I Why the energy inversely propotional to n^2 in Bohr model

[KFC]

After reading some materials on Bohr model, I understand the model is more or less incorrect, especially in terms of "orbital". I just wonder if the energy expression is also wrong or not.

In my text for general quantum theory, the energy about two neighboring level is given as $\Delta E = E_n-E_{n-1} = \hbar\omega$
which is $n$ independent. But in Bohr model about hydrogen, the quantized energy is given as
$E_n = -E_0/n^2$
which is inversely proportional to $n^2$. This will give, for example from level $n$ to $n-1$

$E_n-E_{n-1} = \frac{-E_0}{(n-1)^2-n^2}$
which is $n$ dependent. I am quite confusing why is it. Is Bohr model about energy is correct for hydrogen or hydrogen-like atom?

 

[Simon Bridge]

The formula $E_n = -E_0/n^2$ is correct, and is due to the shape of the hydrogen potential.
The hydrogen potential has a top, so the energies levels get closer together as they approach E=0 from below. For E > 0, the energy levels are continuous and the hydrogen atom is said to be "ionized".

Your quantum theory text should not give $\hbar\omega$ increments in energy for the hydrogen potential - I suspect you have misread it - that spacing is for an approximation for other types of potential called "harmonic oscillator".

Note: your equation for the transition energy is incorrect. Try again.

See:
http://astro.unl.edu/naap/hydrogen/transitions.html

 

[KFC]

The formula $E_n = -E_0/n^2$ is correct, and is due to the shape of the hydrogen potential.
The hydrogen potential has a top, so the energies levels get closer together as they approach E=0 from below. For E > 0, the energy levels are continuous and the hydrogen atom is said to be "ionized".

Your quantum theory text should not give $\hbar\omega$ increments in energy for the hydrogen potential - I suspect you have misread it - that spacing is for an approximation for other types of potential called "harmonic oscillator".

Note: your equation for the transition energy is incorrect. Try again.

See:
http://astro.unl.edu/naap/hydrogen/transitions.html

Thanks a lot. I think I misunderstood some context in the text. I always think the $\Delta E=\hbar\omega$ is universal for all quantum system. Your reply help me to recall the harmonic oscillator. So like what you said, the explicit form of energy for a quantum system is really depending on the potential, is that what you mean?

Thanks anyway.

 

[Simon Bridge]

So like what you said, the explicit form of energy for a quantum system is really depending on the potential, is that what you mean?

That is correct - it is exactly the same for classical physics: the dynamics depends on the specific form of the potential. For instance, a ball rolling around the inside of a bowl - requires more kinetic energy to reach the same distance from the center in a steep sided bowl as for a shallow bowl.

You can see this has to be the case if you consider the case of a repulsive potential, or the free-space potential (V=0 everywhere): does it make sense for the allowed energies in these situations to have the same separation as for bound-states?

 

[KFC]

Thanks for clarifying it. I appreciate your help.

 

[Simon Bridge]

No worries.