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## Homework Statement

Use the Bohr quantization rules to calculate the energy levels for a particle moving in a potential

\begin{equation}

V\left(r\right)= V_0 \left(\frac{r}{a}\right)^k

\end{equation}

where k is positive and very large. Sketch the form of the potential (done) and show that the energy values approach $E_n = kn^2$, where k is a constant.

## Homework Equations

\begin{equation}

L=mvr=n\hbar

\end{equation}

## The Attempt at a Solution

So, first I derived an equation for the force

\begin{equation}

-\vec{\nabla} V = -\frac{kV_0}{a}\left(\frac{r}{a}\right)^{k-1} \hat{r} = \vec{F}\left(r\right)

\end{equation}

Setting the force equal to the centripetal force give me:

\begin{align}

-\frac{kV_0}{a}\left(\frac{r}{a}\right)^{k-1} &= m\frac{v^2}{r} \\

&= \frac{n^2\hbar^2}{mr^3}

\end{align}

Where the centripetal force is rewritten in terms of $n\hbar$ via the quantization rule. Now with a inverse square force, one can easily solve for r, but in this case, after algebraic manipulation, I have

\begin{align}

r^{k+2}&= - \frac{n^2\hbar^2 a^{k+2}}{kV_0} \\

r_n &= \sqrt[k+2]{- \frac{a^k n^2\hbar^2}{kV_0}}

\end{align}

Now, here I have a problem I have no constraints to prevent a complex radius, because $n, \hbar,$ and $k$ are all positive, and $V_0$ or $a$ are not specified in any way.

I have kept the exponents exact, but since k is positive and very large, I'd think $k \simeq k+2 \simeq k-1$, but I haven't used this. Is this reasonable? i would think $x^{1000} \simeq x^{1001}$, etc. . .

So the next step would be to use the quantized orbits to derive the energy,

\begin{align}

E &= \frac {p^2}{2m} + V \\

&=-\frac{kV_0}{2}\left(\frac{r}{a}\right)^k + V_0 \left(\frac{r}{a}\right)^k \\

&= \left(1-\frac{k}{2}\right)V_0\left(\frac{r}{a}\right)^k

\end{align}

Now, if I allow the assumption that $k \simeq k+2$, I could substitute $r_n$ for $r$, and get:

\begin{align}

E &= \left(1-\frac{k}{2}\right)V_0\frac{-n^2\hbar^2 a^k}{kV_0 a^k} \\

&= \left(1-\frac{k}{2}\right)\frac{1}{k}\hbar^2 n^2 \\

\end{align}

We're given that k is very large and positive, which reduces the former equation to

\begin{equation}\frac{1}{2}\hbar^2 n^2

\end{equation}

So this meets the target of $E_n = kn^2$, where $k= \frac{\hbar^2}{2}$. Does this look like a reasonable quantization? Should the energy be positive? Any constructive criticism, etc. would be greatly appreciated. Cheers!