# Are there energy levels when $V(r)\not\propto{r^{-1}}$?

For something like the hydrogen atom $$V(r)\propto{r^{-1}}$$. And there are energy levels described by the equation

$$E_n=-\frac{m_ee^4}{8\epsilon_0^2h^2n^2}$$

that indicate where orbitals are allowed. As understand it $$V(r)\propto{r^{-1}}$$ for any pair of electric charges, not just for the hydrogen atom.

I was wondering if we set $$V(r)$$ to be proportional to some arbitrary function such as having $$V(r)\propto{\cos(br)}$$, $$V(r)\propto{\ln(br)}$$, $$V(r)\propto{r}$$, $$V(r)\propto{r^{-1.3}}$$, $$V(r)\propto{e^{-br}}$$, $$V(r)\propto{r^{br}}$$, or even $$V(r)\propto\Gamma(b(r+1))$$, then are there still energy levels, that describe where orbitals would be allowed? If there are energy levels for other functions of $$V(r)$$, is the equation for energy levels $$E_n$$ dependent or not dependent on the proportionality function for $$V(r)$$?

• Did you study the 1D harmonic oscillator? There is a 3D version. – G. Smith Aug 18 '20 at 18:36
• indicate where orbitals are allowed Each orbital extends throughout all of space. Have you seen the wavefunctions for them? – G. Smith Aug 18 '20 at 18:45

If there are energy levels for other functions of $$V(r)$$, is the equation for energy levels $$E_n$$ dependent or not dependent on the proportionality function for $$V(r)$$?

The states with quantised energy levels $$E_n$$ are so-called bound states.

Only if the potential $$V(r)$$ leads to bound states does energy quantisation occur. Non-bound states are also called scattered states and aren't quantised.

To check if bound states are possible for a potential $$V(r)$$, solve the Schrödinger equation for that potential $$V(r)$$.

A simple example of a potential well with a bound state and a scattered state (schematic):

For $$x<0\Rightarrow V(x)=+\infty$$,

For $$0\leq x \leq L\Rightarrow V(x)=0$$

For $$x>L \Rightarrow V(x)=V_0$$

Because $$0\leq E_L \leq V_0$$, the particle with energy $$E_L$$ is in a bound state. There may be more than $$1$$ bound, quantised state (not shown).

Because $$E_H\geq V_0$$, the particle is a scattered particle.

*A few* typical potential wells with bound states ($$1D$$ schematic):

• a. parabolic potential,
• b. infinite square potential,
• c. finite square potential,
• d. angular potential,
• e. Morse-style potential.
• I see that for a, and e, the steepness of the potential well decreases towards the minimum. Can all potential wells, in which the steepness of the potential decreases towards the local minimum, be transformed into wells that have bound states by simply adjusting the width and/or height of the well? – Anders Gustafson Aug 19 '20 at 2:47
• I'm not sure, TBH. U'd have to try it analytically... – Gert Aug 19 '20 at 4:04