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step 1 Okay, hello everybody. So for this video, I'm going to be talking about problem 6 .9. And in thi

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Precision Measurement and Uncertainty Analysis

Precision measurement and uncertainty analysis are critical in comparing theoretical predictions with experimental data. In atomic spectroscopy, using high precision data and carefully accounting for standard uncertainties allows for stringent tests of quantum theories. The evaluation of uncertainties ensures that discrepancies between theory and experiment are meaningful, guiding improvements in both experimental technique and theoretical modeling.

Atomic Energy Levels

Atomic energy levels are the quantized states of electrons bound to an atomic nucleus. These discrete energy states arise from the solutions to the Schrödinger (or Dirac) equation for electrons in an atom, and their values determine the wavelengths of light that atoms can absorb or emit during electronic transitions. The precise measurement and calculation of these levels are fundamental to understanding atomic structure and spectroscopy.

Fine-Structure Splitting

Fine-structure splitting refers to small differences in the energy levels of atomic states that arise from relativistic corrections to the non-relativistic quantum mechanical model. These include contributions from spin-orbit coupling, relativistic corrections to the electron's kinetic energy, and the Darwin term. Fine-structure splits energy levels into closely spaced sub-levels, which are observable in high-resolution spectroscopic experiments.

Lamb Shift

The Lamb shift is the small difference in energy between two levels (typically degenerate in the Dirac theory) due to quantum electrodynamics corrections. It results from interactions between the electron and vacuum fluctuations as well as the self-energy of the electron. The Lamb shift is historically important as it provided critical experimental evidence for quantum electrodynamics (QED), highlighting the need for corrections beyond the Dirac equation.

Transition Energy Calculations

Transition energy calculations involve determining the energy differences between quantized atomic states. These calculations are essential for predicting the spectral lines emitted or absorbed during electron transitions. Accurate modeling of these transitions requires incorporating both the non-relativistic energy levels and additional corrections such as fine-structure and Lamb shift contributions.

Transcript

00:01 Okay, hello everybody.

00:08 So for this video, i'm going to be talking about problem 6 .9.

00:17 And in this problem, we're basically going to be estimating the partition function of a hydrogen atom at a given temperature.

00:29 Okay, so in other words, we have a system which is a hydrogen.

00:34 Atom that is in thermal contact with some reservoir at a particular temperature and we want to calculate the partition function of the hydrogen atom.

00:48 Okay, so let's go ahead and do part a.

00:53 So for part a, we're given a picture.

00:59 I'm assuming that all of you guys can see the picture, so i'm just going to draw a the picture roughly like this, in units of electron volts.

01:10 We have, on the bottom, we have the energy spectrum of a hydrogen atom.

01:16 I'm sure you guys all know this, but i'm just going to draw it out like this.

01:20 This is one, there's one energy state at negative 13 .6.

01:28 And let's see.

01:30 And there's at negative 3 .4 electron volts, we have 1, 2, 3, 4, and at negative 1 .5 something, we have 1, 2, 3, 4, 5, 6, 7, 8, 9.

01:50 Okay, so this is the energy spectrum of a hydrogen atom, where each line represents a state.

02:00 So for the lowest, the ground state, there's only one state.

02:06 The degeneracy is one.

02:09 For the first excited state, we have four.

02:13 And for the third, excuse me, for the second excited state, there's nine of them.

02:19 Okay, that's the degeneracy.

02:21 Okay, so the partition function is very simple.

02:24 It is a function of temperature.

02:27 For those of you who are not very familiar with the letter tau, it just, it's just notation for boltzman constant times temperature, where temperature t is in si units, which is kelvin units, okay? so z of tau is just going to be a sum, excuse me, it's just going to be a sum over states of boltzman factors, which i'm sure you guys know to be, an exponential of negative energy over temperature.

03:04 So that's all.

03:05 That's the boltzmann factor.

03:08 Now, for part a, we're specifically instructed to only consider the first three energy states, which are the states that you see in this picture.

03:22 So in other words, we're going to ignore all the higher level, higher, energy states.

03:29 We're going to ignore those.

03:30 And there's a reason why we can ignore those.

03:33 But we're not going to talk about that reason until later.

03:35 So for now, let's just calculate that.

03:38 Okay, so we have, let me first of all write this as an explicit sum instead of using the sigma notation.

03:48 Let me just write out the sum.

03:50 We have e to the negative, negative 13 .6 electron volts over tau.

04:01 Right plus excuse me one second guys there's a there's a slight mistake here i forgot to include a factor for for multiplicity okay because as you know each energy state has a multiplicity there could be multiple states with the same energy okay that's what degeneracy is so the g factor this factor or g here is the degeneracy okay okay, so in other words, for example, the ground state has a degeneracy of one, as you can clearly see in the picture above.

04:39 Now, for the first excited state, there's four.

04:42 There's four states.

04:45 So that's the degeneracy factor times e to the negative 3 .4 electron bolts over a towel, plus 9.

04:59 E to the negative negative 1 .5 electron volts over tau.

05:06 Okay, so that's the partition function.

05:11 I have, at this point, i have given you all there is to be known about the partition function for part a.

05:19 Like, given this expression that i have written down, you can go ahead and plug everything into your calculator.

05:25 And you're going to get a number.

05:26 But let me just for the sake of it, let me just go ahead and simplify and beautify this expression a little bit so that you can kind of see sort of a pattern, so to speak.

05:42 So let me go ahead and define epsilon to be 13 .6 electron volts.

05:50 Okay.

05:52 So now, so that's the ground state energy.

05:56 I can write this expression for z as e to the negative, negative epsilon over tau.

06:04 You see, that's prettier than the numbers, right? but anyway, plus four times e to the negative, negative epsilon over four over tau.

06:18 You see, that's much cleaner.

06:20 That's much cleaner.

06:22 And the reason i can write it like this is because i'm sure you know that hydrogen, energy goes like e1 over n squared, right? i'm sure most of you know this.

06:32 And if you don't know this, you can go ahead and just google a hydrogen energy spectrum, and you're going to see the hydrogen energies go like 1 over n squared times the energy of the first, excuse me, times the energy of the ground state.

06:47 Anyway, and you can clearly see that 3 .4 times 4 is equal to 13 .6 electron volts.

06:54 So this is correct.

06:57 So plus 9 times e to the negative, e, excuse me, epsilon over 9 over tell, okay? because 1 .5 times 9 is 13 .6.

07:16 Approximately, of course.

07:18 So this is equal to, let me write it a little bit simpler like this, e to the epsilon over tau plus 4 e to the epsilon over 4 tau plus 9 e to the power of epsilon over 9 tau.

07:41 Okay, so you can see this is a much cleaner expression than the one that we previously got.

07:49 Although it's the same thing, but you know, it's just cleaner.

07:55 So now i can write this even cleaner.

08:00 I can use this, sigma notation, where we sum from n equals 1 to 3, n squared, e to the epsilon over n squared tau.

08:20 You can see this is a very, very clean, expression and you can go ahead and calculate the partition function.

08:31 If you plug all the numbers in, you're going to get a number.

08:35 Excuse me, if you plug all the numbers into your calculator, you're going to get a number, and that number i'm going to leave it to you as a homework exercise.

08:42 By the way, the tau, oh, i think i told you about tau already.

08:47 Tau is just, tau is just shorthand notation for k -boltzman times the temperature.

08:55 Okay, so that's a lot for part of a.

08:58 Let's go ahead and go to part b.

09:07 So part b wants us to calculate the partition function, but not like the way we did in part a, where we ignored all of the higher order terms.

09:20 In other words, we want the partition function that includes literally all of the bound states, like all of them.

09:27 In part a, we only had like this many.

09:32 But we want all of them, okay? so let's explicitly write it out.

09:38 So z is just going to be as a sum of n equals 1 to 3, but not 3 anymore because we want all of them.

09:49 We want infinity, n squared e to the epsilon over n squared tau.

09:57 This is basically the same expression as the one we got before for part a, the only difference is instead of summing over one, the first state, second state, and third state, we're summing over all of them now.

10:14 Okay? so that's the only difference.

10:16 Okay, that's the only difference.

10:19 And let's see what else we got to do for part b.

10:25 Well, so we want to show, we want to show that this partition function diverges.

10:33 It's infinite.

10:33 And it clearly it's infinite.

10:38 Let me show you why it's clearly infinite.

10:41 First of all, let's look at this epsilon.

10:43 Epsilon is 13 .6.

10:46 It's a number and it's a greater than zero number.

10:49 Okay, it's a positive number.

10:51 And tau, tau is temperature.

10:54 Temperature is always going to be a positive number.

10:58 In most cases.

11:01 And so as you can, so, so, this factor here, if you look at it, if you stare at it very, very hard, you're going to see that when n goes to infinity, this factor basically just goes to e to the power of zero, which is, which is one as as n goes to infinity.

11:22 Okay.

11:24 So in other words, at very, very high powers, uh, excuse me, at very, um, uh, high values of n, this exponential, factor drops out.

11:35 It's just one.

11:37 So, so, so, so z just becomes, it just becomes a sum over n squared at very high values of n.

11:45 So it clearly diverges.

11:47 It clearly does.

11:50 So that's all for part b.

11:52 Okay.

11:53 Now let's go ahead and go to part c.

12:02 So, so part c wants us to go back to equation 6 .3 and argue that pdv term is not negligible for the very high -end states, and therefore the result of part a, not that of part b, gives the physically relevant partition function for this problem.

12:20 Well, i'm not going to copy down equation 6 .3.

12:25 I'm assuming that you know what equation 6 .3 is, so i'm just going to give you a verbal argument for why it is that part a gives the physical.

12:35 Relevant partition function.

12:38 Well, first of all, as you remember, the way the way the book derived the partition function is that basically the author used the thermodynamic identity where ds change in entropy is equal to 1 over temperature d -u plus pdv, okay, plus a mudn, but we're ignoring changing particle numbers.

13:13 So the way the partition function is derived is that the book basically just ignored this term, and that's where the epsilon over tau, e to the epsilon over tau, that's where this factor comes from...

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