LOG#152. Bohrlogy (II).

niels bohr

An interesting but relatively unknown variation of the Bohr model is to use a logarithmic potential energy. In that case, we have

(1)   \begin{equation*} E=T+U(R)=\dfrac{p^2}{2m}+U(R)\end{equation*}

(2)   \begin{equation*} E=\dfrac{p^2}{2m}+k\ln\left(\dfrac{R}{R_0}\right)\end{equation*}

(3)   \begin{equation*} F(R)=-\dfrac{dU}{dR}=-\dfrac{k}{R}\end{equation*}

Bohr quantization rules impose

L=mvR=n\hbar

and that

F_C=F(R)

so

(4)   \begin{equation*} m\dfrac{v^2}{R}=k\dfrac{1}{R}\end{equation*}

and then

(5)   \begin{equation*} mv^2=k\end{equation*}

(6)   \begin{equation*} \boxed{p^2=mk}\end{equation*}

(7)   \begin{equation*} \boxed{p=\sqrt{mk}}\end{equation*}

The momentum is NOT quantized in the logarithmic potential Bohr model. By the other hand,

pR=n\hbar

implies that

R_n=\dfrac{n\hbar}{p}

so

(8)   \begin{equation*}\boxed{R_n=\dfrac{n\hbar}{p}=\dfrac{n\hbar}{\sqrt{mk}}}\end{equation*}

The velocities are not quantized either

(9)   \begin{equation*} \boxed{v=\dfrac{p}{m}=\sqrt{k}{m}}\end{equation*}

The energies are easily calculated to be

(10)   \begin{equation*} \boxed{E_n=\left(\dfrac{1}{2}+\ln\left(\dfrac{n\hbar}{R_0p}\right)\right)k}\end{equation*}

or equivalently

(11)   \begin{equation*} \boxed{\dfrac{E_n}{k}-\dfrac{1}{2}=\ln\left(\dfrac{n\hbar}{R_0p}\right)}\end{equation*}

The forces and accelerations are quantized

(12)   \begin{equation*} \boxed{F_n=ma_n=m\dfrac{v^2}{R}=k\sqrt{mk}\dfrac{1}{n\hbar}}\end{equation*}

(13)   \begin{equation*} \boxed{a_n=\dfrac{F_n}{m}=k\sqrt{\dfrac{k}{m}}\left(\dfrac{1}{n\hbar}\right)}\end{equation*}

Areas are also quantized

(14)   \begin{equation*}\boxed{S_n=\pi R_n^2=\dfrac{\pi n^2\hbar^2}{mk}=\dfrac{\pi n^2\hbar^2}{p^2}}\end{equation*}

The angular frequencies and the periods are quantized as well

(15)   \begin{equation*} \boxed{\omega_n=\dfrac{k}{n\hbar}}\end{equation*}

(16)   \begin{equation*} \boxed{T_n=\dfrac{2\pi}{\omega_n}=2\pi\dfrac{n\hbar}{k}=\dfrac{nh}{k}}\end{equation*}

Interestingly, we can modify and enlarge the Bohr quantization rules. The modified or enhanced Bohr rules imply the addition of the quantization of the area/length via an extra condition

(17)   \begin{equation*} \boxed{S_n=L_0^2n}\end{equation*}

and thus

(18)   \begin{equation*} \boxed{R_n=L_0\sqrt{n}}\end{equation*}

Now, we proceed as Bohr himself

L=n\hbar

mvR=n\hbar

p_n=\dfrac{n\hbar}{R_n}=\dfrac{\hbar}{L_0}\sqrt{n}

Therefore, momentum and velocity are again quantized (unlike the usual logarithmic potential with the normal Bohr conditions)

(19)   \begin{equation*} \boxed{p_n=\dfrac{\hbar}{L_0}\sqrt{n}}\end{equation*}

(20)   \begin{equation*} \boxed{v_n=\dfrac{p_n}{m}=\dfrac{\hbar}{mL_0}\sqrt{n}}\end{equation*}

Forces and accelerations are quantized in a different form

(21)   \begin{equation*} \boxed{F_n=ma_n=\dfrac{k}{L_0}\left(\dfrac{1}{\sqrt{n}}\right)}\end{equation*}

(22)   \begin{equation*} \boxed{a_n=\dfrac{k}{mL_0}\dfrac{\sqrt{n}}{n}}\end{equation*}

Energies are also quantized but modified too

(23)   \begin{equation*} \boxed{E_n=\dfrac{\hbar^2n}{2mL_0^2}+k\ln\left(\dfrac{L_0\sqrt{n}}{R_0}\right)}\end{equation*}

This last equation is something more complicated that the first logarithmic potential. We can play with it a bit. Introduce the areas

a_0=L_0^2

A_0=R_0^2

and suppose that

k=\dfrac{\lambda_C^2}{L_0^2}mc^2

where

\lambda_C=\dfrac{\hbar}{mc}

The energies are

E=\dfrac{\hbar^2n}{2mmc^2L_0^2}nmc^2+k\ln\left(\sqrt{\dfrac{na_0}{A_0}}\right)

and algebra provides

(24)   \begin{equation*} \boxed{E_n=\dfrac{k}{2}\left(n+\ln\dfrac{na_0}{A_0}\right)=\dfrac{k}{2}\ln\left(\dfrac{e^nna_0}{A_0}\right)}\end{equation*}

Finally, angular frequencies and periods are also quantized

(25)   \begin{equation*} \boxed{\omega_n=\sqrt{\dfrac{k}{mL_0^2n}}}\end{equation*}

(26)   \begin{equation*} \boxed{T_n=\dfrac{2\pi}{\omega}=2\pi\sqrt{\dfrac{mL_0^2n}{k}}}\end{equation*}

As you can observe, some magnitudes change as we modify the quantization rules. This logarithmic model is useful in some interesting problems in theoretical physics and mathematics.

See you in my next blog post!

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