Quantised energy of a confined system

[CAF123]

In one of my lectures, a graph was plotted of the total energy of an electron as a function of r, it's orbital radius. The function representing this curve was differentiated and set to zero to determine what value of r gave the minimum energy corresponding to the energetically favourable state of the electron, that being the Bohr radius.

My question is: this graph plotted was continuous, but I know energy is quantised in a confined system - so what have I missed?

 

[klawlor419]

What was the energy function?

In quantum mechanics treatment of the hydrogen atom, if this is what your professor was taking about, typically the way the Bohr radius is found is by maximizing the probability density functions with respect to radial values. For the ground state wave functions of the hydrogen atom this result occurs for r=r(Bohr).

Bohrs treatment of the hydrogen atom was classical, so perhaps the energy function you speak of is the total energy of the interaction of an electron orbiting a proton. Perhaps it is something different. So if you could provide some more detail on this more help could be given.

You are right that in quantum mechanical treatment the energy levels are quantized. So if you were to try to maximize one of these values nothing useful would come out of it.

Furthermore, maximizing the energy with respect to a radial value doesn't really make sense in terms of finding the Bohr radius. Let's say that this energy function is the classical expression, the energy of the closest orbit like the ground state should be a minimum.

It is most likely that your professor plotted the probability density as opposed to energy.

 

[CAF123]

We had that $E(r) = KE + PE$, so that $$E(r) = \frac{h^2}{8\pi^2 m r^2} - \frac{e^2}{4\pi\epsilon r}$$ and then this was differentiated and set to zero to attain a value of r which yielded the minimum of the graph. This value of r was $$r = \frac{\epsilon h^2}{\pi m e^2}$$ which corresponds to the Bohr radius. (the graph is essentially the inter atomic potential of two charged particles, so yes, it is like the total energy of an electron orbiting a proton)
Many thanks

 

[CAF123]

Anyone have any ideas?

 

[klawlor419]

Ideas for what?

The method you stated in your second to last post is a classical calculation that happens to produce the correct value of the most probable location of the a quantum mechanical treatment of the hydrogen atom ( an electron orbiting a proton) in its ground state.

The energy is continuous in the classical calculation. It is not continuous in the quantum mechanical calculation. The classical calculation is wrong. Yet it gives a correct value for the Bohr radii and energy levels.

 

[klawlor419]

So it is not wrong. I shouldn't say it that way but it is not nearly as illuminating as the quantum mechanics treatment. i.e. solving the Schrödinger equation.

 

[The_Duck]

This sounds like a semi-classical analysis using the uncertainty principle. By the uncertainty principle we know that the momentum of an electron confined to a region of size r must be at least of order hbar/r, so it must have kinetic energy of order hbar^2/(2 m r^2). So, speaking approximately, the ground state should minimize E = KE + PE = hbar^2 / (2 m r^2) + V(r).

This is an approximate, semiclassical analysis that ignores details like the quantization of energy levels. The point is that it gives you a quick and dirty way of estimating things like the size of an atom and its ground state energy.