 So, when we talk about quantum mechanical energy levels, we're never very far away from talking about spectroscopy, and spectroscopy is the idea of using light of some type to induce the transition between systems in different energy levels, or vice versa, using a transition between those energy levels to generate light. So, what we mean by that is we have a couple of quantum mechanical energy levels on a ladder. So, the system can have this energy or this energy, there's some difference in energy between those states, then if I have a system that's in the lower state, and I bring in a photon, some light with some frequency, or so wavelength, the energy of that photon, if it matches the energy difference between these two states can induce the transition and lift the system up to the other state, or contrary-wise, if the system is in the upper state and it falls down to the lower state losing this much energy, then it can produce a photon of that same frequency or wavelength. So, what we know about spectroscopy is the difference in energy between two states is equal to h times the frequency of a photon that gets generated or absorbed, equal to hc over lambda, where lambda is the wavelength of the photon that gets created or absorbed. So, we also know that there's multiple different frequencies and wavelengths that can be generated, not only in the visible portion of the spectrum, we've seen for molecules that behave like particles in boxes, like perhaps electrons in conjugated hydrocarbons, we can use the differences between these states to generate light of frequencies or wavelengths in the visible portion of the spectrum for molecules like beta-carotene, for example, or perhaps in the ultraviolet part of the spectrum for molecules that are a little bit smaller. But those are not the only two portions of the electromagnetic spectrum, so we've got ultraviolet, ultraviolet, which we often abbreviate as UV, and the visible portion of the spectrum, the wavelengths, we know visible photons typically have wavelengths of about 400 to 900 nanometers somewhere in that range. The edges of this range are a little bit fuzzy, depending on how good your eyes are, you might be able to see a light deeper into the ultraviolet, deeper past the low wavelength edge of the spectrum, and then if your eyes are less good, but generally speaking humans can see light in the range of 400 to 700 nanometers or so. The light has lower wavelengths, so 400 nanometers is beyond the violet end of the spectrum, lower wavelength, so that goes down, generally speaking, and again this is not a firm boundary, but down to roughly a nanometer or so. But on the other end of the spectrum, off the long wavelength end of the visible spectrum, if I make the light less energetic, longer wavelength, that's beyond the red portion of the spectrum rather than the violet portion of the spectrum, we call that spectrum the infrared portion of the spectrum. So here we're talking about light with wavelengths longer than 700 nanometers, and that goes anywhere up to micrometers, all the way up to perhaps a millimeter or so, we would call those photons infrared, and then at some fairly arbitrary boundary, roughly a millimeter or so, we get into a different named region of the electromagnetic spectrum that we call the microwave portion of the spectrum. I'm going to abbreviate that as MW for microwave, and that's, we could talk about the wavelengths of microwave photons, but these for various reasons are not very commonly referred to in wavelengths. And to more typically talk about the wavelengths, talk about the energies of these microwave photons not related to their wavelength, but related to their frequencies. So the frequency of microwave photons is somewhere in the maybe low gigahertz up to hundreds of gigahertz range. So we can of course, that frequency correlates to some wavelengths, so that's somewhere down in the, roughly speaking, the millimeters to hundreds of millimeters range or so. Likewise, we could calculate the frequency associated with these ultraviolet photons or visible or infrared photons, but it turns out the most convenient units to talk about not just microwave photons, but several different portions of the spectrum are not actually either wavelength or frequency, but a different term. So we'll introduce that term now. So just like we can say the energy of this photon is h times the quantity we call its frequency or hc over its wavelength, we're just finding a new quantity now, new with a tilde on top of it. If I define this quantity as the energy divided by h times c, then the quantity I've defined here is called the wave number. And by comparing some of these equations, we can see why it's called the wave number. If we compare, let's say, the wavelength version of this equation, hc times one over lambda, that's equal to hc times the wave number. So the wave number is equal to, by comparing these two equations, is equal to one over the wavelength. So we know what a wavelength is. A wavelength is the length of a wave. It's the peak to peak or trough to trough distance of the wave. It tells you how many units of distance there are in the length of one wave. That's why we call it a wavelength. If we turn that fraction upside down, instead of saying length divided by the number of waves, we have waves divided by the length. So what that tells us is the wave number, how many waves there are per unit length. So instead of saying the wavelength is five meters, we would say the number of waves per meter is a fifth of a wave per meter. So that's the definition of wave number. And if we know the wave number, we can use that to calculate the energy. Likewise, we can convert back and forth between frequencies, wavelengths, and wave numbers anytime we want. And as I mentioned, spectroscopists, people who are in the business of shining light on molecules or chemical systems and using that to induce transitions or letting molecules fall down from upper states to lower states, generating light and then measuring the properties of the light that gets generated. They most commonly talk about light in some portions of the spectrum in terms of its wave number instead of its wavelength or frequency. So we need to get a little bit familiar with wave numbers. So first let's work a quick example. For a photon that we're maybe somewhat comfortable with the units for, let's say we have a visible photon just on the edge of the visible, edge of the spectrum, 400 nanometers in wavelength. If I have a photon with wavelength, 400 nanometers. So we can say that's also just on the border of the ultraviolet. It's on the very violet edge of the visible spectrum. I'd like to know what is the wave number associated with that photon. So that's just one over the wavelength according to our definition. So one over 400 nanometers, one over 400 times 10 to the minus ninth meters. Four times 10 to the minus seventh. When I do that division, I get 2.5 times 10 to the sixth. And now my units are one over meters. So the wave number, the number of these waves, these visible photons that have a wavelength of 400 nanometers. So they oscillate up and down. The electric field is oscillating back and forth with the peak to peak distance, only 400 nanometers or so. So the wave number is telling me that 2.5 times 10 to the sixth, 2.5 million of those waves fit inside one meter. So if I mark out a meter, then the electromagnetic field will oscillate up and down 2.5 million times over the course of that meter. So that's what the wave number is telling us. We most commonly measure the wave number not in units of one over meters. It's going to have to always be a one over some length. But if I say that there's 100 centimeters in one meter, that will convert my units of meters into centimeters. So dividing this number by 100, I get 2.5 times 10 to the fourth inverse centimeters. So same number, but instead of oscillating 2.5 million times in a meter, it oscillates 25,000 times in the length of one centimeter. So that's the wave number associated with this wavelength. And now we've learned the boundaries of these two edges of the ultraviolet and the visible portion of the ultraviolet, yes, ultraviolet and visible spectra. So ultraviolet ranges from 25,000 centimeters, inverse centimeters and higher for its wavelength, I'm sorry, for its wave number up to about a million inverse centimeters or so. Visible light, 25,000 on the upper end. If we did that same calculation with 700 nanometers, we'd find that a red photon has a wave number of about 14,000 centimeters. And if we proceed down to the infrared and the microwave portions of the spectrum, do that same calculation. We'll find that an infrared photon with wavelength of about a millimeter has a wave number of about 10 inverse centimeters. And this is a pretty broad range covering several orders of magnitude, ranges from 10 up to 14,000 inverse centimeters. And then microwave portion of the spectrum, we're at about tenths to roughly 10 or 100 or so inverse centimeters would be the microwave portion of the spectrum. Actually, that's not quite right. Microwave portion of the spectrum can easily drop as low as hundreds of an inverse centimeter. So as we mentioned, for microwave spectroscopy in particular, it's much more common to talk about numbers, the wave numbers of the photons rather than the frequencies of the wavelengths. In part because these are relatively convenient numbers. In fact, for every region of the spectrum, it's not too difficult to talk about wave numbers that have ranges from a few hundreds or a few tenths up to a few thousands of wave numbers. But it's a fairly convenient unit that doesn't require us to talk about 10 to the ninth inverse seconds or something like that. So this introduces the idea of wave number. And just give some stats on the different portions of the electromagnetic spectrum that we can use when talking about spectroscopy for different types of quantum mechanical problems.