Electron Energy Levels

[Mr. Study]

Hey,

Could anyone explain to me why is it electrons only occupy certain energy levels?

My teacher, literally, gave a word for word text book answer.

 

Thanks.

[Lasercookie]

Edit: Sorry for the essay, didn't mean to write so much.

I hope this isn't the same kind of answer your teacher gave you, the crux of it is still pretty much from the textbook. I'll admit my understanding of this isn't all that refined, if there's anything incorrect point it out.

I'll also try to explain as much as possible, but I'll stick in Heinemann page references where some of this stuff is implied in that book, to keep me on topic and also it might help you out with understanding how the book says it.

So are electrons particles or are they waves? You know that moustache man's de Broglie's hypothesis that matter does in fact have wave-like properties. I think it's reasonable to ask, what's the medium of this wave? This isn't something you have to know, I'm not going to pretend that I completely understand this either, but the concepts here will make a bit more sense having realised it (and maybe some of the wording the book uses will make a bit more sense about why they're a bit indirect about things).

The thing is for photons and particles, these waves somehow or other relate to being a probability distributions. (pg 432, pg 444) If there's a bright fringe, that indicates a high probability of a photon hitting that area, dark fringe would mean low probability. A specific electron would be somewhere in that vicinity, but we can't really tell exactly where. I think it'll be worth reading this section from Wikipedia first (and look at picture): http://en.wikipedia.org/wiki/Double-slit_experiment#Interference_of_individual_particles I think you'll be interested in the second paragraph there.

This bit is probably worth noting:

(Note that it is not the probabilities of photons appearing at various points along the detection screen that add or cancel, but the amplitudes. Probabilities are the squares of amplitudes. Also note that if there is a cancellation of waves at some point, that does not mean that a photon disappears; it only means that the probability of a photon's appearing at that point will decrease, and the probability that it will appear somewhere else increases.)

So that's a bit of a tangent about the wave-like properties of electrons (have a look at page 444 of Heinemann too). You won't ever have to discuss probability in VCE Physics (or probably think about it), but I think it clears up a few points, and especially helps when it comes to wondering how the hell it's possible for an electron to... well we'll get to that.

There was Bohr he proposed his model, and we know that had a few problems, namely it viewed electrons like orbiting planets. It did get a few things right though. We're assuming that the atom is stable - Bohr didn't know why though.

Now, when do we get noticeable wave-like properties? Not at scales that we're used to seeing, but when we're dealing with atoms and electrons the wave-like properties are quite noticeable (relative to the size of all the other stuff around it).

So we can view the electrons orbiting the atoms to have wave-like properties. De Broglie suggested that the electrons might actually be forming waves around the nucleus. I'm going to assume you're familiar with the stuff in the Sound detailed study (sorry if you aren't but this is a fairly easy way to understand it --- oh I just noticed that the Heinemann book provides an explanation of standing waves on Page 457 for those aren't doing Sound).

If we're thinking about waves in general, say a sound wave in somewhere closed, there'll be those resonant frequencies that exist. We also know about the 'principle of superposition'. We know when we have these two, we may be able to get standing waves to form, which is what happens with the electrons in the atom. This makes it hard to picture it, but it's also three-dimensional.

Now remember that those different harmonics we can get for standing waves, and that they represent the resonant frequencies (page 581). The same occurs for these electron standing waves, where each resonant frequency is proposed to represent an energy level.

So why do we have to bother with standing waves? Why can't it be any other wave? The electrons are "orbiting" the nucleus, and so their wavelength can only fit into this circumference (thinking of it as a circle apparently). This is really the key statement here, and I suspect this is along the lines of what your teacher told you.

In order to fit around this circumference completely, we can only have integer multiples of the wavelength. I think that's something that should make relative sense now.

This is taken out of the Heinemann book again, but I liked it. So We also know that , so subbing in It's probably not useful having those formulas, but it might help something click.

Energy levels correspond to where the electrons 'are'. The first energy level can be viewed as a standing wave of n = 1. The second energy level can be viewed as a standing wave of n = 2 and so on. You would know with each harmonic how the standing waves begin to split, and these also have to fit around in some kind of circular fashion - these will be those diagrams you'd be used to seeing. Since these energy levels are only stable at these specific wavelengths, we have discrete energy levels.

Apparently this video is how you can roughly visualise it: http://www.ap.smu.ca/demonstrations/index.php?option=com_content&view=article&id=116&Itemid=85

It's worth asking is why standing waves, why won't any other wave be stable? What happens if we don't have an integer number of wavelengths to fit around the circumference? Well, for the integer numbers we get standing waves. If it isn't, then we'll get destructive interference, which going back to the probability discussion before, it's quite unlikely for the electron to be there whilst stable (I think, this is where my knowledge is getting quite iffy, so I'll leave it at that statement). There some other way to word this, (along the lines destructively interfering with itself), but I don't really remember the reasoning/the exact wording.

So to state it again:
Electrons are only stable when they form standing waves, of which there can only be integer multiples of the wavelengths.
These standing waves correspond to the energy levels.
Therefore the energy levels that the electrons occupy are discrete.

Have a go at VCAA 2009, Question 12 of the Light and Matter section. It tests this concept. I'll note that the assessor report points this out as a common mistake: "It was common for students to believe that the electron followed a wave path around the nucleus." This is wrong because the wave represents the probability of where that electron could be, not the actual path it follows.

[Aurelian]

Sooooo... just in case you wanted a shorter response

Basically it just falls out of the fact that if you treat electrons as waves, the energy levels must by formed by standing waves in order to be stable. As a result, the "circumference of orbit" (in a classical picture) must be an integer multiple of the de Broglie wavelength of the electron. Hence there are only certain values each energy level can take - this is what we mean by quantization.