The Hydrogen Atom, Part 2 of 3: Solving the Schrodinger Equation

back for some more hydrogen I presume well you've come to the right place because that's what this video is all about today we've got eigenstates we've got spherical harmonics and Radial functions and the absorption emission spectrum and all kinds of good stuff so come on in here and let's learn about this before we get started I wanted to say a couple of things about this video the format of this video is going to be a little weird we're actually going to start by looking at and exploring the solutions and then only after that are we going to actually solve the Schrodinger equation I think that this approach lends itself better to the video format and I think it'll ultimately be more memorable and helpful and I'll also link you in the description below to papers you can read that have all the equations in a more sequential order if you're interested in that I think that paper is ultimately the better medium for linear deductive processes but if you want to get a feel for something if you want to get the Gestalt and the familiarity with the concepts I think that that's where video can really help out so I'm going to emphasize animations and Concepts in this video all right well let's get started let's take a look at a very special family of functions called the spherical harmonics the spherical harmonics are used all over the place in math and physics they're some of my favorite harmonics they're really wonderful functions so these are called spherical harmonics the word spherical starts with s and harmonic starts with H so naturally the symbol here is a capital Y the subscript is this number L which is an integer and the superscript is a number M which is an integer L has to be non-negative so it could be zero one two and so on and then the absolute value of M has to be less than or equal to l so in other words if L is 2 m could be negative 2 negative 1 0 1 or 2. and we'll see why that is later on in the video but for now just know that we have this family of functions and they're parametrized by these two integers M and L so I want to First Take a particular look at the spherical harmonic when m equals 1 and L equals two I think this is a good example to start with so I have here the equation at the top of the screen but before we unpack the equation I want to tell you something about what I'm showing here so the spherical harmonics don't actually evolve over time but what I'm showing here is a spherical harmonic times this time phase factor that just rotates around in the complex plane the reason I'm showing you that is that the spherical harmonics are actually the angular part of our hydrogen energy eigenstates and so when you multiply in the time dependence you get that swinging around in the complex plane like what I'm showing on the top left side of the screen and that brings the equation to life in a way that I think is important to see so I'm showing that time Evolution here equipped onto the spherical harmonic because I think it gives you relevant information but just know that if you want to see what the spherical harmonic really is you want to pause the video when the phase Factor points all the way to the right that is when it's a real positive number one okay now a couple of observations about this function the first is that if you look at the anatomy of a spherical harmonic there's three parts first there's a constant and that constant is such that the function is normalized and this makes it so that the function is neither too big nor too small we'll get back to that later the second part of the function is a trigonometric function in the polar angle Theta so in this case we have sine Theta cosine Theta some of you might recognize that as one half sine two Theta and if you look at the two-dimensional plot from top to bottom you'll see that function at any given point along the horizontal axis if you just go vertically you see that sine Theta cosine Theta function finally the third part of the spherical harmonic is this e to the I Phi term and that is just going to rotate the phase angle of the complex number as you go left to right okay so if you look at the top right corner of the screen you see we have this spherical thing and on top of the sphere I colored in the colors of the function okay so let me explain what that is so the brightness of the function corresponds to the amplitude of the spherical harmonic so imagine you're on the unit sphere and each point has some latitude and some longitude and you take that Theta and Phi angle and you put it into the function you get a complex number with some amplitude in some phase so on the spherical plot in the corner of the screen I'm using the brightness to represent the amplitude of the number and I'm using the color to represent the phase in accordance with our usual complex number color map the big plot in the middle of the screen is the same function but plotted in two Dimensions where you have Theta and Phi as these two different axes so think of this as a kind of like when you see a map of the Earth and Alaska looks huge and Antarctica looks huge that's kind of what's going on here I just I'm stretching everything out so we can map it in two Dimensions so bear in mind that the poles are going to be warped when you do this so you kind of have to unwarp it in your mind but the advantage of plotting it in two Dimensions is that I can actually show you the complex numbers as these little arrows or you can directly see their amplitude in their phase and I think it's useful to see that even though making that two-dimensional kind of warps it a little bit okay another interesting observation is if you pick any arrow in particular and watch what it's doing it's just rotating around counterclockwise that's all it's doing and yet there seems to be the propagation of this macroscopic wave and that's a result of the phase gradient as a function of space in general whenever you have these kinds of phase gradients and you have this kind of eigenstate time Evolution you tend to see that kind of thing this is the spherical harmonic for m equals 0 and L equals two because m equals zero the Phi dependence of the equation is going to be e to the zero Phi in other words e to the 0 in other words one so because m is zero the function does not change along Phi and if you look at the equation all you see now is a constant term and a trigonometric term as a function of the variable Theta this is one of my favorite spherical harmonics I think it looks really cool now one thing we should notice is that you see how the function goes to zero amplitude when the phase switches if you think about it it kind of has to be that way if the function were non-zero when the phase switches you'd have a moment where it was very sharp and not exactly discontinuous but maybe non-differentiable you know we want a smooth function and why is that well if these are going to be part of our wave function then if we have anywhere that's too sharp or too non-differentiable or somewhere where the laplacian blows up that would take an infinite amount of energy density to create so we should expect that all of our equations we're going to see today are going to be nice and smooth and continuous and well behaved as we'll see later for precisely that reason we get these quantized Solutions we get these integers that give us a family of functions a family of eigenstates and all of that goes back in essence to this idea that the function has to be smooth and that implies that you have to get a certain number of nodes and that it's not just anything goes it's a discrete set of possible solutions which is something that's really fun to think about all right now let's look at another function this time m is negative one so this looks just like the spherical harmonic we were looking at first except that was m equals one this is m equals negative one so in other words the function is exactly the same the only difference is that the way in which the phase changes as a function of Phi is backwards relative to what it was before and so as the time Evolution swings each Arrow around counterclockwise macroscopically it'll appear like the waves are propagating in the opposite direction as before all right we'll look at a few more spherical harmonics this time I'm just going to talk about each one for a few seconds we'll look at the picture and we'll move on if you want to pause it and see the equation you're more than welcome to do so this is m equals 0 and L equals one and because m equals zero you see there's no Phi dependence L equals one so the trig function in Theta is pretty simple which is cosine of theta and we've got that constant out front that normalizes it here we have y zero zero this is the simplest spherical harmonic it's just a constant everywhere on the sphere it has the same value one over two times the square root of pi and now we have m equals negative one and L equals one because L equals one we see this very simple function for Theta it's just sine of theta and then we have a constant in front of the equation okay and now m is one instead of negative one and so it's the same equation but now it goes the other way so here's L equals two m equals two we see that double periodicity as you go around the longitude and you see that L is this kind of second order it's sine squared you know it's not just sine all right and here's m equals negative two l equals two so same thing as we saw a minute ago but just the other way here's the list of all the spherical harmonics we've looked at so far in case you wanted to screenshot this and explore these equations further I'd also encourage you to check out these Links at the bottom of the screen a link in the description below for more information and to see some of these other functions because I'm just showing here up to L equals 2 but you have L equals three and four and five and so on technically an infinite number if you want to explore them but fortunately we only really need to look at the first views for garmonics for reasons that we'll see later okay so now that we've seen a few examples of spherical harmonics I think it's time to look at the general formula so all the general formula is is a recipe that lets us make spherical harmonics so if we put in a number for M if we put in a number for L we'll derive the formula for the spherical harmonics and if we look at the general formula we have some constant term out front there's a square root there's some factorials there's a 4 pi and we can see that there's also this term p superscript m subscript L as a function of cosine of theta that is an Associated legendre polynomial and we'll see later where that comes from and then we have that phase Factor e to the i m Phi another thing we should look at while we're here is the normalization condition of the spherical harmonics so the spherical harmonics are an orthonormalized set of functions meaning that the dot product of any spherical harmonic with itself is one and the dot product of any spherical harmonic with any other spherical harmonic is zero so if we look at this equation to take the dot product of two functions you take one and then multiply it by the complex conjugate to the other and then you integrate that over the unit sphere so this term sine of theta times D Theta D Phi that is the area element of the unisphere and we're going to go ahead and integrate that all the way over the sphere so from 0 to Pi and Theta and zero to two Pi n Phi now in this equation I'm using the over bar notation to signify the complex conjugate normally I like to use the asterisk notation but that gets a bit messy when you have superscripts and primes and the super scripts and it's a whole thing so the over bar looks a little more concise and finally on the right hand side of the normalization equation we have a couple of Chronicle Deltas so chronic or Deltas are the most confusing way of writing either zero or one but think of them as a switch so whenever you see a chronic or Delta the rule is this if the two subscripts are the same number then The Chronic or Delta equals one if the two subscripts are different then The Chronic or Delta equals zero and so these two chronic adult is on the right side of the equation all that is to say is just a concise way of writing that when m equals m Prime and when L equals L Prime then this integral is one in other words the dot product of the spherical harmonic with itself is one and when m is not equal to M prime or L is not equal to l Prime then the dot product equals zero in other words the functions are orthogonal the orthonormality of the spherical harmonics is one of the reasons that we can expand any general function as a function of the variables Theta and Phi as a sum of spherical harmonics so that's a really cool mathematical property of the spherical harmonics and that's one of the reasons you see them all over the place if you've ever worked with like zernicke polynomials for example you can think of the spherical harmonics as kind of the spherical cousin of the zernity polynomials there's a lot of similarities here as well with Fourier analysis if you have some time I highly recommend you do a few exercises to get your hands on the spherical harmonics and make them feel natural and familiar the first thing I'd recommend is look at the e to the IM Phi part of the function and just plot that here what I'm showing in this plot is in a three-dimensional space there's our coordinate system and I'm actually using the variable Phi around the Azimuth and I'm plotting the function the Rainbow Circle shows the complex phase as the color with our usual color map and the amplitude in this case I'm just going to show that as the radial distance from the origin so e to the IM Phi is a constant amplitude of 1 but its phase swings around in the complex plane as we saw earlier here I'm also showing the real parts and imaginary parts and you can see those have a cool like flower petal kind of pattern I always thought that looks pretty now here's something interesting and you gotta see this I want you to notice this in this plot I'm showing an animation where I'm varying the value of M and you can see that for the most part the function is discontinuous when m is not an integer it doesn't really loop back to itself it just kind of goes around and then it stops and so this is really probably the most direct way of showing how continuity and smoothness and yet wrapping around a coordinate that is periodic shows you that you end up getting some kind of quantized number and if you understand where the quantum number M comes from then you understand the essence of the other quantum numbers okay so once you're really familiar with e to the IM5 I'd recommend maybe plotting some of these functions as a function of theta so you know sine Theta cosine Theta or whatever just pick any of these functions and plot them and just play around with them that's the second exercise I'd recommend then the third one would be pick any of these functions and instead of the constant term replace that like with the capital letter C for example and then solve for C using the normalization condition that the function times complex conjugate integrated over the unit sphere gives you one and once you're doing that by then the spherical harmonic should be pretty natural to you I'd also be remiss not to mention the fact that in addition to the spherical harmonics we've looked at there is another kind of function called the real spherical harmonic a real in the sense of real number valued not in the sense of like no cap for real for real but like these are the functions that chemists like to use and if you're into chemistry then you want to become familiar with these but if you're into physics I'd recommend using the functions that we've been looking at because I think they're simpler at the end of the day the spherical harmonics and the real spherical harmonics give us two different ways of saying the same thing in the sense that for those of you who have studied linear algebra you can imagine that both sets of functions are an orthonormal basis that span the space of functions defined on a sphere so in a way if you want to imagine like the spherical harmonics or like the vectors X and Y then the real spherical harmonics are sort of like the vectors X and Y that have been rotated in the plane they still span the plane but they're a different set of basis vectors so when you think in terms of superpositions and everything you'll see that we really have a lot of flexibility as far as whether we want to work with the spherical harmonics or the real spherical harmonics anyway I'm just going to use spherical harmonics in this video I'm not going to talk about the real ones maybe we'll get back to it in some future video but for now we'll just move on the reason we've been talking about spherical harmonics is that they are the angular part of our energy eigenstates they tell us how the eigenstates depend on Theta and Phi now let's look at how the eigenstates depend on the radial coordinate r so these radial functions are fortunately much simpler than the spherical harmonics these functions have real number values they're not complex it's just a coefficient and sometimes a polynomial and then an e to the minus r over some Factor you might wonder exactly where the details of the polynomials come from and we'll get to that in a second first I want to show you that there's a way of simplifying these equations so if you look at this factor a naught that's the bore radius one way to think about the Bohr radius is that it's kind of the most likely distance from the proton that an electron is going to be in the ground state of hydrogen atom and so in that sense it sort of sets the atomic length scale so a hydrogen atom in the ground state would be a couple of bore radii wide about an angstrom wide that's why angstroms are a convenient unit to use in chemistry the bore radius is 4 Pi Epsilon naught H bar squared divided by the elementary charge squared times the reduced mass of the electron and that number actually comes is out of solving for the radial functions so it comes out of Schrodinger's equation because you'll recognize that we used all of those variables when we set up the equation we had an Epsilon naught we had an H bar we had an elementary charge we had a reduced electron Mass so this thing that sets the atomic length scale we can actually calculate from the Schrodinger equation and one thing that's kind of useful to know is that if you imagine a wavelength of green light a bore radius is about one ten thousandth the wavelength of green light so atoms are smaller than visible light okay so what we're going to do is basically imagine that the bore radius a naught is equal to one in other words we're just going to Define our length scales such that our units are in terms of a naught and what that does is that greatly simplifies the appearance of the equations all right let's take a look at some of these functions so first up we have this function it's just a constant and an exponential that drops off here along the x-axis the grid lines are units so one two three four and the units are r divided by a naught because remember we're just saying a naught is one so each grid line here represents an increment of a distance of a naught and moving on to the next function we have R 2 0 so that's basically just an exponential that decays but there's also this function that at first starts off positive for small R and then goes negative and so we get a little sine flip around R equals two that's kind of neat and what that's going to do when we multiply it by our spherical harmonic that means anything beyond R equals 2 to a naught is going to have a phase switch at that point so the phase is going to go from whatever it was when R is less than 2A naught and then beyond to a naught it's going to flip into the exact opposite phase that's what that transition from positive to negative is going to do next up we have r21 and this is kind of a thing where in the limits as R approaches Infinity of course it drops off to zero so the electron is still bound and yet here it's also zero at the proton so when you have r21 as a radial function the electron really doesn't want to be right next to the proton it's a little bit spaced farther out but you'll notice that this is strictly positive it doesn't go negative for R30 we have something that kind of resembles r200 it's basically an exponential decay and the sign flips but now this time we have the sign flipping and then coming back so it goes positive negative positive and when you multiply that by your spherical harmonic what that means is you're going to have an inner spherical Zone where inside of that the phase is one thing and then beyond that there's going to be like a spherical shell where the phase is reversed and then beyond that out to Infinity the phase is going to go back to its original orientation that it was inside that innermost sphere all right r31 is similar to r21 except now there's the sine flip again asymptotically it's zero as you go to infinity and it's zero at the proton but there's a sine inversion that happens around R equals six a naught and R32 at this point things are pretty spread out um by the way one thing about these functions is as n gets higher the average value of these functions is more spread out so as you go into higher energy states the electron is less bound and it's physically going to be fluffier and more spread out from the atom foreign now that we're experts on the radial functions and the spherical harmonics let's look at the formula for the hydrogen energy eigenstates all we do is we multiply these together and we also throw in that time phase factor that has to do with the energy of the state so the interesting thing is we know that the spherical harmonics are indexed by these two numbers l and m and we know that the radial functions are indexed by these two numbers n and l l in both the radial function and the spherical harmonic have to be the same number so those are sort of tied together if you like and n l and m are constrained in the following way n has to be one two three or so on it has to be a positive integer L can be at most n minus one so if n is three then L can be at most two and L has to be non-negative so L could be zero one or two when n is three and then M absolute value has to be less than or equal to l just like we saw before when we looked at the spherical harmonics and because these are eigenstates the only dependence on time is this uniform rotation in the complex plane with angular frequency e over H bar so now the question is what are the energy eigenvalues for each of our energy eigenstates in other words for each state how much energy is actually in the system because hydrogen's bound system the energy is going to be some negative value that is it'll be the amount of energy that you need to remove the electron from the atom so for example the ground state energy is the amount that you have to put in to ionize the atom out of the ground state and send the electron flying away all right and it turns out that the energy level depends only on n and has the value of the reduced electron mass times the electron charge to the fourth power divided by eight times Epsilon naught squared times H squared not H bar squared but H squared so H Bar times two pi and then that whole quantity is divided by N squared the equations that we've seen so far this Schrodinger picture of what a hydrogen atom is is very beautiful math but what makes it physics is that it has to adhere to the strict requirement of being experimentally verified so let's see if our theory is up to the challenge we know that we get a discrete set of energy eigenvalues depending on the energy eigenstate that the electron is in usually it's relaxed into the ground state but every now and then it might be excited with light or maybe even thermally up into an excited state then it'll drop back down into a lower energy state we should expect to observe the emission and absorption of photons corresponding to the differences in energy levels between those States and because we can precisely calculate the energy level of each state so too can we precisely calculate the energy differences between each state then if we know the energy differences we know the wavelength of light or the frequency of the light that goes along with those energy differences because energy and frequency are directly related by Planck's Constant and so therefore from looking at the solutions of the Schrodinger equation we can predict a bunch of frequencies of light that we should expect hydrogen to interact with in a very unique way in a very discrete non-continuous way what I'm showing here in this diagram is a list of energy levels so those horizontal lines are energy levels down at the bottom we have the ground state and here you can see that as n increases as those States get higher and higher energy their energy actually asymptotes to zero because the energy of all these bound States is negative and so as n approaches Infinity yeah the energy gets higher but it's diminishing returns at some point the energy gets so close to zero that the electron is basically no longer bound to hydrogen and this diagram shows that so the dashed line at the very top is zero energy so that's a detached electron and proton okay now one thing we can see by looking at this is that the farther the Gap is between the two energy levels the higher frequency the photon is the way I like to imagine that is imagine that you're in whatever the higher energy state is and you snap down into the lower energy State the farther you fall down that energy axis the more energetic that transition is and the more energetic the transition is the higher frequency the photon is so on the very far left side of the screen that's going to pick up a lot of energy as the electron Falls and so that's going to emit a very high frequency Photon an UltraViolet photon on the other hand if you look at the far right side of the screen you can see that the transition between the two energy states that are very close together that hardly gives the photon any energy at all and so there it's very low frequency it's infrared and conveniently if we look at these wavelengths of light we have an UltraViolet series a visible series and an infrared Series where by series I mean categorizing these wavelengths in accordance with the lower energy state that they end up in so anything that ends up falling into the ground state that's going to be in that ultraviolet series which by the way is called the Lyman series I can never remember the names of these series I just remember that there's an UltraViolet a visible and infrared so the visible series is called the Balmer series and it looks like what I'm showing here there's four lines and there's actually a couple more similar to that far left kind of magenta purple one that are right on the edge of UV if you imagine the higher energy states that are very close to like what I'm showing there but a little bit closer to zero anyway to the right of that we have the infrared series of passion Series so how well do all these predictions actually align with experimental data well the short answer is very well the Schrodinger model of the hydrogen atom is by all accounts very precise and accurate as far as what we're used to when we talk about like everyday accounting of things that happens in the world you really have to nitpick the data to find any deviation from the Schrodinger equation as far as what's actually observed so on the one hand if you want to understand hydrogen all you really need to know is Schrodinger's equation but on the other hand there are some very subtle differences between our calculations and what we actually observe and those subtle differences open the door to spin and antimatter and Quantum field Theory and all kinds of really profound things and we'll explore that further in part three by the way before getting to part three I think I'm going to go off on a tangent and do a video on Lorenz transforms in the Rock equation and maybe a couple of other Concepts to really get us set up for part three part three is going to be looking at how the rock equation applies to hydrogen calculating the differences in energy relating to spin looking at the various corrections to the Schrodinger picture the fine structure and even briefly alluding to Quantum field Theory but I'm not going to actually calculate the lamp shift well anyway we'll get to that later for now back to this video all right well now we've explored the solutions to the hydrogen atom and I hope this has given you some visual insight as to what these Solutions look like and what they can tell us about the absorption emission spectrum of hydrogen but there's one thing that's totally missing from this video so far and that is actually solving the Schrodinger equation to produce these Solutions so far you've just had to take my word for it that this is what the solutions look like but how do we actually get here from the Schrodinger equation well I want to say a couple of things about that the first is now that we have a sense of what the solutions look like it's going to make it easier to solve the equations but also solving the equation is much harder than just having a sense of what the solutions look like all right we begin with the equation that we derived at the end of the last video in this equation we have all kinds of partial derivatives and partial second derivatives and lots of stuff so it's kind of a messy equation fortunately nature is merciful enough that this equation is separable in other words as we've seen Psy can be separated out into the product of a radial function and a couple of angular functions and for now that's all we know we don't know yet that there's spherical harmonics we haven't yet proven that but we're just going to say that PSI can be written as a function R of R radial function times a function f of theta times a function G of Phi and we don't yet know anything about what any of these functions are we just know that it's separable so the fact that the solution is separable means that the partial derivatives become ordinary derivatives because for example if you take partial PSI partial R and PSI is rfg then what you end up with is FG times the derivative of r with respect to R that's just one example but if you go through these other examples here you can see the first and second derivatives all become regular ordinary derivatives and so now what we want to do is we want to replace all of the partial derivatives in our equation with ordinary derivatives this might seem like a step backwards because now the equation is actually a little bit more complicated as far as number of characters but it's actually a huge step forward because partial derivatives are very painful and ordinary derivatives are only an ordinary amount of pain so now the next step that we're going to do is we're going to go ahead and multiply everything by r squared over rfg just to simplify things a little bit just to clean it up after we do that we can go ahead and switch some things around so here what we've done is just addition and subtraction we've just moved things around so we have all of the radial terms on the left and likewise anything that has any angular term is going to go on the right and by angular term I mean Theta or Phi now here is the essence of separation of variables if we look at this equation the left hand side has no dependence on either Theta or Phi but therefore the right hand side can have no dependence on Theta if I either even though ostensibly it looks like it does and likewise the right hand side of the equation doesn't have any dependence on R and so therefore neither can the left side of the equation because the two things are equal so even though in this equation we have R's and thetas and fives actually both sides are equal to a constant you got to meditate on that because that's the essence of separation of variables and it always feels like we're getting something for free when we do that but actually we're leveraging the separability of the solution in order to sort of bootstrap ourselves up into that observation about both sides being constant anyway what that does for us the fact that both sides of the equation are constant is it lets us separate out this differential equation into two differential equations that are coupled via this constant we'll call it capital c for now and so you see we have this first equation which is purely a radial equation it's purely a one-dimensional ordinary differential equation for r as a function of r and the second equation is now a two-dimensional ordinary differential equation for f and g as a function of theta and Phi respectively so now we're going to put that radial equation aside for a second and we're going to focus in on the angular equation again we can apply a separation of variables this time putting the F of theta terms on the left side and putting the G of Phi terms on the right side and again we can do the same old trick we can say well on the left side we have Theta but we don't have Phi and on the right side we have 5 but we don't have Theta and both sides are equal and so therefore neither of them really depends on Phi and Theta they're both actually equal to a constant when we do that oh and by the way this time let's call the constant M squared why I'm foreshadowing you'll see in a moment it's the same M that we've been talking about throughout this video um but when we do that what we find is that we can break up this angular equation into a part involving only F and Theta and a part involving only G and Phi also the part involving G and Phi is very simple negative 1 over G D Squared g d Phi squared equals N squared and so if you rearrange that equation we can immediately recognize it as a one-dimensional hemholtz equation where taking the second derivative of G with respect to Phi is the same as multiplying G by negative M squared and so by inspection this equation has the general Solutions a times e to the i m Phi plus b times e to the minus IM Phi where at first before we apply any boundary conditions or any of that the A B and M can all be arbitrary complex numbers now when we impose continuity and smoothness we find that the function has to come back to itself after a rotation of 2 pi in Phi so the first thing we can say is well if an M has to be real because if M were complex then the imaginary components of M when multiplied by I becomes real and then so you get a spiral going in or out and that's no good because the spiral doesn't come back to itself so M has to be real and for reasons we saw earlier M has to be an integer now there's still the question of even if we know that m is an integer how do we handle the parameter space A and B how do we deal with that and to answer that rigorously you really have to look at it in the perspective of the full angular equation with both Phi and Theta but for now what I'm going to say is a sort of an onslaught we can take the equation that g is just equal to a times e to the i m Phi and we'll find later that a will become normalized by a normalization condition and so on if we now return our attention to the differential equation for f as a function of theta the first thing we'll do is we'll rearrange and simplify it as shown here but now the problem is if we want to solve for f as a function of theta it's not nice having all these Sines and cosines in the equation that's confusing so we're going to do a trigonometric substitution so we're going to define a new variable PSI and x i I don't know if I'm pronouncing that right I can never pronounce this letter right but anyway I assume it's pronounced PSI so PSI is going to be defined as cosine of theta and that will let us immediately replace the cosine of theta term in the equation with PSI at the same time by the relation that sine squared plus cosine squared equals one the sine Theta term in the denominator can be written as the square root of 1 minus PSI squared and if we do those substitutions we end up getting rid of all of the trig functions in our differential equation and now we just have something involving these PSI terms and this PSI squared in a square root kind of thing but here's the problem our derivatives are still in terms of theta and we need to get those derivatives in terms of PSI so if we evaluate the first derivative we get DF D Theta and using the chain rule we can see that that is DF dxi dixiety Theta and we end up with negative sine of theta times DF dxi I really hope I'm pronouncing PSI correctly anyway so now taking the equation that DF D theta equals negative sine of theta DF dxi and noticing that sine of theta is root one minus PSI squared then we can write that DFT theta equals negative square root of 1 minus PSI squared DF dxi and so there you go so now the first derivative of f with respect to Theta is written as a derivative of f with respect to psi now of course this is a second order differential equation so we need to do the same thing for the second derivative of f with respect to Theta and this one there's a couple of chain rules and a product rule and when you figure it all out you end up getting that the second derivative of f with respect to Theta is equal to this one minus PSI squared quantity times the second derivative of f with respect to psi minus PSI times the derivative of f with respect to psi so now we've got our second derivative we've got a derivative and so we can rearrange the whole equation purely in terms of F and PSI and we get this equation shown here this equation is incredibly painful to solve I wasted a weekend once trying to solve it if you're looking for a good exercise set m equals zero so that whole M term goes away and you can probably solve that you got to write out some Taylor series and make a table and do some pattern recognition but that's not too hard when m equals zero when m is non-zero oh my gosh this equation is tough but fortunately mathematicians exist and they've solved this so this equation has a name it's called the associated legendre equation sometimes it's called the general genre equation as opposed to the legendre equation which is the special case that m equals zero this non-zero M term is what makes it the associated equation and this has Solutions the associated legendre polynomials they're symbolized by a P with a subscript L and A superscript M you'll recognize this from our general formula for the spherical harmonics that's where this comes from so now we know that F of theta has the form of an Associated legendre polynomial whose argument is PSI but now remember that PSI is defined as cosine of theta and so therefore F of theta is an Associated legendre polynomial of cosine of theta so when we combine the co-latitude and the azimuthal equation in other words we take F of theta times G of Phi we smoosh them together and we end up with a spherical harmonic because you'll notice that e to the IM Phi term that came from the 1D hemoltz equation for G as a function of Phi and that PML term came from just now the associated the genre equation now we happily glossed over all the details of how the associated legendre equation was solved but at least we gave it a name and at least you can go look it up if you're interested in more details okay now the normalization constant in front of this equation that thing with the square root and all that that just comes from the normalization condition of the spherical harmonics that we looked at earlier and so now what we've proven so far is that our wave function PSI is going to be some radial function that we don't know yet times the spherical harmonic oh and by the way while solving the associated legendre differential equation one of the results that pops out of that equation is that that c constant that couples the radial equation with the angular equations has the value of L times quantity L plus one and ultimately I mean without going into the details of the associated with genre equation you can think about that in essence as a result of the fact that the poles of the spherical harmonics have to be well behaved meaning smooth and continuous and regular like you know pretty functions without any sharp points and so basically what happens is that because m is an integer you get a certain amount of winding around Phi and then when you try to solve this differential equation in Theta in order to keep everything smooth that quantization kind of infects the equation and so L also has to be quantized in a certain way and so anyway you get this two-dimensionally smooth function and that requires having these two quantized numbers l and m because the radial function and the angular functions are coupled via that coupling constant C we can write L times L plus 1 in this radial equation so now the that number actually comes in and infects the radial equation so you see it's all tied together even though the equation is separable okay let's go ahead and rearrange the radial equation we'll subtract the right hand side L quantity Del plus one so we get a zero on that side we'll multiply everything by capital r and we'll divide everything by r squared and after a bit of algebra we get the equation shown here now look inside the parentheses and you see these three terms the first one on the left that is the energy related to the electrostatic potential between the electron and the proton right that's the Coulomb's law in the middle we have the energy eigenvalue and on the right side we have this strange mysterious term involving l and H bar squared and 2 mu and r squared but by dimensional analysis all three of these terms have units of energy so somehow this l number is giving us some kind of weird energy in the equation well as it turns out that's energy from angular momentum and L is the angular momentum quantum number and we can actually show that angular momentum is quantized by thinking deeply about this equation so if we had some angular momentum we would expect it to give rise to a kind of rotational energy that would be equal to the absolute value of the angular momentum L squared divided by two mu times r squared because the electron is in the central potential of the proton so anyway if we take that equation and we try to solve for the magnitude of the angular momentum L we end up with r times the square root of 2 mu times this rotational energy term now if we identify that rotational energy as being that term L quantity L plus 1 H bar squared over 2 mu r squared we can solve for the absolute value of our angular momentum L as simply H Bar times the square root of this term L times quantity L Plus 1. and so what we find is that the magnitude of the angular momentum in the hydrogen atom is quantized when L is zero there is no angular momentum when L is one the angular momentum takes on a value of H bar root 2. when L is 2 the angular momentum takes on a value of h-bar Root 6 and so on so that's pretty cool we can show by looking at the radial equation that angular momentum is quantized because of this number L nice all right so as we saw earlier the radial equations are easier to look at than the spherical harmonics that's just a polynomial and an exponential ironically I think it's way harder to solve for these equations than the spherical harmonics and it's certainly beyond the scope of this video but of course Link in the description papers you can read them you can learn the math but while we're here there are a couple of things I wanted to say about this radial equation so if we look at this equation let's go ahead and Define the dimensionless variable rho as the square root of negative eight mu times our energy eigenvalue divided by H bar squared r are in mind that our inner dragon value is negative so it's actually a positive under the square root when we do that we can transform the equation so we replace the lowercase r with rho and we do all the algebra and we end up with the equation shown here now if you think about in the limit as rho approaches Infinity that is as we go farther and farther out from the proton very far away any term in this equation that has a one over rho will vanish and so in the limit that we're far from the proton the equation will look a lot like the second derivative of r with respect to rho minus a quarter of R equals zero and by inspection that has the solution that R drops off as e to the minus rho over two so that's interesting so now we know that we can look in the definition of that row factor for the Decay rate of the wave function in r as long as we're far from the proton so if we go back to the definition of rho and we insert our energy eigenvalue and we rearrange things we get this equation that rho is e squared times the reduced mass of the electron divided by two Pi n Epsilon not H bar squared r now if you consider the ground state when n equals one that n term drops out and remembering that the bore radius is this term 4 Pi Epsilon not H bar squared over e squared mu we can rearrange that and we can show that in the ground state rho is 2 over a naught times R and so therefore the limiting behavior of the radial function is going to drop off as e to the minus r over a naught in other words a naught is going to be the natural Decay rate of the wave function in the ground state so this kind of reasoning tells us that out of the Schrodinger equation comes the atomic length scale now that of course there's a couple of things I've lost over there how do we know what the energy eigenvalue is right how do we know how relevant this limiting condition is when thinking about the equation right all of that's true all of that's valid again this is not the most rigorous video my emphasis is on animations I'm just walking through the derivation to the extent that I can within a reasonable time frame so all right uh you know I put a lot of time into trying to show the solution of the radial equation in a way that didn't take like many hours and I couldn't find a way to do it so let me just show you the general form of the equation and I encourage you to read the papers to learn more but anyway the general form is that our radial function r with a subscript of N and L is going to be this expression shown here where you recognize a normalization constant you'll recognize the bore radius a naught you'll recognize this polynomial and this exponential decay and then there's this strange function capital L with a subscript and a superscript and that is an Associated lagera polynomial you got to look that one up I don't have time to go into it here but oh it's one of those complicated ones okay and with that I think that'll complete our tour of the math and let me just be totally open about something I actually regret not being able to show the derivation of the energy eigenvalues because that involves really interrogating the radial equation the legare polynomials and how they're derived and I just wasn't able to fit that into this video and I figure look if I've piqued your interest if I've shown you the lay of the land as far as what these Solutions look like and how the equation works if you're motivated you can find in the papers how to calculate those values and it will take some time okay it is you know it's kind of a complicated equation but um anyway so I hope I'm at least pointing you in the right direction but I do regret not being able to show that derivation oh another thing I have to mention I totally glossed over this in the last video and I haven't mentioned it yet in this video but there's actually a lot of nuance involved with the details of these transitions between the different energy eigenstates especially when you think about angular momentum and how the electron interacts with the photons you can imagine there's a lot of detail in there that I'm just totally skipping over in these videos and also of course that has implications vis-a-vis the relative stability of these different states right so the more likely a transition is like that's going to affect how long the atom is going to be in whatever excited state so you know all of that is I think more advanced stuff that I'm not going to get into in this particular series and in addition to that there's a variety of other topics that Branch out of these hydrogen wave functions so for example we can think about the notion of superpositions and the time dependent Schrodinger equation and a wave function evolving over time and kind of sloshing around but then collapsing to an eigenstate when it's observed and so we can use the hydrogen atom as a kind of toy or a kind of sandbox to test and play with these Quantum ideas in a very specific instantiation so I do plan on referring to the hydrogen energy eigenstates in future videos whenever making a broader point about quantum mechanics or getting philosophical these are a very useful concept to have to keep in your back pocket and pull out whenever you need it anyway all that's to say I'm sure there's a lot that I've left out of this video but that's precisely why this subject matter is interesting because it branches off in so many different directions because it has so many implications that's why I think it was really worthwhile to make this video and that's why I hope you got something out of watching it

Channel: Richard Behiel

Views: 138,286

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Id: acN7E7AUHPk

Length: 45min 59sec (2759 seconds)

Published: Thu Jul 20 2023