\theta = \frac {2\pi \nu~\cdot \Delta n}{c~\cdot ~ 2} = \frac{2 \pi^2\rho}{\lambda^2} \tag{43}$$

My question is pretty dumb. If we insert the expression for $\Delta n$ into the equality in (43) we get:

$$\frac{2\pi\nu}{c}\cdot \frac{2\pi \rho}{2\lambda~ n} = \frac{2\pi^2\rho}{\lambda^2}\to \frac{\nu}{c~n}=\frac{1}{\lambda}\to\lambda =\frac{cn}{\nu} $$

We know already--and Rosenfeld mentions this a few lines up and at eq. (21)--that

$$\lambda = \frac{c}{\nu~n}$$

which forces $n$ to be $1$. In general $n \neq 1$.

Can someone tell me what, if anything, has gone awry here?

Edit: After looking at several versions of Rosenfeld's equation I see $\theta$ is proportional to the trace of the rotation matrix...but the constant of prop. can take different forms.

There are a lot of papers on optical rotation which cite Rosenfeld's (German) 1928 paper "Quantum mechanical theory of natural optical rotation..." [Quantenmechanische Theorie der naturlichen optischen Aktivitat von Flussigkeiten und Gasen, Zeitschrift für Physik, Volume 52, Issue 3-4, pp. 161-174] as background. Here is a paid link--

Apologies that I cannot link to the paper. I have a paper copy and can't find a paywall-free version.

The symbols below are: $\lambda$--wavelength; $c$ is the speed of light; $\nu$ is frequency; $n$ is the index of refraction; $\rho$ is a quantity that does not change between equations (42) and (43).

The last two equations in that paper (42), (43), are a conclusion about an angle of rotation $\theta:$

$$\Delta n = \frac {2\pi ~\rho}{\lambda~ n} \tag{42}$$

and

$$\theta = \frac {2\pi \nu~\cdot \Delta n}{c~\cdot ~ 2} = \frac{2 \pi^2\rho}{\lambda^2} \tag{43}$$

My question is pretty dumb. If we insert the expression for $\Delta n$ into the equality in (43) we get:

$$\frac{2\pi\nu}{c}\cdot \frac{2\pi \rho}{2\lambda~ n} = \frac{2\pi^2\rho}{\lambda^2}\to \frac{\nu}{c~n}=\frac{1}{\lambda}\to\lambda =\frac{cn}{\nu} $$

We know already--and Rosenfeld mentions this a few lines up and at eq. (21)--that

$$\lambda = \frac{c}{\nu~n}$$

which forces $n$ to be $1$. In general $n \neq 1$.

Can someone tell me what, if anything, has gone awry here?

Edit: After looking at several versions of Rosenfeld's equation I see $\theta$ is proportional to the trace of the rotation matrix...the constant of prop. can take various forms and is not the focus of this approach.

There are a lot of papers on optical rotation which cite Rosenfeld's (German) 1928 paper "Quantum mechanical theory of natural optical rotation..." [Quantenmechanische Theorie der naturlichen optischen Aktivitat von Flussigkeiten und Gasen, Zeitschrift für Physik, Volume 52, Issue 3-4, pp. 161-174] as background. Here is a paid link--

Apologies that I cannot link to the paper. I have a paper copy and can't find a paywall-free version.

The symbols below are: $\lambda$--wavelength; $c$ is the speed of light; $\nu$ is frequency; $n$ is the index of refraction; $\rho$ is a quantity that does not change between equations (42) and (43).

The last two equations in that paper (42), (43), are a conclusion about an angle of rotation $\theta:$

$$\Delta n = \frac {2\pi ~\rho}{\lambda~ n} \tag{42}$$

and

$$\theta = \frac {2\pi \nu~\cdot \Delta n}{c~\cdot ~ 2} = \frac{2 \pi^2\rho}{\lambda^2} \tag{43}$$

My question is pretty dumb. If we insert the expression for $\Delta n$ into the equality in (43) we get:

$$\frac{2\pi\nu}{c}\cdot \frac{2\pi \rho}{2\lambda~ n} = \frac{2\pi^2\rho}{\lambda^2}\implies \frac{\nu}{c~n}=\frac{1}{\lambda}\implies\lambda =\frac{cn}{\nu} $$

We know already--and Rosenfeld mentions this a few lines up and at eq. (21)--that

$$\lambda = \frac{c}{\nu~n}$$

which forces $n$ to be $1$. In general $n \neq 1$.

Can someone tell me what, if anything, has gone awry here?

Edit: After looking at several versions of Rosenfeld's equation I see $\theta$ is proportional to the trace of the rotation matrix...but the constant of prop. can take different forms.

There are a lot of papers on optical rotation which cite Rosenfeld's (German) 1928 paper "Quantum mechanical theory of natural optical rotation..." [Quantenmechanische Theorie der naturlichen optischen Aktivitat von Flussigkeiten und Gasen, Zeitschrift für Physik, Volume 52, Issue 3-4, pp. 161-174] as background. Here is a paid link--

Apologies that I cannot link to the paper. I have a paper copy and can't find a paywall-free version.

The symbols below are: $\lambda$--wavelength; $c$ is the speed of light; $\nu$ is frequency; $n$ is the index of refraction; $\rho$ is a quantity that does not change between equations (42) and (43).

The last two equations in that paper (42), (43), are a conclusion about an angle of rotation $\theta:$

$$\Delta n = \frac {2\pi ~\rho}{\lambda~ n} \tag{42}$$

and

$$\theta = \frac {2\pi \nu~\cdot \Delta n}{c~\cdot ~ 2} = \frac{2 \pi^2\rho}{\lambda^2} \tag{43}$$

My question is pretty dumb. If we insert the expression for $\Delta n$ into the equality in (43) we get:

$$\frac{2\pi\nu}{c}\cdot \frac{2\pi \rho}{2\lambda~ n} = \frac{2\pi^2\rho}{\lambda^2}\to \frac{\nu}{c~n}=\frac{1}{\lambda}\to\lambda =\frac{cn}{\nu} $$

We know already--and Rosenfeld mentions this a few lines up and at eq. (21)--that

$$\lambda = \frac{c}{\nu~n}$$

which forces $n$ to be $1$. In general $n \neq 1$.

Can someone tell me what, if anything, has gone awry here?

Edit: After looking at several versions of Rosenfeld's equation I see $\theta$ is proportional to the trace of the rotation matrix...but the constant of prop. can take different forms.

There are a lot of papers on optical rotation which cite Rosenfeld's (German) 1928 paper "Quantum mechanical theory of natural optical rotation..." [Quantenmechanische Theorie der naturlichen optischen Aktivitat von Flussigkeiten und Gasen, Zeitschrift für Physik, Volume 52, Issue 3-4, pp. 161-174] as background. Here is a paid link--

Apologies that I cannot link to the paper. I have a paper copy and can't find a paywall-free version.

The symbols below are: $\lambda$--wavelength; $c$ is the speed of light; $\nu$ is frequency; $n$ is the index of refraction; $\rho$ is a quantity that does not change between equations (42) and (43).

The last two equations in that paper (42), (43), are a conclusion about an angle of rotation $\theta:$

$$\Delta n = \frac {2\pi ~\rho}{\lambda~ n} \tag{42}$$

and

$$\theta = \frac {2\pi \nu~\cdot \Delta n}{c~\cdot ~ 2} = \frac{2 \pi^2\rho}{\lambda^2} \tag{43}$$

My question is pretty dumb. If we insert the expression for $\Delta n$ into the equality in (43) we get:

$$\frac{2\pi\nu}{c}\cdot \frac{2\pi \rho}{2\lambda~ n} = \frac{2\pi^2\rho}{\lambda^2}\implies \frac{\nu}{c~n}=\frac{1}{\lambda}\implies\lambda =\frac{cn}{\nu} $$

We know already--and Rosenfeld mentions this a few lines up and at eq. (21)--that

$$\lambda = \frac{c}{\nu~n}$$

which forces $n$ to be $1$. In general $n \neq 1$.

Can someone tell me what, if anything, has gone awry here?

There are a lot of papers on optical rotation which cite Rosenfeld's (German) 1928 paper "Quantum mechanical theory of natural optical rotation..." [Quantenmechanische Theorie der naturlichen optischen Aktivitat von Flussigkeiten und Gasen, Zeitschrift für Physik, Volume 52, Issue 3-4, pp. 161-174] as background. Here is a paid link--

Apologies that I cannot link to the paper. I have a paper copy and can't find a paywall-free version.

The symbols below are: $\lambda$--wavelength; $c$ is the speed of light; $\nu$ is frequency; $n$ is the index of refraction; $\rho$ is a quantity that does not change between equations (42) and (43).

The last two equations in that paper (42), (43), are a conclusion about an angle of rotation $\theta:$

$$\Delta n = \frac {2\pi ~\rho}{\lambda~ n} \tag{42}$$

and

$$\theta = \frac {2\pi \nu~\cdot \Delta n}{c~\cdot ~ 2} = \frac{2 \pi^2\rho}{\lambda^2} \tag{43}$$

My question is pretty dumb. If we insert the expression for $\Delta n$ into the equality in (43) we get:

$$\frac{2\pi\nu}{c}\cdot \frac{2\pi \rho}{2\lambda~ n} = \frac{2\pi^2\rho}{\lambda^2}\implies \frac{\nu}{c~n}=\frac{1}{\lambda}\implies\lambda =\frac{cn}{\nu} $$

We know already--and Rosenfeld mentions this a few lines up and at eq. (21)--that

$$\lambda = \frac{c}{\nu~n}$$

which forces $n$ to be $1$. In general $n \neq 1$.

Can someone tell me what, if anything, has gone awry here?

Edit: After looking at several versions of Rosenfeld's equation I see $\theta$ is proportional to the trace of the rotation matrix...but the constant of prop. can take different forms.