Section 34 of Emily of New Moon by Lucy M Montgomery. This LibriVox recording is in the public domain. Recording by Leanne Fortune. Section 34. Emily's convalescence was rather slow. Physically she recovered with normal celerity, but a certain spiritual and emotional langer assisted for a time. One cannot go down to the depths of hidden things and escape the penalty. Montelisabeth said she moped, but Emily was too happy and contented to mope. It was just that life seemed to have lost its saver for a time, as if some spring of vital energy had been drained out of it, and refilled slowly. She had just then no one to play with. Perry, Ilsa and Teddy had all come down with measles the same day. Mrs. Kent at first declared bitterly that Teddy had caught them at New Moon, but all three had contracted them at a Sunday school picnic where dairy pond children had been. That picnic infected all blay water. There was a perfect orgy of measles. Teddy and Ilsa were only moderately ill, but Perry, who had insisted on going home to Aunt Tom at the first symptoms, nearly died. Emily was not allowed to know his danger until it had passed, lest it worry her too much. Even Aunt Elizabeth worried over it. She was surprised to discover how much they missed Perry around the place. It was fortunate for Emily that Dean Priest was in blay water during this full-on time. His companionship was just what she needed, and helped her wonderfully on the road to complete recovery. They went for long walks together all over blay water, with tweed woofing around them and explored places and roads Emily had never seen before. They watched a young moon grow old night by night. They talked in dim, centered chambers of twilight over long red roads of mystery. They followed the lure of hill winds. They saw the stars rise, and Dean told her all about them. The great constellations of the old myths. It was a wonderful month, but on the first day of Teddy's convalescence, Emily was off to the tansy patch for the afternoon, and Jarback Priest walked, if he walked at all, alone. Aunt Elizabeth was extremely polite to him, though she did not like the priests of Priest Pond over much, and never felt quite comfortable under the mocking gleam of Jarback's green eyes and the faint derision of his smile, which seemed to make Murray pride and Murray tradition seem much less important than they really were. He has the priest flavour, she told Laura, though it isn't as strong in him as in most of them, and he's certainly helping Emily. She has begun to spunk up since he came. Emily continued to spunk up, and by September, when the measles epidemic was spent, and Dean Priest had gone on one of his sudden swoops over to Europe for the autumn, she was ready for school again, a little taller, a little thinner, a little less childlike, with great gray shadowy eyes that had looked into death and read the riddle of a buried thing, and henceforth would hold in them some haunting, elusive remembrance of that world behind the veil. Dean Priest had seen it, Mr Carpenter saw it when she smiled at him across her desk at school. She's left the childhood of her soul behind, though she is still a child in body, he muttered. One afternoon, amid the golden days and hazes of October, he asked her gruffly to let him see some of her verses. I never meant to encourage you in it, he said. I don't mean it now. Probably you can't write a line of real poetry and never will, but let me see your stuff. If it's hopelessly bad, I'll tell you so. I won't have you wasting years driving for the unattainable. At least I won't have it on my conscience if you do. If there's any promise in it, I'll tell you so just as honestly, and bring some of your stories too. They're trash yet, that's certain, but I'll see if they show just and sufficient cause for going on. Emily spent a very solemn hour that evening, weighing, choosing, rejecting. To the little bundle of verses she added one of her Jimmy books which contained, as she thought, her best stories. She went to school next day, so secret and mysterious, that Ilse took offence, started in to call her names, and then stopped. Ilse had promised her father that she would try to break herself of the habit of calling names. She was making fairly good headway, and her conversation, if less vivid, was beginning to approximate two new moon standards. Emily made a sad mess of her lessons that day. She was nervous and frightened. She had a tremendous respect for Mr. Carpenter's opinion. Father Cassidy had told her to keep on. Dean Priest had told her that someday she might really write, but perhaps they were only trying to be encouraging because they liked her and didn't want to hurt her feelings. Emily knew Mr. Carpenter would not do this. No matter if he did like her, he would nip her aspirations mercilessly if he thought the root of the matter was not in her. If, on the contrary, he bade her godspeed, she would rest content with that against the world and never lose heart in the face of any future criticism. No wonder the day seemed fraught with tremendous issues to Emily. When schools out Mr. Carpenter asked her to remain, she was so white intense that the other pupils thought she must have been found out by Mr. Carpenter in some especially dreadful behavior and knew she was going to catch it. Rhoda Stewart flung her a significantly malicious smile from the porch, which Emily never even saw. She was indeed at a momentous bar with Mr. Carpenter as supreme judge and her whole future career, so she believed, hanging on his verdict. The pupils disappeared and a mellow, sunshiny stillness settled over the old schoolroom. Mr. Carpenter took the little packet she had given him in the morning out of his desk, came down the aisle and sat in the seat before her, facing her. Very deliberately he settled his glasses astride his hooked nose, took out her manuscripts and began to read, or rather to glance over them, flinging scraps of comments mingled with her hands, sniffs and hoots at her as he glanced. Emily folded her cold hands on her desk and braced her feet against the legs of it to keep her knees from trembling. This was a very terrible experience. She wished she had never given her verses to Mr. Carpenter. They were no good. Of course they were no good. Remember the editor of the Enterprise? Hinf! said Mr. Carpenter. Sunset, Lord! How many poems have been written on sunset? The clouds are massed in splendid state, at Heaven's unbored western gate, where troops of star-eyed spirits wait. By gad, what does that mean? I, I don't know. Faulted, startled Emily, whose wits had been scattered by the sudden sweep of his spiked glance, Mr. Carpenter snorted. For Heaven's sake, girl, don't write what you can't understand yourself, and this to life. Life as thy gift I ask no rainbow joy. Is that sincere? Is it, girl? Stop and think. Do you ask? Do you ask? No rainbow joy of life? He transfixed her with another glare, but Emily was beginning to pick herself up a bit. Nevertheless, she suddenly felt oddly ashamed of the very elevated and unselfish desires expressed in that sonnet. No! she answered reluctantly. I do want rainbow joy. Lots of it. Of course you do. We all do. We don't get it. You won't get it.  Lines to a mountain cascade. On its dark rocks lack the whiteness of a veil around a bride. Where did you see a mountain cascade in Prince Edward Island? No where. There's a picture of wine in Dr. Booneley's library. A wood stream. The threading sunbeams quiver. The bending bushes shiver. Or the little shadowy river. There's only one more rhyme that occurs to me and that's liver. Why did you leave it out? Emily writhed. Winsong. I have shaken the dew in the meadows from the clover's creamy gown. Pretty but weak. June. June, for heaven's sake girl, don't write poetry on June. It's the sickliest subject in the world. It's been written to death. No. June is immortal, cried Emily, suddenly a mutinous sparkle replacing the strained look in her eyes. She was not going to let Mr. Carpenter have it all his own way. But Mr. Carpenter had tossed June aside without reading a line of it. I weary of the hungry world. What do you know of the hungry world? You, in your new moon's occlusion of old trees and old maids. But it is hungry. Ode to winter. The seasons are a sort of disease all young poets must have, it seems. Ha! Spring will not forget. That's a good line. The only good line in it. Hmm. Wanderings. I've learnt the secret of the rune that the somber pines on the hillside croon. Have you? Have you learnt that secret? I think I've always known it, said Emily dreamily. That flash of unimaginable sweetness that sometimes surprised her had just come and gone. Aim and endeavour. Too didactic. Too didactic. You've no right to try to teach until you're old and then you won't want to. Her face was like a star all pale and fair. Were you looking in the glass when you composed that line? No. Indignantly. When the morning light is shaken like a banner on the hill. A good line, a good line. Oh! On such a golden morning to be living his delight. Too much like a faint echo of Wordsworth. The sea in September. Blue and austerely bright. Child, how can you marry the right adjectives like that? Morning. All secret fears that haunt the night. What do you know of the fears that haunt the night? I know something, said Emily decidedly, remembering her first night at Withergrange, to a dead day with the chilly calm on her brow. That only the dead may wear. Have you ever seen the chilly calm on the brow of the dead, Emily? Yes, said Emily softly, recalling that grey dawn in the old house in the hollow. I thought so, otherwise you couldn't have written that. And even as it is.  You should study the art of titles, Emily. There's a fashion in them, as in everything else. Your titles are as out of date as the candles of New Moon. Soundly he sleeps with his red lips pressed, like a beautiful blossom, close to her breast. The rest isn't worth reading. September. Is there a month you've missed? Windy meadows, harvest deep, good line. Play water by moonlight. Gossamer, Emily, nothing but Gossamer. The Garden of New Moon. Beguiling laughter and old song of merry-mades and men. Good line, I suppose New Moon is full of ghosts. Deaths fell minion well for full of its part. That might have passed in Addison's day, but not now, not now, Emily. Your azure dimples are the graves where million-buried sunbeams play. Atrocious, girl! Atrocious! Graves aren't playgrounds. How much would you play if you were buried? Emily writhed and blushed again. Why couldn't she have seen that herself? Any ghost could have seen it. Sail onward, ships, white wings, sail on till past the horizon's purple bar. You drift from sight in flush of dawn, sail on, and neath the evening star. Trash, trash, and yet there's a picture in it. Lap softly, purple waves I dream, and dreams are sweet. I'll wake no more. Ah! But you'll have to wake if you want to accomplish anything. Girl, you've used purple twice in the same poem. Buttercups in a golden frenzy. A golden frenzy, girl, I see the wind shaking the buttercups. From the purple gates of the west I come. You're too fond of purple, Emily. It's such a lovely word, said Emily.  Emily, the luring voice of the echo of fame. So you've heard it too. It is a lure, and for most of us only an echo, and that's the last of the lot. Mr. Carpenter swept the little sheets aside, folded his arms on the desk, and looked over his glasses at Emily. Emily looked back at him mutely, nervously. All the life seemed to have been drained out of her body, and concentrated in her eyes. Ten good lions out of four hundred, Emily. Comparatively good, that is. And all the rest, balderdash, balderdash, Emily. I suppose so, said Emily faintly. Her eyes brimmed with tears. Her lips quivered. She could not help it. Pride was hopelessly submerged in the bitterness of her disappointment. She felt exactly like a candle that somebody had blown out. What are you crying for? demanded Mr. Carpenter. Emily blinked away the tears and tried to laugh. I—I'm sorry. You think it's no good? she said. Mr. Carpenter gave the desk a mighty thumb. No good. Didn't I tell you there were ten good lions? Jade, for ten righteous men Sodom had been spared. Do you mean that, after all, the candle was being relighted again? Of course I mean, if at thirteen you can write ten good lions, at twenty you'll write ten times ten, if the gods are kind.  I think there's something trying to speak through you, but you'll have to make yourself a fit instrument for it. You've got to work hard and sacrifice. By Gadgirl you've chosen a jealous goddess, and she never lets her votaries go, not even when she shuts her ears forever to their plea. What have you there? Emily, her heart thrilling, handed him her gemmy-book. She was so happy that it shone through her whole being with a positive radiance. She saw her future. Wonderful! Brilliant! Oh, her goddess would listen to her. Emily B. Starr, the distinguished poet, E. Bird Starr, the rising young novelist. She was recalled from her enchanting reverie by a chuckle from Mr. Carpenter. Emily wondered a little uneasily what he was laughing at. She didn't think there was anything funny in that book. It contained only three or four of her latest stories, The Butterfly Queen, A Little Fairy Tale, The Disappointed House, wherein she had woven a pretty dream of hopes come true after long years. The Secret of the Glen, which, in spite of its title, was a fanciful little dialogue between the spirit of the snow, the spirit of the grey rain, the spirit of mist, and the spirit of moonshine. So you think I am not beautiful when I say my prayers? said Mr. Carpenter. Emily gasped, realized what had happened, made a frantic grab at her Jimmy Book, missed it. Mr. Carpenter held it up beyond her reach and mocked at her. She had given him the wrong Jimmy Book, and this one, oh, horrors, what was in it, or rather what wasn't in it, sketches of everyone in playwater, and a full, a very full description of Mr. Carpenter himself. Intent on describing him exactly, she had been as mercilessly lucid as she always was, especially in regard to the odd faces he made, on mornings when he opened the school day with a prayer. Thanks to her dramatic knack of word painting, Mr. Carpenter lived in that sketch. Emily did not know it, but he did. He saw himself as in a glass, and the artistry of it pleased him so that he cared for nothing else. Besides, she had drawn his good points quite as clearly as his bad ones, and there were some sentences in it. He looks as if he knew a great deal that can never be any use to him. I think he wears the black coat Mondays because it makes him feel that he hasn't been drunk at all. Who or what had taught the little J these things? Oh, her goddess would not pass Emily by. I'm sorry, said Emily, crimson with shame all over her dainty paleness. Why, I wouldn't have missed this for all the poetry you've written or ever will write. By God, it's literature, literature, and you're only thirteen, but you don't know what's ahead of you, the stony hills, the steeper scents, the buffets, the discouragements. Stay in the valley if you're wise. Emily, why do you want to write? Give me your reason. I want to be famous and rich, said Emily Cooley. Everybody does. Is that all? No, I just love to write. A better reason, but not enough, not enough. Tell me this. If you knew you would be poor as a church mouse all your life, if you knew you'd never have a line published, would you still go on writing? Would you? Of course I would, said Emily disdainfully. Why, I have to write. I can't help it by times. I've just got to. Oh, then I'd waste my breath giving advice at all. If it's in you to climb, you must. There are those who must lift their eyes to the hills. They can't breathe properly in the valleys. God help them if there's some weakness in them that prevents their climbing. You don't understand a word I'm saying yet. But go on. Climb. There. Take your book and go home. Thirty years from now I will have a claim to distinction in the fact that Emily Bird Star was once a pupil of mine. Go, go before I remember what a disrespectful baggage you are to write such stuff about me and be properly enraged. Emily went still a bit scared but oddly exultant behind her fright. She was so happy that her happiness seemed to irradiate the world with its own splendor. All the sweet sounds of nature around her seemed like the broken words of her own delight. Mr. Carpenter watched her out of sight from the old worn threshold. Wind and flame and sea he muttered. Nature is always taking us by surprise. This child has what I never had and would have made any sacrifice to have but the gods don't allow us to be in their debt. She will pay for it. She will pay. At sunset Emily sat in the lookout room. It was flooded with soft splendour. Outside in the sky and trees were delicate tintings and aerial sounds. Down in the garden Daffy was chasing dead leaves along the red walks. The sight of his sleek striped sides, the grace of his movements gave her pleasure as did the beautiful even glossy furrows of the plowed fields beyond the lane and the first faint white star in the crystal green sky. The wind of the autumn night was blowing trumpets of fairyland on the hills and over in Lofty John's bush was laughter like the laughter of fawns. Wilson, Perry and Teddy were waiting there for her. They had made a trist for a twilight romp. She would go to them presently, not yet. She was so full of rapture that she must write it out before she went back from her world of dreams to the world of reality. Once she would have poured it into a letter to her father. She could no longer do that. But on the table before her lay a brand new Jimmy Book. She pulled it towards her, took up her pen and on its first virgin page she wrote, New Moon, Blair Water, Pe Island, October the 8th. I'm going to write a diary that it may be published when I die. End of section 34, recording by Leanne Fortune. End of solo, Emily of New Moon by Lucy M Montgomery.


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We have been talking about central force tension in the case in which the force center has an infinite mass. So the problem becomes essentially a one body problem with the remaining particle. Today I might take one example of that, a fairly obvious example, which is namely one in which the potential is zero. In that case, there is no force center. It doesn't matter at all. So let's talk about the free particle where the potential is zero. In this case, the Hamiltonian, of course, is just a piece further to M. This is really the three-component square. And so to emphasize Hamilton is the square of a vector. If you want to solve it, the free particle Hamiltonian is really an equation for H psi equal to H psi for the free part of Hamiltonian in three dimensions. The most obvious choice of a complete set of commuting observables is the three components of momentum. These, of course, communicate with each other, and they also communicate with the Hamiltonian so that an eigenstate of the PX, PY, and PZ is automatically an eigenstate of the Hamiltonian with an eigenvalue which is a piece square of a 2M. This is really, of course, the same equation as I had over here for the Hamiltonian, except here the Hamiltonian and momentum are considered operators and here they are considered C numbers that are eigenvalues, but it's the same equation anyway. In this case, we know what the wave functions are. The eigenfunctions in momentum in the X representation as a function of X are just plane waves. They're here in the IP dot X over H bar. And if you want to normalize them with the delta function normalization, you need to divide it by 2 by H part of the three halves. So this is the most obvious solution. This is also the solution you get by separating the wave equation in rectangular coordinates. However, it's more in accordance with what we've been talking about recently to think about a different complete set of meeting those other levels. So I have namely one in which we have the Hamiltonian L spread in LZ, because that's what we normally think of in simple force function. And as I've explained in the previous lecture, we get an eigenstate of L spread in LZ automatically by making the total wave function a product of a radial wave function times a YLM. And in which case the radial wave function satisfies one of two different versions of the radial wave equation. Version one is in the original variable R, and version two is in a modified variable and calling F, just a scale factor of R, the radius between them. For the free particle, it turns out that version one is the more convenient version to use. Before I write it down however, let me note something which actually follows fairly obviously from the solution in the rectangular coordinates, which is the energy of the free particle has to be positive or non-negative. This is clear because the eigenvalues of the square and momentum has to be positive. So the same thing, of course, is true regardless of the coordinate system, so it's going to be true in the spherical coordinates as well. But what it means is that we can define an effective wave number for the solution called K, which is the square root of 2MP. E is the positive eigenvalue. And then I mentioned this, the expected is continuous, so there's no quantization. Divided by h part, this gives us a wave number. It allows us to introduce an dimensionless radial variable called rho, which is equal to K times R. And if we make this substitution then into the version one of the regular Schrodinger equation in which the, this is wrong, this is plus the true potential that you are, in which the true potential vanishes, there's only a centrifugal potential, then the regular equation we get in terms of the rho miracle is this.  It basically, this substitution, rho equals K R, basically gets rid of all physical constants in the version one of the regular Schrodinger equation. It also transforms it into a standard equation of mathematical physics, which is what this is, in which the solutions are the so-called spherical vessel functions. That's to say, one can say that the solution of R as a function of radius R is one true type. The spherical vessel functions are either called J L of rho or else they're called Y L of rho versus the studio convergence. In fact, for a given, see L is a parameter of a regular Schrodinger equation. That's something to keep in mind is really a different radial Schrodinger equation for each value of the angular momentum when you see it here. So the solutions are parameterized by L as well. And this being the second order differential equation, it naturally has two linear and independent solutions, which in mathematics are called as J L and Y L. These are two types of radial vessel functions. Now, if you examine these radial vessel functions in the limit in which rho goes to 0, so it's small rho, you'll find that J L goes like rho to the L power, whereas the Y L goes like rho to the minus L minus 1. And I explained in the previous lecture why this type of behavior of rho to the minus L minus 1 or radius to that power is not acceptable physically. So from a physical standpoint, free particle solutions are only the J L type solutions. And if we do that, then I'll make use of this space over here. If we do that, we now have the solution for the free particle just to plug in the value of J L and what we get is that psi is equal to J L of k rho times the Y L M of theta and pi. And the energy is connected with a k parameter through this substitution we made over here. So I'll just connect you between wave number and energy for free particle. To get a feel for these solutions, allow me to plot the J type spherical vessel functions for you. It looks like this. Let's make this as a function of the dimensionless variable rho and I'll plot J L of rho. For the case L equals 0, it starts off at a constant value, so this is J 0 here, rho. And then it goes down and does some oscillations like this. And the empty oscillations guy's office goes out. For J 1, it rises linearly at first and then reaches a maximum and then oscillates like this, goes on down. For J 2, it's quite valid at first, reaches a maximum and then oscillates on down. I'm probably not drawing this very accurately. But for each increase, so this is the J 1 and this is the J 2. As the L increases, the function lays down ever more flat at the origin. As I explained last time, rises to reach a first maximum. This is really acoustic of the WKB theory. And then after that, there's a oscillation which obviously amounts to a classically forbidden region. The magnitude of the oscillations dies off roughly. It dies off as 1 over rho, 1 over the radius that should go on. And this is the sketch of the, there's just some idea of what these spherical vessel functions look like. I will sketch the Y type solutions. As I mentioned, they're not acceptable as free particle solutions. You have, the particle is free everywhere. But that doesn't mean they're not relevant for quantum mechanics. Because in scattering theory, we have a situation where there's a potential but it has a limited range. So down near R equals 0, the particle is not free. But at a large radius where the potential becomes negligible, the particle is free. And we often need to expand the wave function out of R radius in terms of free particle solutions. Which means, which means spherical vessel functions in YLM. If you do that, then you actually need to Y's. Because out there, as long as you stay away from the origin, that's where the singularity of the Y's is. That's why they're not acceptable if you push all the way down to the origin. But if you don't do that, then they are acceptable. Not only are they acceptable and actually necessary to get a complete set of wave functions in the exterior region. So in other words, they're important in scattering theory. We'll see that come up later next semester when we talk about scattering theory. Alright, so this is the case of the free particle solutions in spherical coordinates. There is, by the way, obviously going to be a transformation. If we have, here is an energy eigenfunction in spherical coordinates with some given energy E. And then over here we've got plane waves, which are also of given energy in the same energy as we make the momentum right. It's obvious that this wave function must be representable as a linear combination of those wave functions. I won't go into that now because it's just special function manipulations to convert one into the other. But we'll actually need that later on in the scattering theory to convert the plane wave solutions into central force type solutions. Alright, now, at this point I'd like to return to this question about the infinite mass of the fourth center, which we needed in order to make this two-body problem into a one-body problem. This is unrealistic. Of course, real force centers are created by particles and other things. So to create a more realistic situation, we have to include the dynamics of the fourth center itself. Let's say there's two of them, they're particles, and that masses M1 and M2. Now we don't have the option of attaching an inertial frame to either particle because if we do it, it won't be inertial. So we need to just choose an arbitrary inertial frame here, x, y, and z. We can call this the lab frame. And then we can draw the radius of vectors in and give us the positions of the two particles called this R1 and R2 vectors like this. And I'll also draw the volume of the two particles. In the case of central force motion, the force between the particles lies along this line, as I drew it. These two particles have a center of mass and it lies somewhere on the line joining them. If the masses are different, it won't be in the center. Let's call the center of mass position, let's give it a name, call it the vector capital R, so it looks like this. Now, this one becomes a two-body problem. And so the Hamiltonian for this is the sum of the kinetic and potential energies. So the kinetic energy has to be the kinetic energies of particle one plus the kinetic energies of particle two, which are expressed in terms of the moment Rp squared over 2 and 1 plus p1 squared over 2 and 1 plus p2 squared over 2 and 2. As far as the potential energy is concerned, we will assume that it's a function only at the distance between two particles, which I'll write this way as the absolute value of the relative vector between the two. So this is the central force motion with two particles now. And we'd like to solve the Schrodinger equation for this. This system has six degrees of freedom. It's the x, y, and z coordinates of both particles. The configuration space is really six-dimensional space now. That means that the wave function, which I'll call capital Psi, is a function of the coordinates of both particles, R1 and R2. So the Schrodinger equation is going to be h-sized into d-sized, which depends on both of these coordinates of R1 and R2. In order to solve the Schrodinger equation, the way to do it is to make a transformation of variables. The transformation is motivated by the physics. We first of all introduce one variable, which is the center of mass, called capital R, as I've done here. The center of mass is just the mass weighted positions of the two particles. So it's M1 times R1 plus M2 times R2 divided by the total mass of the system, which I'll call capital M. So let me put this over here. Capital M is defined in the M1 plus M2 like that. And in addition, let's also introduce the relative position vector between the two systems. Let's say going from M1 to M2 like this. And I'll call that vector lowercase R without any subscripts on it. So lowercase R is defined as R2 minus R1. Now this amounts to an important transformation in the configuration space, as you can see. While I'm writing this down, let me also write down the inverse transformation, which I'll need at this moment. R1 is equal to the center of mass position capital R minus M2 over capital M times the relative vector R. And R2 is the center of mass position capital R plus M1 over capital M times the relative vector R like this. You can easily see for yourselves that if you let capital M1, let's say, go to infinity, you go back to the earlier situation we were talking about before, where one of the masses was infinite. Alright, so this is the transformation on the position coordinates forward and backwards to go to the opening coordinate system. We like to transform the Hamiltonian over to the new coordinates. The first obstacle we run into is the Hamiltonian has momentum in it, as well as positions. So how do we transform the momentum, P1 and P2? Well, we'll do it like this. In the configuration representation, P1 is the differential operator minus i h bar e dr1. And P2 is similarly minus i h bar e dr2. That's what it is in the position representation acting on wave functions like this side here. Now, let's define a momentum called a capital P, which is to be conjugate to the center of mass flow in capital R. And it will be defined as minus i h bar d d capital R. Let's make that definition. Well, this can be expressed in terms of the other two momentum, but it's just by using a chain rule. This becomes minus i h bar multiplied times d dr1 with respect to capital R times d dr1 plus d dr2 with respect to capital R times d dr2. These are really vectors here. So to get it right, you really need to have a, so this thing is really a matrix. You can track the r1 there with the r1 there. If you wanted to put i's on it like this and an i there that introduces some on i, I can do that to clarify what this means. But you can also think of this as a matrix, multiple final vector. Well, in any case, these derivatives, d r1 with respect to R, are easily evaluated by looking at these formulas. d r1 with respect to capital R, you see, is just unity over the identity matrix. So this thing just becomes the identity matrix. And this thing here, d r2 with respect to capital R, it's the same story. That's also the identity. And so the result means this turns into minus i h bar simply times d dr1 plus d dr2. This is just using the chain rule. However, we now see that's the sum of p1 plus p2. And so we get a result which is nice enough that I'll put it on the line by itself. This is that the momentum conjugate in the center of mass transition is actually the sum of the momentum of the two individual particles. Now, let's do something similar for the momentum I'll call the lowercase p, which is conjugate to the relative position. That's the same thing as we define that as minus i h bar and d d r with respect to the relative position. And now you can go through the same kind of a chain rule here. And the difference is that the matrices that appear here are going to be the derivatives of r1 and r2, not with respect to capital R, but rather with respect to small r. So we have coefficients minus m2 over m and plus m1 over m. The result is that the momentum conjugate to the relative position is minus m2 times p1 plus m1 times p2 divided by the total mass dealt. And this then gives us the transformation of the momentum. Work out this way. Now, once we have this, then we can solve these equations for the old momentum to the function of the new. It's just algebra to do it. You'll find that p1 is equal to m1 over capital M times the capital P and then it's minus of the lowercase p. And then you find that p2 is equal to m2 over capital M times capital P plus lowercase p. These are the inverse transformations. And so now we've got a complete transformation between old variables, old position and momentum variables, r1, r2, p1, and p2. And new position momentum variables, capital R, center of mass and relative position and momentum. And having that, we can now take the Hamiltonian and transcribe it into the new operators. Before I do that, let me mention that the commutations work by looking at the commutation relations of the new variables amongst themselves. In the first place, we have the commutation relation of capital R i with capital P i. These are the center of mass position momentum is minus i h bar times delta i j. The reason for that is that we define the capital P as this derivative with respect to R and some usual rules of computing the commutator's work out in the usual way. Likewise, this is also equal to the commutator of the components of the relative position and momentum. Lowercase r ij, lowercase r i with lowercase pj is the same commutator. This should have been a plus by the way, plus i h bar delta i j. These are the main commutators here. And then all other commutators that you can make by combining these four sets of variables are equal to zero. So it is the same structure of commutation relations we had initially when we r 1, r 2 and p 1 and p 2. The common structure of commutation relations is preserved. This is what you call a canonical transformation in classical mechanics. You could call it that in quantum mechanics too. We saw another example of canonical transformation dealing with the probably charged particle in the uniform magnetic field. This is actually a little bit simpler here. But in any case, one of the consequences of this is that any function of the center of mass variables, the capital variables, will commute with any function of the relative variables. So that's a simple rule that's easy to remember. Now, having that transformation, let's now transform the Hamiltonian to the variables. And if we do that, it's some algebra to transform the kinetic energy. We're just plugging these formulas here into that Hamiltonian out at the top of the board. And what you do, here's what you find, is the Hamiltonian now becomes the capital P squared over twice capital M, the square of the total limit of the system divided by twice the total mass, plus lowercase p squared over twice mu, or mu is the deduced mass, plus the potential energy, which is B and B of the magnitude of r 1 minus r 2, which is the magnitude of the relative position vector, which I'll call r like this. The yield here of the reduced mass is defined this way. It's 1 over 2, it's 1 over 1, it's 1 over 22. That's what you call a harmonic mean of the two masses of the two bodies. This just comes from the algebra substituting there. I mentioned one thing about reduced mass. If you're in a situation where one particle is much more massive than the other one, then the reduced mass is approximately the same as the lighter of the two particles. So for example, in hydrogen, when the proton is much heavier than electron, the reduced mass is almost the same thing as the electron mass. However, if two masses are equal, when the reduced mass is equal to the mass of unit particle divided by 2, it becomes comparable to the masses. Anyway, just some rules about reduced masses. So we have two terms here. Let's call it the first term. The first term depends only on the center of mass variables, so let's call it HCM. And then the second term depends only on the relative variables. Let's call it H-relative. So the total Hamiltonian breaks up into a center of mass, a term plus a relative term. Moreover, these two terms commute with one another because they are functions of the different sets of variables. And that means that possessed simultaneous eigenstates. You can diagonalize these two things separately. And once you've done that, you'll automatically have an eigenstated H because H commutes with these two as well. So this is a convenient decomposition. As far as the wave function is concerned, we started out by writing it as a function of R1 and R2. But by doing the change on position variables, which is given right up here, we can re-express it as a function of the center of mass core against the relative coordinates. And if we do this, then the HCM acts only as the capital R, or the H-rel acts only as the lowercase r. And the result is the Schrodinger equation is separable. And it means that solutions exist in the form of a product. Let's call it capital 5 and capital R for the center of mass part and lowercase i of lowercase r for the relative part. I don't mean that every solution for Schrodinger equation has this form, but I mean there exist eigenfunctions of this factored form. If you put this in here, you'll find that just by applying HCM, which acts in the first factor and H-rel acts in the second factor, what you'll find is that you have these breaks up into two eigenvalue problems. HCM acting like 5 is equal to the center of mass energy multiplied by, and you apply the H-relative acting on the side, because that's the relative energy multiplied by 2 times 2 eigenvalue problems like this. And if you solve these, then the total energy of the entire system, the entire Hamiltonian, is just the sum of the center of mass plus the relative energies. The center of mass Schrodinger equation is particularly simple because it's a free particle problem. This corresponds to the classical fact that the center of mass of a system like this moves on a straight line of constant velocity, and so it appears here also. And thus the solutions for these wave functions capital 5R could be taken to be any free particle solution. We could, for example, take it to be a plane wave. This is probably the simplest solution. You need to be on a capital P, got into the capital R over H-bar, and then divide it by 2 by H-bar to the 3-has if you want to normalize it. This is getting the plane wave solutions to the center of mass motion. But of course we can also do this in spherical coordinates in which case you'd have spherical vessel functions. You get to choose how you want to solve this free particle problem. Having done that, we turn our attention to the relative Schrodinger equation, which is down here. And what we see about the relative Schrodinger equation is that it's an effective one-body problem that has the same form as the one-body problems we considered earlier when we considered the force centers having an infinite mass. The only difference is that the variables in question and the positions in moment and in coordinates are no longer the coordinate in particular is no longer the coordinate of the particle relative to an inertial frame. Instead, it's the relative position between the two masses. And in fact, to refer back to these equations up here, you see it's actually a function of the positions of both particles. Likewise, the momentum which occurs here is also a function of the momentum of both particles. It's worth all remembering that. If you're dealing with hydrogen, for example, you're talking about the radius, this is really the distance from the proton. And as operators, they are, they are operators that satisfy the usual operator, the polar commutation relations. But physically, they're actually functions of the positions and momentum of both the electron and the proton. Well, in any case, mathematically, it has the form now of a one-body problem. Oh, another important change is that the mass of the particle, which is moving in the field, is not ever replaced by the reduced mass. Except for those changes, it becomes a one-body problem, just like we were talking about in the movie. So solving one-body problems is enough to solve two-body problems when the force is the central force. All right. Plus a three-particle solution for the center of mass, which is pretty easy. Okay. So now what I'd like to do is to turn to some examples of central force motion. Let me begin with a simple example, which is that of a rigid rotor. I'll talk about it classically first. So we get some ideas about it in a picture of it. A rigid rotor we can think of as being two masses, let's call it L1 and M2, which are joined by a massless rigid rod. In other words, it has a fixed length, let's call it R0 connected with 2. This is obviously an idealization because there are no rigid rods in nature. Let's say somewhere here is the center of mass positioned like this. And if we compute the moment of inertia of this rod and the rotation is about the center of mass, what you'll find is that it's the reduced mass mu multiplied by the square of the distance between two particles. That's a classical calculation of what you do to see if this is true. Also, the motion, as we know, is going to be just rotating. So the two particles rotate about the common center of mass and they're going to do so in a circle, which therefore lies in a plane. The angular momentum factor will be perpendicular to the plane. And so will the angular velocity be in the same direction. And so, just by elementary mechanics, the energy in the system is going to be one half of the moment of inertia times the square of the angular velocity. We also know the angular momentum is the moment of inertia times the angular velocity. And if you're writing energy in terms of L instead of omega, it turns into an L squared divided by twice the moment of inertia. Now, here's an interesting fact. If we take this expression for the moment of inertia in terms of the reduced mass and plug it in, we get L squared over twice mu times r not squared like this. And what this shows you is that this, as you see, is actually the centrifugal, this is actually the centrifugal potential. I remind you the centrifugal potential is physically a part of the kinetic energy. In fact, physically, it's the angular part of the kinetic energy. For a rigid rubber, there is no radial part because there's no radial motion. So this is all the kinetic energy there is. And I'm not including any potential here. So in fact, that's the total energy. It's the effect that comes to the Hamiltonian in the system. All right. Now, the configuration of the classical rigid rubber is determined by, let's say, a unit vector that points from, let's say, let me swap these things out one and two so that I have a vector that points from one to two going out like this. And this has the direction between which we specify the spherical coordinates by theta and phi. Basically, if you tell me what theta and phi are, it's functions of time. You've got the complete evolution of the rigid rubber. This suggests that when we go to quantum mechanics, that the wave function should be a function of the spherical angle stated in phi. It should be a wave function on the unit sphere. Likewise, this expression, either one of these two expressions with the classical energy, suggests that the quantum Hamiltonian should be elsewhere over twice the moment of inertia, where L is now the angular momentum operator, which we know how it acts on functions on the unit sphere. Particularly, it brings out, well, we know the eigenfunctions of YLM. So the energy eigenfunctions should be YLMs of theta and phi. And if we let this Hamiltonian act on that, then the L squared is going to bring out L times L plus 1 h bar squared. So this is the energy, then, that the energy depends on the quantum number L and it's L times L plus 1 h bar squared divided by 2i, which if I replace i by a reduced mass R0 squared, you can see, again, it's really the same as the centrifugal potential frozen at the radius R0, the fixed radius. Okay, so this is some of the quantum mechanics of the rigid rotor. There's a degeneracy you can see because the energy doesn't depend on the magnetic quantum number. This is the 2L plus 1, this is 2L plus 1 called the degeneracy because of the possible values of the magnetic quantum number. And the usual reason is that the magnetic quantum number indicates the orientation of the system and the energy doesn't depend on that. All right. Now, the rigid rotor is an idealization and we've been making a lot of guesses in terms of writing down these quantum mechanics relations. They're kind of obvious, but they're still guessing. If you wanted to make this more rigorous, you'd have to make the rigid rotor as a kind of a limit of a central force problem in which you end up finding potential and it keeps the two particles at very close to a constant radius. I'm not going to bother you because it's not what I want to do today. But I just want to say that this involves some guesses to what we've done. Yes. So when you go down like this, you have to do know that it's dependent on the size of the atom, 8 and 5. But why do you know that it's the YLNs? Because the energy eigenfunctions, the eigenfunctions of L squared are the YLNs. Okay. And then how can you say or how do you know that it's independent of N? Because the energy only depends on the L1 number and not on the N1 number. Because the Hamiltonian doesn't involve the ALZ. That's because it's rotationally invariant. This is the normal situation with rotationally invariant Hamiltonians. All right. Now, rigid rotors are idealizations of non-existent nature, but there are biotomic molecules which exist in nature that provide nearly the same features as a rigid rotor. So I'm going to tell you about a diatomic molecule as examples of central force motion. So a particular diatomic molecule, take a specific example, look at the carbon monoxide molecule. The mass of the carbon, well, first of all, the atoms, the carbon monoxide atoms are composite particles. They're not elementary particles. That's because they're made up of sub-components electrons and nuclei and so on. But it is possible in quantum mechanics to treat composite particles as if they were single particles under certain circumstances. I won't go into that now, but we'll just go ahead and do that. As far as carbon atom is concerned, the mass of the carbon is about 12 times the mass of hydrogen. The mass of the oxygen is about 16 times the mass of hydrogen if we take the normal isotopes. The mass of hydrogen is about, something like 1800. I'll make it very roughly about 2,000 times the mass of the electron. Let me also talk about the reduced mass of the carbon oxygen system, which I'll call u. This is about 7 times the mass of the hydrogen if you work it out. Momentary net by 2,000, we get about 14,000. I'm going to make a very rough estimate of this and just replace it by about 10 to the fourth times the mass of the electron. I need to eat off the lowercase sample standard of the mass of the electron in the following discussion. In the first place, there is a large disparity between the reduced mass of the system and the electron mass. It's a ratio of 10 to the fourth. Let's give this already some small parameters, which are all right like this. First of all, u is the reduced mass of the carbon-oxygen system. If I divide this by the electron mass, I get, let me do it the other way, put the small number on top, and then divide it by a mu. This is about 10 to the minus 4, a small parameter. Another small parameter that we'll run into is the square root of this, as we'll see. This is about 10 to the minus 2, and yet another small parameter will be the fourth root of this, which is about 10 to the minus 1, or about 10%. We'll see these small parameters occurring. We're going to treat the carbon-oxygen molecule as a central force problem, where the two atoms interact by means of a central force potential. I'm going to tell you something now about the potential between the two. I'll start by drawing a picture of it. If we draw the potential like this as a function of radius r, and here's v of r, so this is the carbon-oxygen potential. It has a gentle structure like this. It rises very sharply. It's small-radiated. It decreases down and creates a well, and then the well has a shallow tail that goes out like this. In fact, it was a curve of the same shape, which applies to essentially all biotomic molecules. The physics of this is that at large distances out here, the two atoms, the electron clouds and the two atoms polarize each other to create dipole moments. It's a dipole interaction. It's only a prosthetic interaction between the two atoms, and that's what causes the attraction that pulls them in, just like when you're pulling your hair up. However, at short distances, we start to feel a repulsive force, and that arises when the two electron clouds start to overlap substantially, because you're trying to force electrons into the same region of space, just like taking a particle in a box and keep on adding more and more electrons. They have to go on the higher energy levels. And so the repulsion which occurs here is in the following sense is being due to the pressure of a degenerate electron gas that's being squished into a smaller and smaller volume. And it's a competition between the two these two effects, which gives you the well of the bottom. Now, there's no simple analytical formula for this curve, so I don't even attempt to write one down, but there are some obvious parameters. One of them is the radius of the equilibrium position. Obviously, this is the equilibrium position, and the carbon and oxygen molecule can undergo vibrations around this equilibrium position. This is the potential in which it's moving. So one obvious parameter when it's in dotted lines here is the radius which I'll call r0 out to the middle of the potential well. Another obvious parameter is the depth of the well which I'll call v0 like this. In the case of the carbon and oxygen molecule it turns out that v0 is about 11 electron moles and it turns out that r0 is about 1.1 angstroms. Other molecules have different numbers but very roughly they're all in the same order of magnitude. There's some variations in it, but very roughly this is fairly typical of numbers. All right. Now to understand this in a slightly more quantitative way there's no, as I said, analytical formula for this. So I'm going to rely on order of magnitude estimates using some dimensional analysis. As I pointed out the forces between the atoms are determined by the properties of the electron clouds alone, whether it's or dipole-dipole interactions. Now the physics of the electron clouds is governed by three fundamental constants. The charge of the electron, the mass of the electron, and the plane's constant h bar. The speed of light is not near because the forces between the electrons, the electrostatic forces are electrostatic and not electromagnetic. Magnetic forces are not important. There's no seclusion to one's law. Another thing is that the velocities of the electrons are essentially non-wilderistic but very small comparisons to the speed of light. So they're already relative distinct effects either and that's why, at least not the lowest order. And so that's why c is not here. So given that we've got just these three constants to work with it becomes possible to take these two dimensional analysis and to construct a fundamental length a fundamental distance time, the length is the same as distance, length, time, energy, etc. And if you do this here's what you get. So let me call the fundamental length let's call the fundamental length a0 and this turns into h bar squared divided by e squared. And if you work this out the variably it's about 0.5 angstroms. This is the only this is the only quantity of distance you can work out by using these three physical constants. And as far as energy goes there is let me call it kgon as the energy this is contributed to the force divided by h bar squared and this turns out to be about 27 electron volts. If we talk about velocity it's called v0. This is equal to e squared over h bar. I'm going to take this e squared over h bar and write it as e squared over h bar c times c. I said a moment ago the speed of light is not on the list but I have modified it and modified it so c's really not there even now. But it's still useful to write it that way because e squared over h bar c is the fine structure constant and you can see that this turns into alpha times c which is the same thing as 1 over 137 times the speed of light. So this confirms what I said earlier that the typical velocity of the electron is small compared to the speed of light it's less than 1 percent of the speed of light. It's fast and it's not relativistic. There's a frequency that comes in fundamental frequency and this is basically the energy divided by h bar so this is m e to the fourth over h bar q the reciprocal of this gives us a fundamental time t zero which is 1 over omega zero which is equal to h bar q over m e to the fourth and this is about 2 times 10 minus 16 seconds it's a pretty it's a pretty small amount of time okay so these are these are the quantities which enter now these fundamental units of physical units of distance energy and so on in fact they're exactly the same ones that arise in the hydrogen atom and the reason is the same is you've only got these three physical constants to work with so for example this a zero is the same as the Bohr radius over here is actually twice the ionization potential of hydrogen which is 13.6 electron volts this velocity d zero is basically the velocity of the electron in the ground state of hydrogen and so on so these are basically hydrogen numbers that are used here so just on the conventional analysis along we would expect that the depth of this well the carbon oxide molecule as well as the radius should be in the same order magnitude of these numbers given here if you look you see that p zero the depth of the well is something like 40% of k zero so it's not too badly off and you see that the radius is about twice, a little more than twice the Bohr radius so within a factor of two or so actually not too bad it gives you the right order of magnitude you know if you want to change the problem around the management you may need a molecule like carbon oxide but instead of using electrons let's say mu minus some muons which have the same charges of electrons that are heavier and you want to know how big would the molecule be the dimensional analysis would just tell you that you just need to replace the mass of the electron but the mass of the muon and the answer will come out now that means you'd like to do some things because it would be faster and so on and still you need to check to see that it's still not all the distance but roughly it would be the correct answer nobody makes muonic molecules like that there's too many muons required but they do make muonic molecules that have one or two muons replacing electrons and there's actually quite some interest in that alright there's no such thing as muonic carbon and oxide as far as I know alright alright so so this is the dimensional analysis now given this we can start to do some other things alright you can see that the that the vibrations of the molecule are going to be represented classical by motions in this well and therefore quantum mechanically they'll be quantized and there'll be energy levels in this well these will be the vibrational energies also you can see that if we approximate the bottom of the well by parabola that the ground state and hopefully the first few excited vibrational states can be roughly described as a harmonic oscillator so let's talk about this harmonic oscillator let's make a harmonic oscillator approximation to this well that would mean that there's an approximate potential here we look like this there would be one half the mass mu which has to be the reduced mass of the carbon monoxide system which is like put up there it's 10 to the 4 times the electron grass times the vibrational frequency square times the radius minus the equilibrium radius square this is the harmonic approximation now we know what the reduced mass is but we don't know what the vibrational frequency is that depends on the curvature at the bottom of the well but it's possible to estimate the vibrational frequency and we do this in the following way we'll say that as a as a guess or a rough order of magnitude we'll say that when the when the distance between the atoms moves a distance which is comparable to A0 which is comparable also to R0 and that far we must be up to the point where we're at the association energy so in other words let's do this let's equate one half mu of a mega vibrational square times A0 square to the square of the deviation away from equilibrium equals A0 which is an estimate of the dissociation energy of V0 and I'll just ignore factors of 2 and as long as I ignore factors of 2 let's put away the one half here as well again it's just a potential analysis but you see it now allows us to solve for a mega V in terms of things we know so we find a mega V then it's equal to B squared it's equal to 1 over A0 times the square root of K0 divided by the reduced mass mu like this now let's plug these things in this becomes equal to H bar squared over MB squared or excuse me it's 1 over A0 it's equal to B squared over H bar squared and then we get the square root of K0 over mu so there's the K0 so this becomes M over mu times E to the 4th over H bar squared and so you can take the square root of E to the 4th over H bar squared and you get E squared over H bar multiplied here so you get M E to the 4th over H bar Q times the square root of M over mu but the M E to the 4th over H bar Q is the polynomial frequency we have here since the left hand side is dimensions of frequency the right hand side has to be equal to this omega not times something dimensionless and in fact here's what we find so we find the mega V is equal to the square root of M over mu times the omega not so this is an important relation the square root of M over mu is the square root of the ratio of the mass of the electron of the mass of the nucleus which is really the reduced mass of the 2 in the eye which as you see is about 10 to the minus 2 roughly so this means that the vibrational frequency is about 10 to the minus 2 times omega not and this basic factor of 10 to the minus 2 roughly speaking applies to most diatomic molecules some variations but that's the basic idea it's a rough order of management this is how it works this frequency of omega not as I mentioned earlier is the same as the orbital frequency of the electron and hydrogen this typical orbital frequency of the electrons in the carbon and oxygen molecules and so H bar times this is the typical energy of a photon which is emitted in an electronic transition in an atom these electronic transitions as you know are in the optical or also in the ultraviolet regime the 13.6 electron volts for hydrogen and the ionization energy is in the ultraviolet some of the other transitions to hydrogen are in the visible part of the spectrum but what this shows is the vibrational energies are downed by a factor of about 100 the vibrational frequency is the F4H bar of omega which is the H bar of omega B which is the delta E on a vibrational transition of the molecules downed by a factor of 100 it's a square root of mass ratio what that means is the vibrational transition of the frequencies or the photons that are emitted and absorbed and changing the vibrational state are not in the optical they're down in the near to middle infrared this is important for example in the greenhouse effect optical radiation in the sun well the sun puts out most of its power in the optical range of frequencies and the atmosphere is obviously transparent to the light but when the light hits the ground it warms up the ground with something like 300 degrees Kelvin which then radiates how a formal radiation is about that temperature it's not exactly a black body but it's the idea that's the right temperature range and that as it turns out that's in the infrared so the radiation that has to cool the air off get back out of the space has to pass back through the atmosphere but as it does so it's in the right frequency range of the vibrational transitions of molecules now part of the monoxide is not one of the greenhouse gases this part of the monoxide is the usual culprit that's a triatonic molecule which I know I want to talk about but still the basic physics applies that it's vibrational transitions of those molecules is what is it swerving those infrared photons and then they get reemitted as well in the atmosphere of course nitrogen and oxygen are also diatomic molecules in the atmosphere and they also have a vibrational spectrum it's very similar to this there's a question about why those infrared photons don't interact with them in other words why are oxygen and nitrogen greenhouse gases as well the reason for that is those are what you call homonuclear diatomic molecules those atoms are identical they're identical the result is they don't have any electrical dipole moment there's no charge separation carbon monoxide has got a nice it's one way or another I'm not a chemist so I don't know which way it is but there's a charge separation on it and this produces a dipole moment which strongly interacts with radiation at the right frequency with O2 and N2 that doesn't happen because they're neutral that's why the O2 and N2 are not greenhouse gases so the fact that although the N2 and O2 have vibrational transitions in the same range as just CO2 but it's the other fact that there's no interactions between the photons they don't interact with the electromagnetic field but the infrared photons produces an electric field that vibrates at the right frequency to stimulate the bonds the nuclear bonds of molecules such as CO2 alright so that's a little bit of the physics of the greenhouse effect the I mean if nitrogen and nitrogen were greenhouse gases it would be a trouble because it would be like Venus which is instead of most of the CO2 so it's really hot okay alright so this story's not quite over yet there's other interesting things we can do if we want to calculate the number of vibrational eigenstates for dissociation one of the ways of doing that is just to take the dissociation energy which is v0 here which we'll estimate as an order of magnitude by k0 so let's take k0 and then let's just divide it by h bar we'll make it v which is the vibrational energy so this is the separation between the vibrational energy levels if you work this out which you find the square root of u over n it's the inverse of that square root so it's an order of order of about 100 and the result is the typical diatomic molecules have about 100 vibrational states before they break and dissociate alright now this potential that I drew here is only the true potential and I've been ignoring the reciprocal potential before I get to that though there's one more thing to say the vibrational energy the ground state vibrational energy is of course one half h bar who may not be the vibrational frequency and from many molecules this is common to somewhat argument in the room temperature so at ordinary temperatures the diatomic is common for diatomic molecules if we mostly in the ground state vibrational ground state well perhaps it's just in the low line problem numbers, vibrational ground state, you get an idea of how big the vibrational amplitude is so that's fairly easy, we go back to the so I call this delta r here it's called this delta r as the width of this it's a Gaussian wave pattern you see the width of this Gaussian wave pattern what is it equal to? well if we go to the the story of harmonic oscillators you look in the notes on harmonic oscillators and you look at the characteristic length of scaling for harmonic oscillators it's sort of a h bar over h omega if you look in the notes you see that and this delta r is really the same thing except the h bar is the same but the mass is replaced by the reduced mass and the omega becomes the vibrational frequency omega v like this now this has dimensions of length so it's got to be proportional to our a0 times the dimensional slumber and if you work it out that dimensional slumber is the mass ratio n over u to the one quarter power times a0 which is very roughly something like 10% of the more radius and so very roughly the idea is that in the ground vibrational state the molecule is undergoing zero point oscillation which is magnitude is about 10% of the equilibrating radius and in that sense it is very close it is roughly a rigid roller it's got some small vibrations the radius doesn't change very much and I see I didn't hear the bell it's so interesting well it's too bad I've just got a couple more comments to do to finish this story of molecules and then we'll go on to a hydrogen atom which we'll talk about in a minute maybe that's all


So sustainability is very important to our family on this farm. We have solar on our barn and we have now started using plug-in electric harvesters and this year I have committed to moving entirely to electric in the within the next two years. I'm a steward of this land and I think it's really important that I pass this land on in better condition than it was when I was here to start.


Hi everyone, Sean with jaspyscasebrakes.com here doing 2021 Bowman Sapphire five box break pick your team number nine Where we're giving away a free box of sapphire as well So how to be entered in the free box giveaway by two or more teams in this break and entered to win No max entries two teams one entry four teams two entries three the six teams three entries Etc. If you only buy one team good luck with that team, but you'll not be entered for the box giveaway Top name wins the sealed box at the end all card ship And obviously this is the configuration for the break So there you go. Here is our list of customers from Steve Wright and the Diamondbacks down to Steve Wright and the Nationals And Here's the box we're gonna give away at the end Those are the five boxes. We're gonna rip right now and here's that final list as well Good luck, so Steve Wright with last spot mojo Diamondbacks and Nationals very cool man. Congrats Very cool, that is a pretty great luck there man all in one box, huh? Good luck everybody box one wander Franco for the Rays Chris Garosi and Mick Abel to 99 for the Phillies Eric Jock Eddie Yeen Otto for the Nationals Steve Wright last spot mojo man That is awesome Jason Dominguez for the Yankees Michael Pacifico and Jordan Bozovic to 50 for the twins Eric Sanzala Jefferson Espanyol for the Diamondbacks last spot mojo again Steve Wright Well tonight 100% Steve 100% both of them Jared Kelly to 75 for the White Sox and Eric Ainge Austin Martin Blue Jays Matthew Shira Marielle Batista for the Reds Dennis Foley and Cade Cavalli to 99 For the Nationals Steve Wright last spot mojo again You also had a Jeremy De La Rosa base in there and wander Franco for the Rays Chris Garosi Jordan Adams to 125 for the Angels Chris Garosi Kevin Alcantara for the Yankees and Michael Pacifico and Eddie Yeen again for the Nationals Steve Wright last spot mojo It's your night. It's your night and last box. Good luck folks Christopher Morales for to 125 for the Cubs and Eric Ainge Maximo Acosta for the Rangers Matthew Shira and Daniel Montana for the Rockies Ika Pola Blaze Jordan for the Red Sox Eric Sanzala And there you go guys. Those are the boxes So got to give away one box To between the folks about two or more teams No CJ pulled 11 Dowsky stained glass, but no on Sufati Not sure what they pulled over on personals earlier today though So pop back over here to the list copy the customers Over into a new spreadsheet alphabetize Anthony gets one Chris gets two Darren, thanks for getting in Dennis gets two Eric a gets one J gets one And Eric S gets one Ika Pola. Thanks for getting in the same as Joshua Lucas Matthew Shira gets one Michael P gets one PJ Peter, thanks for getting in Steve right gets one and Steven George gets one There you go. So from Anthony down to Steven. We got 13 entries name on top after Five and a three eight times Eight and final time and that's gonna go to PJ eight times on the randomizer PJ on top Eight times on the dice and there you go PJ. Congrats. That was 2021 Bowman Sapphire five box break picker team number nine from Jasper's case breaks.com. I'm Sean Jasper's case breaks.com. Thanks for hanging out. We'll see you next time


You, for the rest of your lives, will be Fallujah Marines.  I mean, years from now, people say, hey, see that guy? He was a Fallujah Marine. You answered your nation's call.  Our camaraderie is what brought us together. Our camaraderie is what made us succeed. Our brotherhood is what allowed us to achieve what we achieved. I am in the company of heroes.  God bless.


Live from London, England, it's theCUBE. Covering Discover 2016 London. Brought to you by Hewlett Packard Enterprise. Now, here's your host, Dave Vellante and Paul Gillis. Welcome back to the docks of London, everybody. This is theCUBE, the Worldwide Leader in Live. Tech coverage, I'm here with Paul Gillin, my co-host. This is day two of HPE Discover. Bobby Patrick is here, he's the CMO of HPE. Cloud, software defined, you know, you guys are now sort of blending those together, so welcome back, it's good to see you again, Brian Gartner, who's a technical director at SUSE is here. Brian, good to see you, thanks for coming on. Bobby, what's happening? Yeah, well, we had a pretty big announcement today, really excited about this part, kind of fits in our overall software shift to the new stack, and that is we announced with SUSE that they're going to acquire our Helion OpenStack distro and our Helion Staccato distro. They were the first really distro in OpenStack back in 2012, amazing technical team. They were not investing in Cloud Foundry, now they're going to invest in Cloud Foundry. Essentially, we're shifting from code contributor over the last few years of kind of cropping up OpenStack to Market Maker. So we're hiring and customer-facing engineering and implementations and support, and then we're going to partner and let SUSE do the kind of deep work with the code.  What's happened? You've completed that task and now you're saying, okay, let's turn it over to the guys that really want to maintain this. But we need to get OpenStack to a state where you're beyond the rocket scientists, who deploy it, right? And so we were building out a number of the projects. We were consistently number one or number two in code contributions, and we realized about nine months ago that it was an inflection point. While not quite mainstream, it's now broadly deployed across retailers and telecom and others. The use cases now are very real, the returns are real. We realized our biggest challenge now in the worldwide kind of in the field is helping customers integrate in their data centers, right? The uniqueness in putting feet on the street, having a really strong technical support staff, and now with the community as large as it is, we're on the 15th release coming up with Okada.  It's not us to build the code anymore. It's us to take it to market, help customers deploy it in traditional enterprises, and help them get to cloud faster. So how does this make life better for OpenStack customers? I think, well, because we focus on the hardest part, which is getting it implemented in their environment, training their staff, helping them shift even to culture, right? We're like, we're becoming culture warriors. How do we help them shift to continuous delivery? This is really what HPE's role is about. Plus, we'll then match it to the right infrastructure, right? So we put cloud system, which is built on OpenStack on Synergy, announced it here at Discover as well. You can run OpenStack across our scale out servers with a cloud line and with ProLiant. And so we're going to bundle those up. We're going to test it. We're going to put the resources in the market and help get adoption to the next level. I mean, together, what we're really saying is we want to accelerate the adoption of OpenStack. All right, you guys have an OpenStack distro that you're going to merge with? Yeah, we've had an OpenStack distribution since 2012. So you'll bring those two together? Yeah, we're bringing the technology and also the talent over to our side too to help us grow that and take advantage of all the business cases that he's talking about, all that learned knowledge to improve OpenStack even further and make the distribution even more enterprise-appealable. What does that mean to your existing OpenStack customers? I'm sorry? What does that mean to your existing OpenStack customers as you merge these two distros? I think initially there's going to be a little bit of worry, what things are happening, but clearly we've both talked about this and the release even talks about that SUSE OpenStack Cloud will remain. It's as infrastructure as a service. We're acquiring that infrastructure of the service component from Helion and also the platform as a service as well. So will you have a migration path for customers?  Do you want to take that? Yeah, I think the plan is to put a roadmap together over time with our customers involved, right? Roadmaps existed whether we're going to SUSE or not and I think that's what you've got over the next two years, roll that out and make sense, but we have significant OpenStack customer base, big banks and telcos and others. I'm certain SUSE wants to make sure they're happy and we want to make sure they're happy so it'll be a good joint collaboration I think. Yeah, I'm very much echoing the same thing. We have large customers using OpenStack as well so we're going to have to give them that migration and that's always been one of our goals with the distribution is to have a steady roadmap that's appealing to an enterprise with regular update cycles and teacher requests and all of that. And long-term, I mean, it's goodness that you consolidate the distros, right?  Yeah, I mean, we don't need to have 30 distros in the market now, right? The projects are getting very mature. We want a quality preferred vendor that's going to maintain it and test it, but we're going to focus on getting it adopted in the market, getting it adopted in on-premises and data center, 74%, actually 79% of all workloads in Europe over the next two years are going private cloud. There's really only one real option when it comes to a private cloud platform for IaaS and that's OpenStack, right? It is definitely the de facto standard. Yeah, I would hazard a guess that a lot of people don't know what Suze's position is in the market. They know you're a Linux company but they don't really know what your forte is. Can you just define that?  Our forte is really about, and we have a small size of our company, we automate ourselves out of the jobs that don't make sense, so we really focus on Enterprise Linux, doing the testing, doing the QA, doing the support, and then we expanded to do that with OpenStack as well and we do that with all of our products. We make sure that those Enterprise products are Enterprise-grade, whether it's security updates, in fact, that's what the first thing we did in OpenStack was go fix a lot of the security issues that existed in the code base. That was job number one for us because we did want to ship something insecure. Let's start, go ahead, please. Well, can you distinguish yourself from Red Hat? Sure, you know, one of the things that I always say when I get that question is we tend to provide technology sooner. So again, OpenStack was a classic example but we've done that with file systems, we've done that with hypervisors. We adopt the kernel versions faster than anybody else. So that's one of our distinguishing factors. And of course, when we do that, again, we focus on making an Enterprise ready. Whether it's a file system, we know how important that is. Data can't go away, there's got to be integrity there. So we make sure that the matrix of capabilities that we promote are Enterprise-grade. So Bobby, let's talk about the stack you put out a tweet, I responded to it. It was the new stack and all these cool logos on it.  Right, right. And now we're here. It used to be, exactly, good question. So for you, Spear, we're seeing a docker at this show. We've got new programming languages, Chef and Puppet, Ansible, et cetera, et cetera. Talk about what your intention was with that tweet and that picture that you drew and what's behind it.  Think of it as a pats and cattle, actually. But it's actually really multi-cloud, you're welcome to, let me thank you for that. There you go. Which kind of defines our strategy a little bit. It's a gift tier. There always is a t-shirt. Discover a multi-cow world, right? With the pats and cattle analogy, right? Multi-cloud, yeah. So, you know, we see a new stack evolving now. You know, our shift of our software, largely proprietary software to micro-focus is part of this. We have an ability now to offer our partners and customers a much broader expansion of offerings. In the case of the new stack, we think we're going to pick from the winners. We're going to pick the winners, like Chef that we've invested in, right? Important works or Mesosphere and our Docker-ready servers. And we're going to build these stacks on top of synergy and our infrastructure. And we're going to support the ecosystem, you know, like OpenStack with funding and in the market. We're going to have higher support and services. And our partners and our customers are saying, tell us which ones to go with. This is a really important part of our strategy. So you're going to see over time, us picking the winners. You know, the database layer, you've got, you know, React for Key Value and Cassandra for Columner and MongoDB for Master Slave and Neo4j for Graph.  And so we pushed just Vertica. So Vertica is a really important option, but there are many more. So we see this open, kind of open source new stack taking it, so there's really three stacks in there. There's the open source new stack we're just going to win. There's the VMware stack, which, you know, over the next two years, I think we'll go smaller and smaller and smaller. It's proprietary, you're locked in expensive. I mean, then there's the Azure stack that's emerging. And I think Azure stack's interesting. It can really open up a private cloud option and kind of the Azure, you know, .NET shop, VisualSea, Visual Studio's kind of shop. So we're committed to all three, but I think this new stack one is pretty compelling. Excellent. So, you know, we were talking about Azure before. Couple of things we talked about. And Bobby, you and I have been speaking now for several years in the queue. We saw the sort of brief attempt for public cloud. You guys chose that said that's not the right option for us. And that opens up a whole new set of options. We saw Microsoft on stage yesterday. Talk about that relationship a little bit, what it means and how it fits into the strategy. Yeah, I mean, we do have as part of our strategy, we want to power public clouds, right? So whether it's, you know, our infrastructure help powering Azure, it's part of our partnership or Dropbox, we announced the last discover, if you remember going off AWS, that's a big part of the strategy. But, you know, really what we see is every company, every government wants to move to their own right mix of cloud. Everybody has their own unique right mix, right? A bit of private and a bit of public. We think certain workloads like email make sense in public, right? Certain workloads that are more mission critical with sensitive data belong in private clouds. And, you know, there's really three then options that we see. Microsoft side's very interesting to us because Azure has had tremendous momentum in the last two years. You've seen it right. Since Satya took over, the focus of that company's been unbelievable, right? And the execution there is huge. But they want a private cloud option as well. And note, Microsoft on stage talking about the right mix of Azure and endorsing hybrid, recognizing don't exit your data centers. You know, AWS is the ultimate and vendor lock-in. You are locked in on the entire stack, right? And there is no private option. Now we do support nine of their core services through our Eucalyptus product within Helium. But I think Microsoft's vision is very different, much more like ours around hybrid cloud. So we're doing joint engineering. We're hiring out our Azure centers of excellence. We're putting feet in the street to get up to deploy it. And Azure Stack, you know, we're targeting next summer for version one. Got, you know, close to 100 companies now who are testing and deploying Azure Stack right now across a wide variety of industries. And for us, it opens up a much bigger part of the private cloud opportunity, grows the market. But I think for those shops that are really, really focused on Microsoft, these dotnet shops just gives them a real option to go to cloud and have both private and public. How about the developer strategy? Does HPE need to own developers? No, we don't. In fact, what we need to help is we need to help CIOs help own their developers, right? Take back their developers, you've kind of gone around them, give the developers that same experience, but give them that on a private cloud, right? That abstracted infrastructure, you know, the instant provisioning, the APIs and the self-service, that's what we're doing. We're helping CIOs remain relevant, transform from ops efficiency to value creator and cultural warrior. We're doing that with our technology and services so that they can enable their developers.  In two years, I've got to be flipped that. I've got to be 80% dev and 20% ops. That's what our portfolio and our strategy delivers on. Brian, does this acquisition change the rules for SUSE at all? Does this portend new directions for your company? Actually, no. I mean, everything that he just described is the same thing we've been observing, right? With the public cloud, the private cloud, the hybrid cloud and models that everybody's moving to. And even internally within our own organization, we've moved to a dev ops even for our development and for all of our testing, our CICD kinds of things. As we develop products, we do exactly that thing. And we're getting exactly the same things you are as far as the customers wanting to move that way. In fact, when we started doing OpenStack Cloud, the IT groups would come to us and say, we want to be a service provider to our internal customers. Right, and they wanted to manage that infrastructure so that their developers could do exactly what they needed to do. I think we have very shared interests, you know? I think what I was really excited about with our discussion with SUSE over the last few months has been their willingness to, I ask this, engineers want more than I ask, right? And that's where PASS comes in and Cloud Foundry became a really good expansive option for your portfolio. And I think this is good for the Cloud Foundry or a foundation as well. And like I said, we're going to continue to invest in those ecosystems, but on the customer-facing side, they're going to invest in the distros. How about the micro-focus acquisition of HPE software? I mean, micro-focus, if I understand it, largely runs a portfolio of companies. Do you see that changing or do you see opportunities to now collaborate across those sets of companies or do you expect it's going to be largely sort of independent? Honestly, I think a lot of collaboration, right? When I've talked with a lot of the folks that are probably heading towards micro-focus, they're very anxious to start working and doing that collaboration. Yes. Is that different than the way you guys operate today or, I mean, just smaller scale? Now it's going to be substantially larger. And how about the developer question? Same developer question for Susie and micro-focus. Do you need to own the developers? Actually, we do because we're part of the open-source community, right? And open-source is in our blood, right? It's our heritage, and that's a passion that we have within the company. I mean, everything we ship is open-source, which means we have to collaborate with the community, which means we have to develop there. We have to contribute, we have to collaborate. Those are two core values. What are you guys doing specifically around developer outreach today? About developer. About developer outreach? Well, certainly, we're growing considerably. And not just through the acquisitions like this, but organically as well. We're hiring lots of folks. And honestly, when we hire folks, we look for the cultural aspect. Do they understand open-source well enough? Because we know we can teach the skills. And if they have the skills, that's a bonus coming in, right? Like our talent that we acquired. There's some real bonuses too. Like, ArcSight's an amazing product, right? For threat detection, right? Every hybrid cloud should have ArcSight in it. We create, for example, like DreamWorks, who rave about having that threat detection across their hybrid cloud, right? And I think, so I think there's important pieces that are needed in their portfolio that'll be important parts of our recommendation. So you become an ArcSight reseller. That's right, that's right. I mean, Microfocus becomes, I think, the sixth largest software company in the world. I might be off by one, but it's a big company in software. So with some great assets. Vertica is an amazing database. It's great for big data, it's distributed, it fits well on OpenStack. We're going to push that. The only difference here is we're going to also push other options. We're going to have the more use case specific, right? Being able to match the right kind of database, the right kind of technologies. You know, in some cases, I ask the answer, and you may just need Docker on bare metal or container on bare metal. Great, and so we're, we want to expand the number of options. And our partners, once they understand and see that, like the slide you mentioned, right? With all the logos on it. You know, they go, wow, this is great, I'm really excited. But they want to know, tell me which ones. I can't invest in everything out there. Pick the winners. We're picking the winners. And just to kind of jump on that too, I mean, SUSE is very partner centric, right? And a lot of things that we do in that space. So like the ArchEye thing, what security, you know, as we go through OpenStack deployment, they'll ask for exactly those extra features up and beyond. So we're going to use that. And we're going to leverage all of our partners to get those pieces in too. And, you know, if you have recommendations, awesome. So as being part of MicroFocus, give you access to resources that you did not have prior to the acquisition or the pending acquisition of the HPE software portfolio? Yes, that acquisition has been really important for us at the MicroFocus level and also the one for the SUSE one. Both of those give us access to those resources that were hard to engage before, right? Once they've freed up and become kind of software entities on their own, they're much more willing to work with folks like SUSE, okay? Or other partners that we have. So how might that affect your roadmap going forward? I'm sorry? How might that affect your roadmap going forward? So the beauty of our roadmap is it's partner and customer driven anyway, right? So as we get requests from partners like HPE, we put in a feature request that drives our roadmap. We mesh that together and prioritize that for what we're going to contribute upstream. Because again, we've got to have it open source. So all of these inputs are valuable to us and we prioritize them and we have a regular cadence for all of our products. Is everything you do open source? Yes, absolutely. Everything we ship is open source. And their business structure is interesting, right? And the smart, right? They kept that like wholly owned group, right? It's got its own, and I think that's important. So you're part of this kind of mothership but you've got that real autonomy, right? So to engage a community differently. Well, the winds are changing, the winds are shifting. Bob, we'll give you the last word. Yeah, no, I'm excited. I think, you know, I'll pull a plug in for a site called Prepare for Billions we launched today. I think, you know, when you match up these cloud platforms and technologies and technology synergy with the Internet of Things and this disruption, I read today that the GDP for global will go from 4.1 trillion in 2016 in nine years to 11.1 trillion. That's all driven by IoT. I'm really excited about the fact that these technologies we're building are designed to help customers take, you know, actually capitalize on that. And I think you'll see more there PrepareForBillions.com, check it out. PrepareForBillions.com. PrepareForBillions. Floral, okay, cool. Gentlemen, thank you very much for coming to theCUBE. It was great to see you. Thank you. All right, keep it right there, buddy. We'll be back with our next guest. We're live from Excel London. This is theCUBE, HPE Discover 2016. Right back.