LOG#241. Flatland & Fracland.

Flatland is a known popular story and book. I am going to review the Bohr model in Flatland and, then, I am going to strange fractional (or fractal) dimensions, i.e., we are going to travel to Fracland via Bohrlogy today as well.

Case 1. Electric flatland and Bohrlogy.

(1)   \begin{equation*} F_c(2d)=K_c(2d)\dfrac{e^2}{r} \end{equation*}

Suppose that

(2)   \begin{equation*} E_p(2d)=K_c(2d)e^2\ln\left(\dfrac{r}{a_0}\right) \end{equation*}

Then, we have

(3)   \begin{equation*} m\dfrac{v^2}{r}=K_c\dfrac{e^2}{r} \end{equation*}

and thus

(4)   \begin{equation*} v=\sqrt{\dfrac{K_c}{m}}e \end{equation*}

Moreover, imposing Bohr quantization rule L=pr=mvr=n\hbar, then you get

(5)   \begin{equation*} r=\dfrac{n\hbar}{mv} \end{equation*}

(6)   \begin{equation*} \tcboxmath{r_n=na_0=n\dfrac{\hbar}{e\sqrt{mK_c}}} \end{equation*}

Total energy becomes

(7)   \begin{equation*} E=E_c+E_0=E_m=\dfrac{1}{2}mv^2+K_ce^2\ln\left(\dfrac{r_n}{a_0}\right)} \end{equation*}

(8)   \begin{equation*} \tcboxmath{E_m=E_0\left(\dfrac{1}{2}+\ln n\right)-E_0\ln\left(\dfrac{\hbar}{e\sqrt{mK_c}}\right)} \end{equation*}

where E_0=K_ce^2.

Case 2. Gravitational flatland and Borhlogy.

(9)   \begin{equation*} F_N(2d)=G_N(2d)\dfrac{e^2}{r} \end{equation*}

Suppose that

(10)   \begin{equation*} E_p(2d)=G_N(2d)m^2\ln\left(\dfrac{r}{a_0}\right) \end{equation*}

Then, we have

(11)   \begin{equation*} m\dfrac{v^2}{r}=G_N(2d)\dfrac{m^2}{r} \end{equation*}

and thus

(12)   \begin{equation*} v=\sqrt{G_Nm} \end{equation*}

Moreover, imposing Bohr quantization rule L=pr=mvr=n\hbar, then you get

(13)   \begin{equation*} r=\dfrac{n\hbar}{mv} \end{equation*}

(14)   \begin{equation*} \tcboxmath{r_n=na_0=n\dfrac{\hbar}{m\sqrt{mG_N}}} \end{equation*}

Total energy becomes (up to an additive constant)

(15)   \begin{equation*} E=E_c+E_0=E_m=\dfrac{1}{2}mv^2+G_Nm^2\ln\left(\dfrac{r}{a_0}\right) \end{equation*}

(16)   \begin{equation*} \tcboxmath{E_m=E_0\left(\dfrac{1}{2}+\ln n\right)-E_0\ln\left(\dfrac{\hbar}{m\sqrt{G_Nm}}\right)} \end{equation*}

where E_0=G_Nm^2.

Exercise: Gravatoms. 

Suppose a parallel Universe a where electrons were neutral particles and no electric charges existed. In such a Universe, 2 electrons or any electron and a proton would form a gravitational bound state called gravatom (gravitational atom for short). The force potencial would be V_g=Gm^2/r. And we could suppose that electron mass and G_N are the same as those in our Universe.

a) Calculate the ratio between the gravitational potential and the electric potential in our universe. Comment the results. 1 point.

b) Compute the analogue of Bohr radius in the gravatom. Comment the result. 1 point.

c) Compute the analogue of Rydberg constant for the gravatom. Is it large or small compared with the usual Rydberg constant? 1 point.

d) Compute the period of the electron in the lowest energy level. Compare it with the age of our Universe. 1 point.

e) Imagine a parallel Universe B, where the electrons were indeed supermassive. Higher the electron mass is, lower the size of the gravatom. If big enough, the size of the gravatom is smaller than the Compton wavelength of a free electron, measuring the size of the irreducible wave function of the electron. In that limit, there is no free electron but a bound state of a black hole. Compute the critical mass for the cross-over. Compare that scale to a human lifetime. 1 point.

Case 3. Welcome to Fracland, land of fractional Bohrlogy. 

3A. Fractional H-atom.

Consider the potential energy

(17)   \begin{equation*} U(r)=-\dfrac{Ze^2}{r} \end{equation*}

and the hamiltonian

(18)   \begin{equation*} H_\alpha=D_\alpha\left(-\hbar^2\Delta\right)^{\alpha/2}=D_\alpha\left(-\hbar \Delta^{1/2}\right)^\alpha \end{equation*}

where, in principle, we allow only 1<\alpha\leq 2, but a suitable analytic continuation could be feasible somehow. Then, \alpha\overline{E_k}=-\overline{U} and pr_n=n\hbar provide

(19)   \begin{equation*} \omega(n_1\rightarrow n_2)=\dfrac{E_2-E_1}{\hbar} \end{equation*}

such as

(20)   \begin{equation*} \alpha D_\alpha\left(\dfrac{n\hbar}{r_n}\right)^\alpha=\dfrac{Ze^2}{r_n} \end{equation*}

And, finally, you get the radii and energy levels for the fractional H-atom as follows

(21)   \begin{equation*} \tcboxmath{r_n=a_0 n^{\frac{\alpha}{\alpha-1}}\;\;\;  a_0=\left(\dfrac{\alpha D_\alpha \hbar^\alpha}{Ze^2K_C}\right)^{\frac{1}{\alpha-1}}} \end{equation*}

(22)   \begin{equation*} \tcboxmath{E_n= (1-\alpha)\overline{E_k}}\;\;\; \tcboxmath{E_n=\left(1-\alpha\right)E_0 n^{-\frac{\alpha}{\alpha-1}}\right)} \end{equation*}

(23)   \begin{equation*} \tcboxmath{\omega_n(\alpha)=\dfrac{(1-\alpha)E_0}{\hbar}\left(\dfrac{1}{n_1^{\frac{\alpha}{\alpha-1}}}-\dfrac{1}{n_2^{\frac{\alpha}{\alpha-1}}}\right)} \end{equation*}

and where now

(24)   \begin{equation*} \tcboxmath{E_0=\left(\dfrac{Z^{\alpha}(K_Ce^2)^\alpha}{\alpha^\alpha D_\alpha \hbar^\alpha\right)^{\frac{1}{\alpha-1}\right)}}} \end{equation*}

Note that E_k=D_\alpha p^\alpha. Virial theorem implies \overline{E_k}=\overline{U}(n/2) if U=\alpha r^n.

3B. Fractional harmonic oscillator (in 3d).

Consider now

(25)   \begin{equation*} H(\alpha,\beta)=D_\alpha(-\hbar^2\Delta)^{\alpha/2}+q^2 r^\beta \end{equation*}

In the case \alpha=\beta you get

(26)   \begin{equation*} H_\alpha=D_\alpha(-\hbar^2\Delta)^{\alpha/2}+q^2r^\alpha \end{equation*}

For a single d.o.f., i.e., if D=1, you can write

(27)   \begin{equation*} E=D_\alpha p^\alpha+q^2x^\beta \end{equation*}

The energy levels can be calculated

(28)   \begin{equation*} \tcboxmath{E_n=\dfrac{\pi \hbar \beta D_\alpha^{1/\alpha} q^{2/\beta}}{2B\left(\frac{1}{\beta},\frac{1}{\alpha}+1\right)}\left(n+\dfrac{1}{2}\right)^{\frac{\alpha\beta}{\alpha+\beta}}} \end{equation*}

and where B(x,y) is the beta function. Remarkably, only the standard QM simple HO has equidistant energy spectrum!

3C. Fractional infinite potential well.

Let the potential be

(29)   \begin{equation*} V=\begin{cases}V(x)=+\infty, x<-a\\ 0,-a\leq x\leq a\\ V(x)=+\infty, x>a\end{cases} \end{equation*}

Then, the energy spectrum becomes

(30)   \begin{equation*} \tcboxmath{E_n=D_\alpha\left(\dfrac{\pi\hbar}{\alpha}\right)^\alpha n^\alpha} \end{equation*}

The ground state energy is

(31)   \begin{equation*} \tcboxmath{E_0=D_\alpha\left(\dfrac{\pi\hbar}{2\alpha}\right)^\alpha} \end{equation*}

3D. Delta potential well.

Consider 1<\alpha\leq 2, and the \delta-function potential V(x)=-\gamma\delta(x), with \gamma>0. The energy spectrum is, for the bound state,

(32)   \begin{equation*} \tcboxmath{E=-\left[\dfrac{\gamma B\left(\frac{1}{\alpha},1-\frac{1}{\alpha}\right)}{\pi \hbar \alpha  D_\alpha^{1/\alpha}}\right]^{\frac{\alpha}{\alpha-1}}} \end{equation*}

3E. Fractional linear potential.

Consider the potential

(33)   \begin{equation*} V(x)=\begin{cases} Fx, x\geq 0, F>0\\ \infty, x<0\end{cases} \end{equation*}

The energy spectrum will be

(34)   \begin{equation*} \tcboxmath{E_n=\lambda_n F\hbar \left(\dfrac{ D_\alpha}{\left(\alpha + 1\right) F\hbar}\right)^{-\frac{1}{\alpha + 1}}} \end{equation*}

and where \lambda_n are solutions to certain trascendental equation, with 1<\alpha\leq 2.

Hidden connection with the riemannium. Some time ago, I posted in physics stack exchange this question https://physics.stackexchange.com/questions/60991/mysterious-spectra

Thus, fractional H-atoms and oscillators, with care enough, can also be seen as riemannium-like.

See you in another blog post dimension!

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