I have infinite annular potential well (scheme in the picture). scheme of annular potential well

Schrodinger equtation in the anullus (for $R_1 <r<R_2$ is $V=0$) with polar coordinates is \begin{equation} - \frac{ \hbar }{2m} \left[ \frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial \psi}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 \psi }{\partial \phi^2} \right] = E \psi \end{equation} I assume separable solution $\psi = R(r) F(\phi)$. So after modification I get equation $$ - \frac{ \hbar }{2m} \left[ r \frac{R'}{R} + r^2\frac{R''}{R} \right] -r^2 E = \frac{ \hbar }{2m} \frac{F''}{F} = - \epsilon_{\phi} $$ From the side with $F$ i got $F(\phi) = e^{i l \phi}$, $\epsilon_{\phi} = \frac{\hbar^2 l^2}{2m}$ and $l$ is integer (from periodicity)

From the side with $R$ I get after modification Bessel equation: $$ r^2 R'' + r R' +\left( k^2 r^2 - l^2 \right) = 0, $$ where $k = \sqrt(2mE / \hbar^2)$.

The general solution should consists of Bessel's functions of first kind $J_l(kr)$ and second kind $Y_l(kr)$: $$ R(r) = A_l J_l(kr) + B_l Y_l(kr) $$ I have got two boundary conditions $R(R_1) =0$ and $R(R_2) =0$ However I need to find 3 constants: $A_l$, $B_l$ and $k$, to get eigenfunctions $\psi$ and eigenvalues $E$.

Did I forget some other condition? I can't find out how to continue.

How to continue with analytical solution?


2 Answers 2


The problem isn't fully solvable in exact form, since it requires the solution of a transcendental equation to get $k$, and with it the energy eigenvalue. This shouldn't be surprising $-$ it's exactly the same situation as for a particle in a ring (i.e. without the internal cutout).

You are correct in all the steps that lead down to your radial eigenfunction in the form $$ R(r) = A_l J_l(kr) + B_l Y_l(kr), $$ and now the task is indeed to solve for the coefficients $A_l$ and $B_l$ and for the wavenumber $k$. From the first two, one is easy, and you can get it by setting $R(r)$ to zero at one of the two borders, say, $$ R(R_1) = A_l J_l(kR_1) + B_l Y_l(kR_1)=0, $$ or in other words $$ B_l = -\frac{ J_l(kR_1)}{Y_l(kR_1)}A_l, $$ so that's $B_l$ down and $A_l$ to go. Shifting notation slightly by defining $C_l=A_l/Y_l(kR_1)$, you get your full eigenfunction in the form $$ R(r) = C_l \bigg[Y_l(kR_1) J_l(kr) - J_l(kR_1) Y_l(kr)\bigg], $$ and your second boundary condition in the form $$ R(R_2) = C_l \bigg[Y_l(kR_1) J_l(kR_2) - J_l(kR_1) Y_l(kR_2)\bigg] = 0, $$ and this is where things get tricky.

The reason that things get hairy at this point is that you are now solving for the wavenumber $k$ (itself a proxy for the energy eigenvalue $E=\hbar^2 k^2/2m$) in the transcendental equation $$ Y_l(kR_1) J_l(kR_2) - J_l(kR_1) Y_l(kR_2) = 0, $$ and this simply does not accommodate any exact-form solutions (much in the same way e.g. the finite square well will reduce to a transcendental equation of the form $\tan(ka) = \sqrt{C+k^2b^2}$ with no exact solutions). This is analogous to the situation for the no-inner-cutout case, where you'll just be left with $$ J_l(kR)=0, $$ where you know that the product $kR$ needs to be a Bessel zero, but that is as far as the analytical approaches can tell you.

In the no-inner-cutout case, of course, you're in luck: the Bessel zeros are extremely common and well-studied objects, and they are included in most tabulations of special functions (e.g. Abramowitz and Stegun, and the like) and they are a standard component of most mathematical software.

For your case, with a nonzero inner cutout, you're less lucky, because the situation is less common and it is therefore less widespread in both tabulations and mathematical software. The keyword to search for is the cross-product Bessel zeros: we do know a lot about them, but that knowledge doesn't always trickle down to applications.

When I work with those objects, I tend to do so in Mathematica, which does have a BesselJZero in-built function in the core language, but for which the rest of the BesselZeros package, including the cross-product zeros, didn't quite make it into the core system. The package is still available and it does work (though with some reliability problems on the cross-product zeros!) if that's the way you want to roll.

And, finally, that third constant: for the normalization constant $C_l$, which is fixed by requiring that $$ \int_{R_1}^{R_2} |R(r)|^2 r\,\mathrm dr = 1 $$ or some similar requirement $-$ yeah, there's no chance of that being analytically integrable, much like the no-inner-cutout case. You just integrate it numerically when it comes down to it.

  • $\begingroup$ Thaks for help. However, from $Y_l(kR_1) J_l(kR_2) - J_l(kR_1) Y_l(kR_2) = 0$ I will got roots $c_{l,i}$. When I want to calc. corresponding $k$. How can I do that? Because I can use, $c_{l,i} = k_{l,i} R_1$ or $c_{l,i} = k_{l,i} R_2$. And i will get differnet $k$. Am I missing something? $\endgroup$ May 21, 2018 at 20:12
  • $\begingroup$ @PavelBlazek The cross-term Bessel zeros of that form (as you can see from the links already given) are normally quoted (or calculated) as zeros of the function $Y_l(x) J_l(\lambda x) - J_l(x) Y_l(\lambda x)$, where $\lambda = R_2/R_1$ is a fixed parameter and $x=k R_1$ (or vice versa). Thus, you don't get numbers (as with the zeros of $J_l(x)=0$) but functions $c_{l,i}(\lambda)$ from which you obtain the wavenumbers through $k_{l,i}R_1=c_{l,i}(R_2/R_1)$. Conventions vary, so check your source / software docs, but the core is that you now get one 'shape' parameter. $\endgroup$ May 21, 2018 at 20:22
  • $\begingroup$ This is at play in the linked question, where the plots with respect to $b=R_2$ while fixing $a=R_1$ to unity are essentially plots of the behaviour of the zeros as a function of $\lambda = b/a=R_2/R_1$, modulo trivial dimensionalization factors. $\endgroup$ May 21, 2018 at 20:24

So, the potential is defined by next statement.

$$V(r) = \begin{cases} V = 0 &\text{if $a < r < b$ annular zone}\\ V = \infty &\text{if $r < a$ or $b > r$ out zone} \end{cases}$$

The excess charge quanta may lay on the annular zone. The non-time dependent of the Schrödinger equation must be applied $(1)$ in cylindrical coordinates as follows.

$$-\frac{\hbar^2}{2m} \left[ \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \psi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \psi}{\partial \varphi^2} + \frac{\partial^2 \psi}{\partial z^2} \right] = E \psi \tag{1}$$

where no potential in annular zone is considered, $m$ is the electron mass, $\varphi$ is the angular azimuthal coordinate, $z$ is the long coordinate. Wave function can be rewritten as product of its separable variables as:

$$\psi = \psi_z R(r) \chi (\varphi)$$

Next, simplified expressions can be written from equation $(1)$, as soon as the coordinates are self-independent.

$$A_1 + A_2 = -\frac{2 m E}{\hbar^2}$$

And not so much evident:

$$\frac{r}{R} \frac{\partial R}{\partial r} + \frac{r^2}{R} \frac{\partial^2 R}{\partial r^2} - A_1 r^2 = - \frac{1}{\chi} \frac{\partial^2 \chi}{\partial x^2} = B$$

where $A_1$, $A_2$ and $B$ are constants of integration. The solutions to these equations are published for cylindrical potential well $(1)$ as follows:

$$\chi(\varphi) = C_1 \exp{\left( +i \sqrt{B} \varphi \right)} + C_2 \exp{\left( -i \sqrt{B} \varphi \right)}$$

where $\sqrt{B} \in \mathbb{N}$, so the natural number $n_\varphi$ azimuthal. Next function is the solution of the radial equation, that is different to publication [1] because of the boundary conditions.

$$R(r) = C_3 J_\sqrt{B} \left( -i \sqrt{A_1} r \right) + C_4 Y_\sqrt{B} \left( -i \sqrt{A_1} r \right) \tag{2}$$

where $J_n (x)$ and $Y_n (x)$ are Bessel functions of first and second order respectively [2]. Boundary conditions are as follows, $R(a) = 0$ and $R(b) = 0$ and non-trivial solution of the $R(r)$ functions arouse fulfilling next condition.

$$J_\sqrt{B} \left( -i \sqrt{A_1} a \right) \, Y_\sqrt{B} \left( -i \sqrt{A_1} b \right) - J_\sqrt{B} \left( -i \sqrt{A_1} b \right) \, Y_\sqrt{B} \left( -i \sqrt{A_1} a \right) = 0$$

Some math equations are needed to obtain $A_1$, $(2)$. The next equation shows the solution.

$$A_1 = -\frac{4 {n_r}^2 \pi ^2}{(b-a)^2 {n_\varphi}^2} \tag{3}$$

The equation $(2)$ can be rewritten from the result of equation $(3)$ as follows.

$$R(r) = C_3 J_{n_\varphi} \left( \frac{2\pi}{b-a} \left( \frac{n_r}{n_\varphi} \right) r \right) + C_4 Y_{n_\varphi} \left( \frac{2\pi}{b-a} \left( \frac{n_r}{n_\varphi} \right) r \right) \tag{4}$$

where $n_r$ is the quantum number in radial coordinate and $J_n (r)$ and $Y_n (r)$ are Bessel functions of first and second kind respectively [2]. The real constants $C_3$ and $C_4$ are chosen for cylindrical normality condition, as result, second summand of the last member is negligible in equation $(4)$ for low values of radii.
Finally standard box free particle between $-L/2$ and $L/2$ shows the quanta of the long $z$ coordinate by $A_2$ as next equation.

$$A_2 = - \frac{4 {n_z}^2 \pi^2}{L^2}$$

The result of quantization of energy is as follows.

$$E = \frac{4 \hbar^2 \pi^2}{2 m} \left( \frac{{n_r}^2}{(b-a)^2 {n_\varphi}^2} + \frac{{n_z}^2}{L^2} \right)$$


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