Effective Nuclear Charge

by Jared Rovny

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    00:01 Now that we've discussed what the Bohr model of the atom is as well as the basics structure of atoms and then how electrons behave in a slightly more complicated way with the quantum numbers for the atom as we discussed.

    00:13 We have few more advance concepts to discuss about the electronic structure of an atom.

    00:18 First of all we're going to have a brief discussion on the effective nuclear charge.

    00:23 So just sort of imagine yourself as an electron here in one of these orbitals.

    00:27 Suppose you are in the higher orbital, below you between the proton and yourself, there are other electrons.

    00:34 But just going back to our discussion about the electric force in coulombs law, we know that while you have an attractive force from the proton that's in the center, you will also feel a repulsive force in those other negative electrons since like charges repel.

    00:49 Because the electrons that are in the orbital, orbitals below you are going to be sort of spread out because they are moving.

    00:55 We have sort of an electron clot as we call it.

    00:57 It's not as though you have one directly opposing, even stopping you from orbiting.

    01:02 But it will slightly change the attractive force that is experienced between the electron in the outer orbitals and the proton in the center.

    01:11 So now for this reason we have an effective nuclear charge that is slightly lower than the charge it would experience otherwise.

    01:19 And this is due to what we call "Shielding." Shielding from the inner causing that shielding for the outer electrons.

    01:25 Now we are going to get into a topic that is one of the most talked about and very commonly discussed.

    01:32 But it is very tricky to may be think about it first.

    01:35 So this is the Heisenberg Uncertainty Principle.

    01:38 We of course will not get into all of the quantum mechanics or the complexity of the Heisenberg Uncertainty Principle.

    01:44 But since we are talking about the world of the very small things we do need to understand there's very important aspect about how things behave on the quantum level.

    01:53 We'll just do a basic introduction here.

    01:55 But how it works is basically like this.

    01:57 Suppose, we have a particle.

    01:59 In the quantum description of things it turns out that rather than knowing that this particle is at a particular location, we are actually uncertain about where it is.

    02:09 And this turns out not just be an experimenter's error where we're just not sure because may be our instruments aren't good enough.

    02:16 This is actually a fundamental aspect of nature itself, that particles, the idea of location of a particle becomes a little more washed out as we zoom into a smaller and smaller scale.

    02:27 So what we say instead is we have a probability of a particle being at a particular location at a particular time if we tried to measure where it was.

    02:36 So on this graph like this what we have is the position just on the x-axis.

    02:41 So this is literally just left to right.

    02:43 Just where is our particle.

    02:44 But then in a vertical axis where we are plotting is the probability or something in fact related to the probability, whether the particle is at that particular location.

    02:53 The reason I say it's related to the probability, but you can just think of it as probabilities, is that these negative numbers, when we go below the axis in this curve, doesn't mean that there's a negative probability.

    03:04 So that's just a brief caviar.

    03:06 Just so that you are comforted that we're not talking about negative probabilities.

    03:10 The way we would actually calculate the probability is something related to squaring this wave.

    03:15 So we would only will have positive numbers.

    03:17 So again that's a brief detail sort of a caviar.

    03:19 But you can think of this in general as just the probability wave for your particle.

    03:25 The strange thing is what if we do assume that we have this probability wave.

    03:30 What happens to other aspects that we can measure about the particle.

    03:34 It turns out also for reason that we can't discuss.

    03:37 So this slide is the sort of assumptions that I'm going to have to ask you to make.

    03:41 That the momentum of the particle, how fast it's moving through space is related to the wavelength of the uncertainty wave.

    03:49 So really just looking at this wave there are two things that you should be familiar with.

    03:54 One is that this particle has a particular position which is spread out in space because we don't really know where it is.

    04:00 So we give it probability of being in certain locations.

    04:03 And two is that the wavelength of this wave tells us something about the momentum of the particle.

    04:08 Again for more complicated and in depth reasons that we won't be getting into here.

    04:13 So with this position and momentum and with these starting assumptions that we begin with, we can ask ourselves how the position and momentum would be related if we try to measure the two.

    04:24 If we had some wave like this one, we would of course have complete uncertainty in where our particle was.

    04:31 Because look at this wave, it could be at this location or this location, or this location.

    04:36 It could really just be anywhere.

    04:37 Specially since we're assuming that this wave would just keep going off the page to the right and off the page to the left.

    04:42 We just have absolutely no clue where this particle is.

    04:46 On the other hand, the distance between the two peaks of the waves as I've drawn it here, is completely determined.

    04:54 There's this wave has no ambiguity in the distance between the two peaks.

    04:59 So in other words the wavelength.

    05:00 So we have complete uncertainty in where this particle is.

    05:04 Which is why we draw this probability curve.

    05:07 But we have complete certainty about what the wavelength of the wave is.

    05:12 And therefore, we have complete certainty about the momentum of this particle.

    05:15 On the other hand, what is if reverse this situation.

    05:18 What if I said, I was uncertain about the wavelength of the wave.

    05:22 What if I did not know what the momentum was.

    05:25 In that case all of these waves that are drawn at the top here would be possibilities.

    05:29 As each one has a slightly different wavelength than the other ones.

    05:32 And so each wavelength is a possible wavelength.

    05:35 And again we don't have certainty.

    05:37 We don't know which of these wavelengths it could be.

    05:39 The interesting thing is that if we don't know what the wavelength of our wave is, so we have all these different possibilities for the wavelength of our wave.

    05:48 Look what happens when we add up, we simply take all of these top waves on the top here and add them up point by point using the same kind of mathematics we discussed when we were talking about interference with sound waves.

    05:58 We just add them up point by point.

    06:00 You can see that right at the middle here, where all the waves coincide with each other, if you add them all up, you will just get a much bigger number.

    06:08 Because they are all the same position, they are all positive numbers.

    06:12 On the other hand, the tails of the waves, near the far edges of the wave, because they are different wavelength, those different frequencies, they've spread out from each other.

    06:20 And they start becoming less and less organized.

    06:22 So as you go towards the tails of this wave, all of your waves are disorganized and some might be positive and some might be negative.

    06:29 And on the whole on the average if you add up all the waves at that particular point on the outside of your wave, as many will be positive as there are negative.

    06:39 And so you'll end up with just a number close to 0.

    06:41 In this effect where they are becoming less and less organized as you go out, causes the sum of all of these waves to be constructive in the middle.

    06:50 And more and more destructive as you go out.

    06:52 So that it levels out towards the edges.

    06:54 The really interesting thing about this behavior, is that when we added all these together we now have more certainty in where are particle is.

    07:02 So just by giving ourselves some uncertainty about the momentum or the wavelength of our wave, we've put them together and found that we now sort of localized our particle to a particular location because it can't so probably be at these very far distances as we had with our first wave.

    07:18 So putting these ideas together we now have what's called "A Wave Packet." We say that this sort of centralized wave packet is in fact what we experience in our sort of macroscopic, large scale world as particles.

    07:32 A very localized wave which can still exhibit some of these probabilistic behaviors.

    07:37 But for our intensive purposes acts more like a classical particle.

    07:42 So putting this together in an equation what we would say is that the delta a x or the uncertainty in our positioned or the uncertainty in x, times the uncertainty in momentum have to be greater than or equal to some constant number.

    07:57 And we already discussed this number is 'h' bar.

    08:00 It's a letter 'h' with a bar through the vertical line.

    08:03 And this is called the Reduced Planck's constant.

    08:06 We introduced the Reduced Planck's constant before but it is simply Planck's constant h divided 2 pi.

    08:12 And don't forget the actual value of Planck's constant which is a very, very small number.

    08:17 The 6.6 times 10 to the minus -34th.

    08:21 And this gives us sort of an energy scale or a size scale that would see quantum effects.

    08:25 The fact that this number is so small is exactly the reason that we haven't noticed quantum effects historically until relatively recently.

    08:32 But notice what the uncertainty equation is telling you about the momentum and the position of your particle.

    08:38 If the product of these two uncertainties has to be greater than or equal to a particular value, that means we can't pinpoint both of them together.

    08:46 So in another words as we saw and explored if my uncertainty in x is very, very small.

    08:52 That must mean my uncertainty in momentum by delta p is very big.

    08:56 Because we have this requirement that these numbers are greater than a particular threshold.

    09:01 And by the same token for very certain in our momentum meaning the delta p is very small because we don't have any uncertainty.

    09:08 Delta x must compensate and be a very big value to again ensure that this product of the uncertainty in the post ion and the uncertainty in the momentum is still greater than this fixed value h bar.

    About the Lecture

    The lecture Effective Nuclear Charge by Jared Rovny is from the course Electronic Structure.

    Included Quiz Questions

    1. The electrons present in the inner orbitals repel the electrons occupying the outer orbitals.
    2. Some nuclei have an effect from the neutrons in the nucleus, which causes a slightly changed charge.
    3. Each electron orbital has interference between the two different electron spins.
    4. The energy of the electron energy level can interfere with the orbit of the electron.
    5. The motion of an electron in orbit causes a slight change in its real charge due to relativity.
    1. The inherent uncertainty in observable quantities of a particle, like its position or location.
    2. The experimental uncertainty in observable quantities of a particle, due to the constant inadequacy of our measurement tools.
    3. The uncertainty about which objects we are observing when we look at minuscule structures.
    4. The uncertainty in what a particle is before it has been measured.
    5. The quantum mechanical uncertainty associated with the many types of fundamental particles and their probabilities.
    1. The wavelength of the wave.
    2. The height of the wave.
    3. The number of waves available.
    4. The square of the wave at each point.
    5. The phase of the wave.
    1. When we know its momentum (wavelength), we can no longer localize the position.
    2. When we know its location (wavelength), we can no longer constrain the momentum to a particular place.
    3. When we know its momentum (probability), we can no longer localize the wavelength (which encodes location).
    4. When we know its location (probability), we can no longer adequately measure the height of the wave.
    5. Momentum and location are fractional inverses of each other (mathematically); so that if one becomes zero, the other is unknowable.
    1. Planck’s constant is very small.
    2. Planck’s constant is very big.
    3. Planck’s constant was not measured with enough accuracy.
    4. Planck’s constant is always greater than the uncertainty, which had not been noticed before.
    5. Planck’s constant, when reduced, cannot be known without measuring the uncertainty principle.

    Author of lecture Effective Nuclear Charge

     Jared Rovny

    Jared Rovny

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