We're looking at one national WKB theory, so we want to solve the short-term equation in one dimension, pretty simple equation, with some energy keys. We're looking at energy eigenstates. If you imagine a potential energy scaling that's called capital L, and I've kind of sketched here what that means, we're interested in a situation in which the buoyant wavelength of the wave culture, which we call lambda, is much less than the scale. So we're interested in the condition of the wavelength of WKB theory as a lambda that's much less than L. The lambda, of course, is relatively tuned in momentum by the de Broglie relation that's 2 pi h bar divided by t. By use of de Broglie relation, one can say that the lambda is small if h bar is small. And frequently, the WKB approximation is described by saying that the valence of h bar is small. It isn't really dimensionally correct to say that. If you want to put it in dimensionally correct language, you can say that the wavelength is much less than the scale of a potential. Nevertheless, the WKB expansion isn't, in a sense, the formal power series expansion of powers of h bar. Now, by the way, the momentum that comes from the Broglie relation is now a function of position because the wavelength of this wave actually changes as you move around. Now, the need to interpret this will have a function. Yes, that's something you should figure out. Pay attention to. In any case, this leads by looking at a picture like this. This leads to the WKB-onzox of the wave function, which is that it's a slowly varying amplitude, and it's an rapidly varying phase. Slowly, there may be a very little scaling of the potential that you see by this amplitude I've drawn here. And a rapidly varying phase means that in a short change of x, the phase undergoes some sort of thermal oscillations. In fact, in a single label, which is regarded as a short ring here, the phase changes by, of course, by 2 pi. If you think of h bar as being small, the rapidly varying phase is achieved by the fact that since you've got this h bar in the denominator here, the small change in s produces a large change in the phase. In any case, by refining the change in s equal to 2 pi, delta x equals lambda at the end of the hour last time we derived this relation that said that s prime of x, the derivative of the function that appears in the phase, is equal to this momentum function p of x that comes from the de Broglie relation. Now, pay attention to this function p of x because it's being defined here in terms of the local de Broglie relation of the wave length in a common way. But it also has a classical interpretation which is going to come out in a minute. So these two things will be connected together. It's also, as I said, a derivative of s. Alright, so I think that takes us up to where we were at the end of the hour last time. Now, this wkb-onsatz, so the name of the next step is to use this wkb-onsatz in effect, plug it in the Schrodinger equation and give equations to the A and S and then we've got the solution in this approximation. The wkb-onsatz by itself is not a systematic expansion in powers of h-bar. It's rather just something we look down by integration by looking at a picture like this. However, we can put it into the form of a systematic expansion which I think clarifies the structure of the theory in the following way. First of all, let me just rewrite the derivative of k-b-onsatz by bringing the amplitude A up into the x-colon. So I'll write it as e to the i under h-bar times k-b-onsatz plus the logarithm of A. That's no change, of course. But when we do this, you can notice that when Mel appears in the x-colon, he's starting to look like a power series in expansions in powers of h-bar. So the first term goes as h-bar to the minus one and the second term goes as h-bar to the zero. And so what this suggests is if we replace this by a systematic expansion that looks like this, e to the i over h-bar times the series, which I'll write up this way, let's say w-zero at most order plus h-bar times w of x plus h-bar squared times w of x plus dot, dot, dot like this. So if we replace the wave function psi by this expansion in x-colon, you can see the idea is that we're making a power series expansion of the logarithm of the wave function in powers of h-bar, where we're thinking of h-bar as being small. Now, why don't we expand the wave function itself if you're going to study this in the classical limit? You'd think h-bar is small, you'd want to expand psi in powers of h-bar. The answer is that when h-bar goes to zero, if your whole classical formula is fixed, psi does not approach a limit at all. Instead, it just starts oscillating more and more rapidly because of the void wavelength getting shorter and shorter. So it's not suitable to expand psi in powers of h-bar. But instead, we're going to expand the logarithm of psi factoring out the 1 over h-bar, which is what takes care of the rapid oscillations. So in any case, this is what we get and it's a systematic expansion.  So I'll copy it again here to say psi of x is equal to e to the power of h-bar times w of x, this becomes a definition of w now. So the next step what I want to do is to take this version of psi and put it into the Schrodinger equation and get an equation for w. Oh, by the way, maybe before I do that, let's notice that comparing this series here to w of zero is the same as s. So let's write that down. w of zero of x is the same as s of x. It's just another notation for it. And w of one is equal to the logarithm of a except we have to multiply by i's. So w of one of x is equal to as you'll see minus i times the volume of a. And then there's higher order of corrections too which were actually further refinement of the original WKB pharmacist. All right, so to get back to where it was just a moment ago, let's take this expression for psi and plug it into the Schrodinger equation. We have to evaluate the kinetic energy term of the second derivative of psi. So let's start taking derivatives. So psi prime you see is equal to the i over h bar times w prime multiplied times psi. First derivative. Taking the second derivative, psi double prime is equal to first I differentiate with w so I get i over h bar w double prime times psi. And then the difference in psi brings out another factor of i over h bar times w prime. So we get minus one over h by squared times w prime squared times psi. And it will allow me to do a bad thing which is to use my eraser here. I'll just factor it out and come into factor of psi in both these terms and write it like that. All right, so that's just the algebra. Now if I plug this psi double prime into the Schrodinger equation since you now see psi double prime as proportional to psi and the potential energy term is also and total energy are also both proportional to psi. And I make this substitution the size to cancel from all three terms. And so what's left over is this and it's minus h bar squared over 2m times this coefficient in the round brackets. I'll reverse the order of the terms minus one over h bar squared w prime squared plus i over h bar times w double prime. And then add the other two terms which is the potential energy v of x is equal to v. This is what we get. And let me now just clean this up, multiply through by minus h bar squared over 2m. And what we get then is to set one over 2m times w prime squared minus i h bar over 2m times w double prime plus v of x minus e, excuse me, is equal to v. Like this. And I'll put a box around this because it's a useful intermediate step to look at. This is the Schrodinger equation which has been transformed from psi into its order. In fact, there's no approximation at this point. It's exactly equivalent to the original Schrodinger equation. However, what I would do now is to substitute this expansion in for the w here. So this becomes one over 2m times w prime and becomes w zero prime plus h bar times w one prime plus dot dot dot quantity squared. That's what it says here. Then we get minus i h bar over 2m times w double prime which is the same as w zero double prime plus h bar w one double prime plus dot dot dot. Plus potential energy v of x equals e. So this is to take this series and to collect it order by order in powers of h bar. So h bar to the zero power which is a leading term. As you see, we just take the value zero prime here which is squared. So the first term is one over 2m times w zero prime squared. And then the next term is already order h bar so it doesn't contribute to h bar to the zero. And there's potential energy which don't depend on h bar so this becomes plus v of x is equal to e. And take back the soft little bit from the h bar to the zero so you can see the equation. So this is the lowest order equation to get. Now, at first order in h bar we're going to go to the first two orders of this. At first order in h bar there's a cross term here and I multiply these two terms together so I get 2w zero prime times h bar times w one prime divided by 2m. So this gives me one over m times w zero prime times w one prime. Over here, like, they do get a order h bar term from the w zero double prime so it becomes minus i into 2m times w zero double prime. And then v and e don't contribute to h quarter h bar so the whole thing is equal to zero. And this is the beginning of a hierarchy of equations. I'll stop there, but if you promote it, you can go on to the second order to get another equation. And observe the general structure of these equations. The first one is an equation for w zero alone. It doesn't involve any higher returns. And if you solve this for w zero then you can plug that into the second equation here and here for w one. And likewise you go to the next order so you solve this one for w one. Then you go to the next order you can plug w zero and w one to get an equation for w two. It's a hierarchy that goes on this way and you can solve it order by order. We're only going to be able to do the first two quarters. Now, let's take these equations here in terms of w zero and w one and let's put them back in terms of the original W Gaby-Hansatz. So with the first equation that's easy it becomes one over two m times s prime squared plus v of x equal to b. And for the second equation let me do some algebra on that. First of all we have to cancel what you see here. So the w zero prime becomes s prime. The w one prime since w one is minus i log a this becomes minus i times a prime divided by a. The second term is minus i times s double prime that whole thing is equal to because w zero is s. So it's minus i s w prime to s double prime equals zero. And you change the two minus sides to a plus and cancel the i's to make it go away. And this can be written now as a prime over a times s plus one half s double prime equals zero. And since that's sort of money out of the rules we carry out over here. And if I take this equation and let's multiply it times twice a squared which clears it up. Then the first term becomes two a a prime times s which should be s prime. And the second term becomes a squared s double prime equals zero. This in turn is an exact derivative of the left-hand side. In fact it can be written as d d s of the quantity a squared times s prime equals to zero. And that's the most convenient way of writing this first order equation after we've converted back into these s and a variables. So over here on this box then is the version of the equations that I want with the function s and function a. Which are the two functions that have to do with the phase which are the w-p and the on socks. I really started writing the wrong formula this morning so I'm afraid I can cover this up. I'll write out these two equations once again. And here they are. So the first equation over h part of zero is this. It's one over two m s prime squared plus d of s equals e. In the order h part of the first equation is d d s d squared times s prime equals zero. Pretty simple equations actually. The first equations is actually a version of the so-called Hamilton-Dakota equation which has a long history in classical mechanics that goes way back before the invention of quantum mechanics in 1830s and 1840s. The second equation is sometimes called an amplitude transport equation and that's what I'll call it today. So we need to solve these two equations. Now let's we need to solve the let's solve the Hamilton-Dakota equation first. Obviously the way to do that is this is s prime dot s 12 s prime squared. The way to do this is to solve algebraically for s prime. And if we do, we find that s prime of x is equal to plus or minus just bring two m across and take the square root. Plus and minus the square root of twice the mass times the energy v minus v of x like this. Let's just take the plus sign of this for now and we'll worry about the minus later on. I'll remind you from the previous board that we figured out that s prime of x was the momentum which was the momentum determined by the local de Broglie wavelength. If we take that as this function, definition of this function of p of x, what we now see is that we have a little additional information now as we see what we have in formula for p of x of that de Broglie wavelength, local de Broglie wavelength of the wkd wave function. What does it mean, however? Let's get an interpretation of that. In order to do that let me take an example. Let's suppose we have a potential energy that looks like this. It's still a one dimensional problem. Let's suppose the potential is essentially zero for large negative x but that is the movement of the region of positive x that rises up like going up to a mountain and let's say it stays up forever so there's no possibility of entertaining through with this mountain and let's suppose there's an energy which looks like this. Now the energy that I want to talk about is the energy that appears in the Hamilton-Chicobbi equation and actually that energy is the same as the energy eigenvalue in the original Schrodinger equation. That's where that came from. So but in a picture of the potential it looks like this. Now, classically a particle coming in from negative x would originally be a free particle and it would be a natural zero. This would be an x term here. But as it comes in it starts to feel the potential and it slows down. It reaches, it comes to a standstill when the energy is at energy zero just right here. This is the class of turning point which I'll call XTG and then of course it reverses and comes back. If I take some x-value from a typical x-value here like this then you can see a particle going in the forward direction and one is going in the backward direction and the magnitude of the velocity of course is the same in both cases but the sign is changed exactly for it. Now, on the same scale allow me to make a phase speed a phase speed of classical phase space position momentum phase space. Let me bring this x turning point down here also to the lower graph The orbit in phase space of large negative x is a straight line in a constant momentum because the particle is going free so the momentum is constant and it just keeps moving forward in that. However, when it starts to get to where the potential is active it slows down so the momentum goes down like this and then goes right through momentum equals zero at the turning point and it turns around like I sketched this well and it comes back out like this so the orbit in phase space looks like this, it's kind of a leaf plane. This is really a scattering problem it's a one dimensional scattering problem of an impenetrable barrier. Likewise, if I take this x to the typical x scale here I bring this dotted line down like this and again here's some typical x scale I'll do what you see once again is the velocity of the particle passes this x point twice once going in the momentum is positive and the momentum is negative like this. Now, this curve in phase space what is the equation of this curve? The answer is it's just given by conservation of energy. If you write down the total energy of the system which is p squared over 2n plus v of x this is now the classical energy and instead of equal to the total energy which of course is conserved then you get an equation for a curve and the x p plane and it has to be the orbit the orbit is just a plot of a curve which you get by setting the classical Hamiltonian equal to the energy but in particular this allows us to solve for the momentum values that correspond to a given x position and that momentum just solves this equation for p and that momentum gives us p as a function of position as you see and it's the square root of twice the mass times the energy minus v of x in the Hamiltonian well that agrees with the same p of x that came out of solving the Hamilton-Jacobbi equation from the WKB approximation so the local momentum connected with the local de Broglie wavelength from the WKB wave function is exactly the same as the momentum of a classical particle which is some moving moment together potential with two sides plus and minus so that's the interpretation of this function of p of x and you can see now right away that the momentum is going to be shorter where the momentum is higher where the velocity is higher and if we longer as the particle slows down I didn't exactly sketch that very well on this diagram but it would mean that as we rise up and the potentials of the velocity decreases this wavelength is going to get longer down here in this lambda now so that's the interpretation of this function p of x which is also s prime of s solution for s of x is easy it's just an integral with an upper limit of x of p of x prime d of x prime which to be a little more explicit is with an upper limit of x of the square root twice the mass energy minus p of x prime times of d of x prime of this what about the lower limit of this integral the lower limit of the integral is nothing but a constant of integration to help the Jacobi equation because you can set it equal to anything you want because it's a constant of integration we just call it x0 you can see that if you change x0 to something else x0 prime all it's going to do is just add a constant to s since s is the phase of the wave function let me write the wkv onslaughts down again so that you have here on this board if I change the phase s by a constant actually the phase is s over h but if I change s by a constant you can see an ounce of just changing the phase of the wave function so this lower limit acts as related to the phase convention for the solution and right now let's worry about phase conventions later so we can set this x0 to anything that does convenient anyway that's just the meaning of that lower limit there's also a geometrical meaning now if I take x0 let's say over here just to choose a value for it s of x is the integral of this p function between x0 and x which is to say this is the area here so the area here the area here under the classical curve is the same thing as this function s of x what I'm saying is the classical part moves along you can visualize that s of x is a little integrator with it and it integrates pdx and as it does so what it's accumulating is this function s of x which is divided by h bar is the phase of the quantum wave function so you can imagine that it's also creating oscillations so I'm not going down as it moves along it's a way of thinking about the solution now now I'd like to shift the tension now to the quantum interpretation but with this WKB Onsatz in particular to examine the probability density and probability current this is only a one-dimensional problem so the current is just a scalar and only a vector of one component but if we use the definitions we've talked about before the probability density is the square of psi and the probability current which I'll call j I'll remind you it's the real part and then you have the velocity operator which is minus i h bar over mass m acting on psi this is the definitions we get oh excuse me it's minus h bar on the m times the derivative pdx acting on psi this would have been a gradient in three dimensions but since this is one of the probabilities of the definitions we have now if I take the WKB Onsatz this is only an approximation of the number but in this approximation let's take this and plug it in here and see what we get when you can see right away the rub is just equal to the square of the ampers room that's assuming that s is real but s is real as you see up here in these pictures that I've stacked and so that's easy and so a squared is just the square of the wave function what about j well that's equal to the real part here I'll do it out for you, psi star is equal to the amplitude times either the minus i s over h bar and then we have minus i h bar over m which is a constant and then we have pdx applied to psi which is way up there there's two terms one comes to differentiating a and the other one from the exponent so one term is this is an a prime and the next term is a plus i times a i over h bar and this is all multiplied by times d by s over h bar and I close all of that so this is just a square of the algebra calculating j now this term i here times the amplitude a as you can see is purely imaginary and this term here times that second term that has an i and it's going to have a minus i times the plus i so this is purely real so the result is a prime term goes away because we're taking the real part and what's left over is is that this turns into an a squared which takes care of that a and that a there the phase here and here of two phases cancel out as a complex conjugate minus i times plus i cancel there's one over md h bar's cancel so what's left over is just s prime divided by m okay however we decided that s prime was the same thing as this momentum function p of x which we have an interpretation for up here and this is divided by m let's call p of x let's call it a velocity let's call it v of x so it's a logical definition here if p is a momentum of particle I can't reach it I'll just say v of x is v of x over m so this is just the velocity of the particle that goes with x position really one of the two velocities because there's plus and minus but it's part of the momentum of the directions let's take this to be the positive velocity and so to summarize then we get j, rho is equal to a squared and we get j is equal to a squared times the velocity v which is the function of x like this okay so in the wk of the approximation this is the quantum density and current at least we have to satisfy a continuity equation this is the continuity equation which we talked about the last hour I believe except in one dimension instead of a divergence it's just going to be working on the derivative so well it's going to be partial derivative partial of j with respect to x equals zero so this is the one dimensional continuity equation however nothing here depends on time everything's, rho for example is a transition that's because of the original shorting equation they're looking at is the time-independent Schrodinger equation so we're really interested in stating our solutions so this term goes away and so the continuity equation just turns into derivative of j in fact let us call it an ordinary derivative since the original time derivative anymore and so the continuity equation reduces just dj v axis equal to zero however j is a squared times v so this becomes d dx of a squared times v of x is equal to zero and apart from the factor of 1 over m you'll see somewhere here yes up here that's the amplitude transport equation so the meaning of the amplitude transport equation is conservation of probability it's the continuity equation for that given that fact so now we have an interpretation of both of these equations and also for the function we'll have an interpretation of s in terms of the classical mechanics we'll have an interpretation of a let me address that question next let's say a classical interpretation it works like this to be specific let's take the case of an oscillator suppose I have an oscillator like this and in face space this is the orbit so the particle is going around in circles like this this topological circle for a harmonic oscillator this is an ellipse but for other oscillators not linear oscillators will be before we look like an egg or something but topologically it's a circle now let's take some small interval dx like this by the way let's let capital T here be the oscillator a lot of time it takes for the classical part of it to go around let's define let's define a I'll call it a classical probability density which is a function of x and let's define this way we'll say the classical probability at x times dx which is a small interval this of course is the probability of finding the particle in this small interval let's declare that this is equal to the amount of time dt which the particle spends in this small interval divided by the total period it's just the fraction of time the particle spends in that small interval then if we do this this clearly is normalized to 1 around the oscillator once the light thing to change however it's a question of the interpretation of dt the total time the particle spends in this interval is a dt from one above so this dt but it stands for the sum of those two small time intervals it might be preferable to let dt stand for the time but it takes the particle to cross this small interval on just one of these two crossings and if we do that we're going to factor a two here to double back with that new interpretation of dt this becomes the classical expression now we can then solve for the max explicitly it becomes 2 divided by the total period times dt divided by dx which is equal to 2 over the period times 1 over the velocity now why is that interesting it's because it agrees with the solution of the amplitude transport equation here it is using amplitude transport equation really can't really compute any equation so the dx of this thing is equal to 0 it says that a squared times the velocity of u of x is equal to a constant over a squared is a quantum probability instead of plugging us in we see the quantum probability loaded back to seem to be a constant fighting line with the velocity of u of x so it agrees with this classical calculation this quantum probability density in this semi classical interpretation is the fraction of time which a particle spends well times dx is the fraction of time a particle spends on that number of dx here it's explicitly normalized of unity for the case of an oscillator if we had a scattering problem which is the kind of thing I have in the company board here I don't even need to draw it again where the orbit goes out to infinity then in that case it's not possible to normalize the classical probability distribution because the time the period of t goes to infinity but it's still possible to talk about the classical density which is just now not normalize we can just in this case we can just say the classical density is equal to some constant divided by the velocity v of x and this is in fact precisely what we get from the solution of the amplitude transport equation so this thing gives an interpretation of the classical interpretation of the amplitude a in terms of particle density as the quantum row you see classical row and quantum row are the same the consequence of this result is that when the velocity gets low the density gets high in the case of this oscillator here if I plot again this now has two training points let me plot row classical as a function of x for this oscillator it looks like this it actually goes to infinity as the velocity goes to zero you will find classically this is still normalizable the intertural area of the curves is far right but nevertheless there is divergence in the classical classical density here's another way of thinking about this classical density as a measure that we populate this orbit with classical particles in such a way that individual particles are separated by the same amount of the last time you see each particle is following an orbit it's the same orbit they're just displaced they're just in different places so they're all moving around the orbit at the same time when I say that this place by a certain amount of fixed amount of time there's some delta t here and the idea is that after delta t each particle moves into the position of the next particle if you think of delta t getting smaller and smaller then the particles are all moving but they're moving in such a way that the density of particles that they create is constant in time so one can say that it's an ensemble of classical particles whose density is stationary in time that's what's occurring in the solution of the eigenvalue equation stationary state problem in quantum mechanics the particles pile up here at turning points because you see there's no pile up in the orbit and phase space but they pile up in the density and next space because we're getting projected down like this and there's slope of this curve coming down that's the reason for that okay so most of what I've been saying here has to do with classical interpretations and solutions of these two equations to help the Jacobian amplitude transport equation it's actually easy to solve the equations it's just a matter of knowing what they are now so we have these two equations still here so to summarize the solution then we've got s prime of x is equal to p of x is equal to the square root of 2 nanotimes e minus p of x so s of x is the integral of that of the limit x of p of x prime of dx like this with some lower limit and as far as the amplitude transport equation is concerned it says that a squared is whatever s prime is whatever the momentum so a of x is equal to 1 over the square root of the momentum p of x times the constant here like this and that's the solution for the amplitude transport equation if I factor a mass out of the square root of mass out of the constant it inverts this momentum into a velocity it takes impact these classical densities that I'm talking about here this constant that appears here is just the constant generation of the amplitude transport equation and you can see that it also was just related to the normalization of the quantum wave function in which you can sort of lost it then the wk beyond spots is the amplitude times the phase and so obviously a constant multiplying a is just an overall multiplicative constant inside so we can likewise as I said x0 is related to the phase of the wave function we can worry about normalization and phase later so these two constants we can set as something which is convenient of a later point all right Delvin this means that if I go back to this case of the scattering problem where the potential rises like this here's the energy B here's the turning point sketch it again here's the turning point here's x here's the phase phase orbit coming down like this all right so this is the solution for a and s here there's a couple of things that I need to a couple of minor points I need to pay attention to one is that in solving the amplitude transport equation there were really two solutions for s' having to do with a plus or minus on the square root and I just took one of the two solutions from writing these equations down but the other one that's in there is just the opposite side so s is equal to plus or minus the integral of the effect of the two solutions the result is we actually get two solutions of the WKD equation so if I write psi of x out explicitly there's an amplitude for one of them is one over the square root of p of x I'll set just one there for this constant and this is all the y times e to the i over h bar s of x but there's another solution which is one over the square root of p of x times e to the minus i over h bar s of x there's really two linearly independent solutions this of course is what we'd expect because the Schrodinger equation is the second order and has two linearly independent solutions so the general solution is going to be a linear combination of these two now for one of these solutions the function s is increasing as I move to the right because it's the area under the curve so the phase is increasing as we move to the right and that corresponds to a wave just traveling in the forward direction just like in a positive rental whereas the second term s is the phase is decreasing as x increases which means there's a wave here which is moving in the other direction and so the linear combination is a linear combination of forward and backward going waves which corresponds to the two particles which are crossing into an exposition like this if you think of this in terms of one particle you don't see it because you see a positive velocity at one point and then a negative velocity if you think about this curve as being populated by a whole ensemble of classical particles and then there's always one particle above and one particle below where their phases are increasing in opposite directions here's something I mentioned a moment ago the WKB theory expresses an approximation expresses the solution of the quantum problem in classical terms but the physical interpretation of quantum mechanics of course is very different from classical mechanics and in particular quantum mechanics is statistical fundamentally and classical mechanics is not so how does this very different physical interpretation come about in this approximation it's one of the questions to ask when we start this business and we can already see part of the answer to that is that the statistical aspects of the quantum solution is contained in this classical ensemble of particles that are orbiting around and around this orbit or in the case of an oscillator there's an infinite number of coming in scattering going back out again it's not a single classical particle but a whole ensemble of them they all however have the same energy because they all have the same orbit alright in any case these two solutions one is moving to the right and one is moving to the left so let me give this a coefficient of a call CR and this one a coefficient of a call C L where R and L just stand for right and left going waves and if we have this then this is the solution this is the WKB solution that begins alright now there's another point that I've been admitting up to this point which is that these solutions that I've worked out here actually only apply the classically allowed region in this diagram here for example the classically allowed region C A R let's call it is the region which is the left of the turning point this is where the actual particles are moving but there's also a classical combination that's called a CFR which is over to the other side there are some classical particles there may not be any classical particles over there but there are solutions that the Palomkin to Covinae have to do transport equation the classically allowed region the energy is greater than the potential energy that's what the definition of that whereas the classical convenient region is one where the energy is less than the potential energy that means that in the classically convenient region you would say the kinetic energy is negative which would mean the momentum is imaginary that doesn't have a meaning in classical mechanics but it does have meaning in solving these quantity equations and so in the classical convenient region what's going to happen is this quantity here is negative so this little box here let me just say that in this box that I've got here applied in the classically allowed region and let's write down some things that apply in the classical convenient region let's just do it this way let's define this let's write p of x p of x now has to be purely imaginary let's write it as i times the square root of 2m times p of x minus e this is just a way of defining one of the two the two with solutions plus and minus for the function p of x and let's define an integral on k of x which is an integral with some of the limit and it would be the absolute value of p of x prime p of x prime which is the same thing as the integral with an upper limit x it's a lower limit of the square root of twice the s times the potential of p of x minus e of x prime minus e of x prime okay and if we do this then s of x which is is equal to excuse me, pi of s of x is equal to minus k of x so over here in the classical so this solution I wrote down in the book this board is for the classical in the classical convenient region we can say this that sine of x is going to be a linear combination of a term which is one of the square root of x the absolute value of the moment can make it real and then we get e to the k of x over h bar and then another solution is one of the square root of the absolute value of p of x times e to the minus k of x over h bar in the classical convenient region there again are two many independent solutions k is increasing with x and so the term that has e to the k of x over h bar is a term which is diverging exponentially as you move in the positive x direction so let's call this c g growing the minus sign is a solution which is decaying exponentially as you move to the right so let's call this damping solution c v growing in damping solutions as you move to the right and the result of this is that we have now the two solutions here and the classical and the classical and the regions now to proceed with this let's take the case as I sketched it here exactly as I already sketched in which the classical and the left of the turning point in the classical convenient region is to the right we can also call these regions and let's do it this way in the classical area of the the lower limit of integration let's make a turning point which I'll call the XTP let me call it XR but R is the center of the right means the right turning point let's call it XR as a right turning point let's integrate S as I mentioned any lower limit can be chosen it's equivalent to a phase convention for the way function so let's just make it the turning point it's the obvious place to choose anyway then that gives us a let's take the lower limit here to be XR that means both K and capital S manage exactly that and so those are definite conventions for that alright so now what we have then is definite functions here with unknown coefficients actually to four definite functions and four unknown coefficients when applied in two different regions the Schrodinger equation is second order in time in space excuse me, second order in space so it has two linearly independent solutions and so the general solution of the Schrodinger equation can only involve two arbitrary constants and not four so it has to be a connection between these four constants to the right and the left and the growing and the dabbing constants in different regions of space so what we need is some way of connecting together the solution on the left side with the solution on the right side these are called connection rules now it's not possible to extend the class that allowed solution right up to the turning point and the class that forbidden region right up to the turning point and then that is the matching of there in order to get the connection rules connecting to coefficients the reason for that is that neither one of these solutions is valid in an immediate neighborhood of the turning point in fact it can be shown that there is some small region around the turning point where both of these solutions are invalid why is that so? they're invalid there because the momentum lost in the momentum are going to zero at the turning point but remember that the Devois white point is 2 pi h r divided by the momentum p so this actually goes to infinity at the turning point and the result is the condition of Lydian WPB which is that the wavelength should be small compared to the scale of the potential breaks down so if you look at this more carefully you can actually estimate what size of the region around here the plunger is not valid I won't go into that but let's just say because of this small region around the turning point where neither one of these solutions is valid it's not possible to find the connection coefficients by just bringing them both up to the turning point it would be nice if you could so instead of what do we do what we need is to connect with yet a third solution on top of those two a third solution which is valid in an immediate neighborhood of the turning point this is a stuff like this if I sketch my potential again the events let's take this case here here's our energy here's the turning point like here like this I'm calling XR now let's approximate the potential of my straight line in the neighborhood of the turning point so let's start here V of X is approximately equal to V of X at the turning point XR what's X minus XR times V prime of XR and if we plug that in in the Schrodinger equation then it becomes minus H bar squared over 2M times psi whole prime plus this thing V of X so it's V of XR and it's copied at X minus XR times V prime of XR times psi equals E psi V of XR however is equal to is equal to the energy E because that's the definition of the turning point and the energy is equal to the potential so this term V of XR cancels the E term on the other side and the result is the beginning equation which now just has this linear term this of course V prime of XR is a constant here just has this linear term on my side the right hand side is 0 this equation can be cleaned up and put into a standard form if we do a change in variables that's why X equals XR plus the constant A times Z so that Z is a new variable A is a constant here C is a new variable which has been advantaged exactly at the turning point and it will be negative on the left side and it will be positive on the right side and otherwise it's just X scaled by a certain factor A so we do a linear transformation in this result of the equation now we choose the constant A in order to clean up the equation all the constants that are in here and if you do this you'll find that the nicest place for A is this it's h bar square whereas here over 2M times V prime elevated at the turning point to the 1 third power and if you make that choice then the joining equation becomes this it becomes Z squared psi VZ squared minus Z times psi and it's been cleaned up and all the physical constants disappear this is one of the standard equations of mathematical physics you can read about it in books like The Member of Itz and Stegen and it has two solutions psi and Z what is called the airy function A, I, Z and then it's called the airy function B, I, Z and the general solution is linear combination of these two look in the books on special functions and get all the properties of the airy and airy functions but in order to understand what you're reading from a physical standpoint it helps to have a physical model to understand the results the way that I derive this is the airy equation is by using an approximation of the potential of a straight line approximation of course it's possible the potential really is a straight line and not just a continuity point could be like this for example if we had a particle in the gravitational field then B of Z let B of X that's called it would be equal to Mgx or the strictly linear in the position and so the airy and airy functions are the approximate solutions that can be approximated in the neighborhood of a continuity point of an arbitrary potential but there are also the exact solutions for a particle in a uniform gravitational field or for that matter a charged particle in a uniform electric field which is the same story because it's linear and x so thinking about the physics of a particle the quantum mechanics of a particle in a uniform electric or gravitational field helps us to understand the mathematical properties of the airy and airy functions and that's where I'll pick up the next slide because that I I I I I I I I I I I I I I I I I I I I I I I I I I I


Hello friends, welcome back to another episode here on the channel. I hope you're all doing great and for anyone new to the channel, my name is Lee. In today's episode, we're going to be kicking off our VGC Series 9 content. So the rules updated on the 1st of May. We saw the rules change from Series 8 with all those restrictives like Kyogre Groudon. We said goodbye to those and now we're moving into Series 9 which is reverting back to a Series 7 rule set that we've already played, already quite familiar with. Now if you need a refresher on the rules update, I did a video on Friday last week just kind of overviewing the new rule set for Series 9 and kind of overviewing the format that met again going into it because obviously it's a format that we've played before and I ended up with a bunch of I think five rental teams in that video to help everyone get a head start. Now that'll be linked down in the description if you do need that refresher or just want to grab some of those rental teams that we've already got up on the channel then be my guest to check that out. Hopefully it is helpful but today we're going to be kicking off with probably one of the most stronger teams from Series 7 at least in my opinion especially from a probably one of the best Japanese players that we've had in the last year or so. So all their socials are going to be linked down in the description below. This team obviously there's a rental code. We'll have a couple of games with the team today. Very exciting obviously going back to a format without restrictives. It kind of opens things up a little bit more. We've got some nice options on this team. We'll have a couple of games with the team and end up with the rental code. So hopefully you enjoy this one today. We've got a nice mixture of Pokemon to hopefully kick us off into this format in good fashion. Hope you're enjoying the new format as well. Do let me know your thoughts down below in the comment section. Let me know what you're thinking about Series 9, how you're finding it and what changes you like compared to Series 8. If you do like it at all I do really like to hear how players are kind of adapting and enjoying new formats but without further ado friends let's get into our first match of today. First up today we have a team of Reggie Alecki, Glastria, Cinderace, Togokiss, Rillaboom and Ferromosa. It's kind of weird going into Series 9 now. This is literally my first game in Series 9, going back without those restrictives. But what's going on in this team? Okay you've got some really fast defensive Pokemon straight off the bat. You know the Ferromosa and the Reggie Alecki are going to have to be pretty much everything on our team. But we've got screen support on Grimstone we can utilize and we've got ways to kind of get around those particular Pokemon and hit them for decent damage. Things like Moltres, obviously the Cinderace is a little bit of an issue because it can threaten our spectra pretty hard. But we do have Tappukini to kind of give us a little bit of a leg up there but we've got to watch out for that Rillaboom which is going to be a bit awkward for us to deal with. One of the things we could potentially do here is go Silvian. It gives us access to Jorn so we can really disrupt and also with the Hyper Voice as well we can put that spread damage onto the field pretty quickly and not worry so much about a lot of the other Pokemon that are going to be popping up. Let's just kind of give ourselves a refresh of this Grimstone. See what support options we've got. And Thunder Web Lights Greens, yeah so it's a kind of standard set. I think we'll need with Grimstone always a nice option here. It's whether or not we want to go with something like Moltres because it's a decent option. But I think Metagross is probably the one that we want. I'm going to go with Silvian, Metagross and I think we'll go Finny as well. Okay, double fairies for this first one today. So here we get on in this first one. It's always nervy kind of going into your first match in a new series always. Doesn't matter what you're doing. I mean I've pledged on Showdown quite a bit so I can't say it's my first match of course. But on cart on the ladder here we go. We're premiering our series nine right here. So going up against Feremosa and Glastria. And you can know what's coming. The speed swap onto Glastria straight off the bat is what we're going to see. So I think we have to be probably go for a reflect because I'm worried about Max Steel Spike and I think we go for a yawn into the Glastria because there's a good chance that it will potentially Max here especially after the speed swap. And you can see that's kind of what my opponent's probably going to go for in this turn. I mean it kind of screams that doesn't it. So I'll be very surprised if it's anything other than that. But that feels like what we're probably going to see. And we do have Metagross that we can bring in. But the problem with the Metagross in this team is it's weakness policy. So we haven't got that kind of instantaneous kind of attack buff to really put some damage on the Glastria. If we can put it to sleep though that makes things a lot easier for us going forward. Of course we can waste some of those Max turns and then you know it might not be as effective. Yeah there's a Steel Spike. But the Bibi Riberi the reflect is going to help us out a bunch here. And the big thing is just being able to get this yawn onto. Wow. I mean we take that pretty well don't we. So that's nice. That's nice. Very nice. Very nice. The Steel Spike not ideal because obviously we're kind of planning if we're going to use Metagross in this match to kind of hit it on that physical side which is not ideal. Okay so we've got a what can we do here. I mean they're going to go Steel Spike again so we could bring in Metagross here. But one of the things I prefer to do honestly is probably just we could spirit break the pheromosa. The pheromosa is so slow though. So it's likely the pheromosa switches out here. The walk kind of comes in for the pheromosa. Maybe something like Toakies potentially. I think we're pretty safe going first spirit break here. We're probably going to take Max Steel Spike into Grimmsnall. They're probably going to they're probably going to predict us protecting Sylveon. I'd imagine and go after the Grimmsnall. Yeah it makes a lot of sense. And they think oh no they go after Sylveon again. But the thing is like we can't go from the Thunder Wave into the the Glastria because then it like nullifies the ability for us to put it to sleep which is such a waste of that first turn which is always tempting you know. Not to waste that first turn but tempting to kind of get that a Thunder Wave onto it especially after this this speed swap. So pheromosa taking a decent amount of damage there. They're going for the Poison Jab doubling up into the Sylveon and the Glastria. It's finally Slooping which is great. So that means we can Hyper Voice here. Probably switch into something like better to switch in Metagross at this point. I do worry about a weakness policy as well on the Glastria you know. I mean the other option is we've got Tapu Fini trying to get some calm mines under our belt and then kind of approach the match from that angle. I'm going to go Hyper Voice here. We should still outspeed the pheromosa pick up knock out there get some decent well we'll get some chip damage. Well it's not so decent damage because definitely not going to be decent damage onto the Glastria is it. But we don't have access to Yawn or Thunder Wave anymore which is a little bit of an issue of course unless they've got any floaty Pokemon which I doubt they do. Problem is with Tapu Fini we've got to kind of watch out for something like Regi Alecki which could enter the field now which we know my opponent's got and gives us a few issues of course. But we do have Mystic Fire that we can take advantage of with Silvian and now we know that the you know the Glastria is kind of just taking a nap. Glastria not max anymore which makes it slightly easier. Do we just max with Fini and go after the Cinderace? I do worry about a bounce here but like what are the options? What is the Cinderace going to do? I would like to get a calm mind up if I'm completely honest. I think getting a calm mind up might be a nice option. Then we just go for a Hyper Voice again. It's not really going to hit so hard but they could go poor. I mean we could switch in Metagross but it's so risky against something like Cinderace you know. We could just protect. It's so passive though but it kind of I guess catches the Cinderace out because max Steel Spark, well the Iron Head I guess is the best option you would imagine or bounce. The Glastria taking another Tennis Leap bounce yeah. So I didn't really want to attack into the Cinderace here because the bounce is like you know and obvious that they're going to go for that. We do get the calm mind which makes things a lot easier for us to kind of manage going forward because I think well we've got the option now where we can go after potentially the Cinderace this next turn. We can max go after the Cinderace, get the rain up and then we can go after the Glastria to turn after that which is probably the best way to approach this and maybe at the same time switch in Metagross because you can imagine a heavy slam coming out from the Glastria it'll help us preserve the Sylvian. Going into this next turn just give us a little bit of room and then probably switch back into Grimsnall the Fallen turn because we kind of want to keep Metagross as healthy as possible for an end game because of the Assault Vest it does give us a better way to get around things like Regilec which could be kind of sweeping around in the back. So Finny going to come out as our maximum like I say the reason for going into Cinderace here is because we're going to get a bit more power than the following turn on to the Glastria especially with the rain up so Glastria staying asleep again really turns asleep we get very fortunate there as a bounce comes out revealing the life orb which is probably quite a common item on something like Cinderace but we are going to get the guys we've got to keep in mind as well that you know the Glastria is 100% going to wake up this next turn and the only issue I could kind of foresee would be something like Regilec coming in now which would make things a little bit awkward but we could max guard potentially thinking that my opponents may be going to go for the double up into the Finny with the Glastria oh but it's not it's the it's the Tograkus okay well this is actually all right this is this is actually fine I think we stay in now and I think we just go full of punch and we go max guys are again because if they redirect then we get it onto the Tograkus and then it's pretty easy to kind of wrap up after that we get Grimstarling, Thunderwave and then go from there so very good game for my opponent we get fortunate with the three turn sleeps but it's how it goes sometimes isn't it so we'll take that and we'll jump straight into our next match of today except we have a well like kind of a rain team Polly Todd, Dracovish, Tog of Tomorrow, Mantine, Lola Mawak and Stack Attacker so lots of wide guard, lots of Lightning Rod and lots of rain that's pretty much wrapping this one up um okay what are we looking at here we've got speed control potentially Nuzzel I would say 100% Nuzzel from the Tog of Tomorrow got to be careful around that for sure but we do have Tap of Finny's terrain to kind of help out with that a bit um wide guard I would say probably on Mantine um does it get tailwind I want to say yes I want to say yes but I'm not 100% sure um but regardless you've got kind of airstream as well and it's like fully protected from things like electric type attacks it would really kind of hinder it a little bit but we haven't got any so we don't need to worry about that in that respect and you've got a trick room switch with the Stack Attacker predominantly going to be supporting something like the Mawak you would imagine right well well well how do we best approach this I feel like Metagross is quite good here especially with screen support but it's not really got the offensive kind of um prior to that kind of want Sylvion's not bad um you know what I think we'll go Sylvia and Grim Snarl again I think it gives us a lot of options uh we probably want Tap of Finny for sure and I think Metagross again you know I do feel like Metagross is gonna be it's not really threatened too much you know like there's not really too much outside of the the Mawak which is you know if we're not in a trick room environment it's pretty easy to deal with with Tap of Finny especially if the rain's up you know so I think one of the big things it's jumped out to me um I was being quite surprised and like not surprising but it's being very noticeable for me is just uh how much it's like I've enjoyed kind of going back to something like Tap of Finny and really appreciating it it's uh as you use in this format I think it's just it's such a good pokemon you know um now what do we do here we just hyper voice do we zone do we protect uh I think getting damage onto the board is probably a good idea you know um I'm gonna just hyper voice and I'm worried about the tug of tomorrow going for something like Iron Head but the likelihood of it having Iron Head is is not I just can't see it we'll go for light screen it probably if anything's gonna max it's gonna be the man time but I can't oh well here we go it's gonna be the man time it's gonna go stop that airstream abuse I reckon um I can't see it going for max guys I just airstream I think makes more sense but I've not really come up against too many man time so I'm a little bit a little bit oh I know what they're doing they're gone weakness policy aren't they weakness policy into the man time volt switch there we go see it a bit too late and then the Dracovish comes in and then we we get nailed oh but the thing is right if it is Dracovish gonna take a heck of a lot of damage from a hyper voice from the sylvian um which which is all right which is all right it is not the end of the world and we got thunder wave as long as the tug of tomorrow kind of stays off oh it's going to be towed okay they're going for the geyser the rain boosted weakness policy boosted uh we need to put a yawn down onto that thing okay they're gonna have to grim snarl wow forget us wow okay well and we do absolutely zip like man time's uh special defenses phenomenal do we have room to calm mind because they're definitely going after sylvian this next turn I think we could get a yawn we could get a yawn calm mind um but the the yawn's gone now with the terrain of course could we do what can we do in this situation we can mystic fire but I don't really think that's gonna do too much change the rain that means max and sylvian on I'm not really that fond of that idea I think we did they just airstream finny here that's the the big question you know actually we can yawn we can yawn because they're flying right so we we yawn and we calm mind and we get something out of this turn so there's a protect which we don't mind we like to see that good old polytod the gun geyser into sylvian if we can take this this is huge for us I don't know if we're gonna be able to though yeah there we go sylvian taking it like a trooper trooper there we go that's perfect because now I'd still gonna be very difficult you know like extremely difficult I think the thing to keep in mind is that we really need to kind of try and keep sylvian active because as soon as we go down to one I like the two pokemon then something like polytod really becomes a problem with something like um perish song I think we go for we'll protect sylvian here don't know if it's worth it but you know we're so low health anyway and we'll hope that we can get the toad plus one max starfall if we can we remove the the perish song threat and it looks like it's gonna be kind of maybe a tapu finny show today but we'll see how we get on this opening for us it's not the best start we're not out of it by a long way we still got met across in the back I think the big thing to worry about first is definitely a perish song from the polytod so we get to protect off with sylvian you'll see the man time go for a geyser into sylvian it will probably pick up the knockout here and I would imagine we'll probably see a perish song if we miss the knockout onto polytod which could happen polytod's such a bulky pokemon but we've got to believe in finny that's so it's literally so close if they get the perish up here we're done we're done because I don't think metagross can can deal with this by itself they go for a skull though so maybe they don't have perish we'll take that all day long and now the man time will go to sleep the polytod will be um in a knockout position for metagross to come in so we're not sitting in the worst of spots in all honesty and they've not went for a single max airstream so we know the man time it's definitely 100% stay in a sleep this next turn uh we can max starfall into that and we can just get rid of this polytod the stomping tantrum just in case something like tug of the morrow comes in again we'll go for a starfall into the man time it might have been better on in all honesty kind of rock sliding here just to get rid of the man time because the threat of it it is quite it's quite a scary pokemon obviously with swift swim you know it's gonna be moving first still got the the weakness policy boost you know so it's still hitting like you saw how much damage it did to grim snow so it's still gonna dent anything it really hits our topic finish not in the worst spot here but yeah come on be tug of the morrow please tug of the morrow let it be oh it's marowak there we go we like to see that okay i think we can win this one from this point depends if we can pick up the knockout here like i said man time special defense is pretty good pretty good but we do have bullet punch that we can take advantage of the next turn just go for the guys that into the marowak because bullet punch should get the man time from this range right like a hundred percent it will 100 percent it's definitely getting it and i'm gonna go max guys that into marowak and then after that we've got tug of the morrow and polytod with a muddy water i'll be able to deal with with pretty much both of us i would imagine in the rain does the rain does the rain stop this turn i think i'll i switch oh dear oh dear forget about that forget about these these these moves okay well man time stay in the sleep again and the guys are going to be the thing that finishes it off okay so this kind of makes it a little bit you know it makes it a bit more tricky for um my opponent like if they bring in tug of the morrow here and if that marowak has got light and rod that's very tricky for my opponent to actually get any attacks off with them with tug of the morrow because they're not they only got single target like they don't have discharge right so yeah that's this is the thing you know polytod coming in yeah we're just a you know we're going to be able to just go muddy water and stomp and tantrum and that that should be enough to clear the field here again i feel a bit like rock slide could could also be a decent option but in marowak gonna protect i don't know why why why why you're protecting the marowak surely you want to protect the polytod over everything else you don't want tug of the morrow to kind of come in it's kind of weird okay the accuracy drop might help us that depends on the speeds but metacross gonna add speed and um yeah i mean i don't really surely out of everything there you want the the the marowak to kind of go away because now tug of the morrow tug of the morrow can come in for sure and just click the fake out button but we just double in like we just muddy water and stomp and tantrum into the marowak because it's the biggest threat to metacross right now um and they can only fake out one target so there's not really yeah i mean we do pick up the wins so two really convincing wins and tapu finny i will say has definitely been mvp of today's episode 100% very good game to our last opponent i hope you've enjoyed today's episodes we didn't really get to see the glarium altres and stuff like that but you know this team is so versatile it's kind of we've got that really solid core slowing things down and then you've got the the finny to come in and just clean up if you want and then metacross being the assault vest gives you a lot of stability there so we'll hop over now remind you all of the rental code for today's okay friends here is today's rental team like i say we've had a good couple of games with the team we've really seen tapu finny shine in this one a bit unfortunate that we didn't get to see the glarium altres really the spectra in today's episode which is a bit of a shame because they're two big hard hitters but it just kind of goes to show that the team can function super well without these pokemon in it and when the kind of the terrain or the the matchup kind of feels right then these pokemon can come in and do a really good job for you especially with the screen support and speed control support from the grim snarl which is likely you know i mentioned in that preview kind of video four series nine how predominant how good that is going to be as a support option going forward in in this meta game and having answers for it and ways around it like psychic fans like brick break things that get past it you know infiltrate our ability critical hits from crit case there are things to look at to try and get around grim snarl and prevent these screens going up and being able to kind of support team members so well but big shout out to um to the team it's been a lot of fun playing it hope if you do try it out you have a lot of fun with it let me know down the comment section as always i'm really looking forward to getting our teeth into series nine and kind of seeing how this format develops so we've got a long way to go before we stay another format kind of pop up but we should get some news on that and i'd say a month's time to see what the next series will be but in the meantime we've got a lot to kind of digest and play around with there's lots of different things that we can do in this format i've got a really spicy team that will be throwing up in our next episode so do keep an eye out for that one thanks as always for tuning in i hope you have a great rest of your day and i'll catch you all for another episode very soon something like friends take care of yourselves and bye bye


This is a narcissistic society, where the losers are pushed to have less and less and the winners take more and more. Without regard for consequences, now such societies inevitably always explode. It's a question of time. They always end in social unrest, bloody rebellions, bloodshed, massive wars, tens of millions dead. Always happen, always will happen.  And we start again and again and again. This is the cycle of narcissism. Narcissism is not linear. It doesn't lead from point A to point B. Narcissism is cyclical. It brings us back to the starting point. Always we have to start from zero with narcissism. Consider for example Adolf Hitler. Adolf Hitler took a relatively strong or relatively big Germany. Look what he left behind in 1945. Look what the Germans had. The Germans started from zero. There was nothing there, total obliteration. Nothing was left. That's narcissism. So that's the first problem. The second and even much bigger problem is that narcissism is a religion. That is something no one understands. No one understands. I'm trying now, not six months, to make people understand this. Narcissism is not only a mental health disorder. It's not only a principle according to which we organize society. It's not only a principle which can explain to us many things that are happening around us in our relationships, in our politics and so on. It's not only these things. But it's a new religion similar to Islam or to Christianity or to Judaism or to Buddhism. It's a new religion. But it's an extremely malicious, toxic and dangerous religion. Never before has there been anything remotely similar to this religion because it has a few features that no other religion ever had. But before I go into that, do I have permission to talk about this? Before I go into that, I want to explain why I'm saying it's a religion. Consider how narcissism is formed. How it becomes. What happens is there's a child. The child is abused by the parent. Abuse can have many forms. Abuse means when the child is not allowed to separate from the parent. The child is not allowed to become an individual. When the parent subsumes the child, takes over the child. So it could be via sexual abuse, physical abuse or verbal abuse. But it could also be by using the child to realize the parent's unfulfilled dreams. Or by using the child for some other form of gratification, for example, parentifying the child. Or by triangulating with the child, teaming up with the child against the other parent. Whenever the child is used as an instrument or extension of the parent, whenever the parent does not respect the child's boundaries, this is abuse. So some children react to abuse, a minority react to abuse by developing narcissism. But what is narcissism? The child creates a separate entity, the false self. The false self is everything the child is not. The child is helpless, the false self is omnipotent or powerful. The child cannot predict the behavior of his parents, his capricious, violent, aggressive, abusive parents. The false self is omniscient, knows everything. The child is told by the parents that he is imperfect, not brilliant, unworthy and a bad object. The false self is perfect and brilliant. But when we have an entity that knows everything, has all the power in the world and is perfect, what is the name of this entity? It's a God. This is God.  The child creates a God. The false self is a God. It's a religion. It's a religion with one God, the false self, and one worshiper, one adherent, the child. So the child forms a religion, creates a private religion where there is a God called the false self. But you know, when you have a religion, you need to sacrifice something to God. You need some sacrifice. So in the old times, we sacrificed cows, we sacrificed lambs. But in some religions, in some locations, people sacrificed their own children. The Moloch, Abraham almost killed Isaac, killed Isaac in the Bible. So child sacrifice is pretty common. What happens in narcissism? Same, child sacrifice. The child sacrifices his true self to the false self.  I'm killing my true self for you. And you will be from now on, my God. Narcissism starts as a private religion. And then every narcissist is a one-man religion, a one-man cult. When we have a society full with narcissists, we have a society full with gods. We have a society full with worshipers. We have multiple cults. Narcissism is therefore a non-monotheistic religion. So going back very long time, it's paganism. It's a form of paganism. We're going back very long time before monotheism. It's a non-monotheistic religion with as many gods as there are adherents, as there are worshipers. But still it's a religion. It has all elements of a religion. It has sacrifices, it has worship, it has perfection, it's a religion. So it's the first network religion. It's the first distributed religion. It's not a religion with a central figure, which we all worship. But it's a religion where the worship is distributed, the gods are distributed, everything is distributed. It's a modern religion. It's the first modern religion. Because in today's world, the ruling metaphor, the ruling organizational principle is the principle of the network. We organize our computers as networks. We think of our brain as a network. We interact socially in networks. We, in business, we network. There is a concept of networking in business. We go to conferences to network. It's all about networking. The dominant metaphor is networking. Even marketing is made via networks, MLMs and so on. The dominant metaphor is networking. The new religion, of course, would be a religion that uses the dominant metaphor. It will be a network religion. And there's only one type of religion that is distributed and networked, narcissism. And so it's a new religion. Within a few decades, the dominant religion will not be Islam, will not be Christianity, will not be anything else. It will be narcissism. Why do I say it? Because the old religions don't fit the modern world. The old religions are hierarchical religions. There's hierarchy. There is God and there is you. And you are inferior to God. We live in a society that is totally egalitarian. We live in a society that is totally networked. And we live in a society where everyone believes themselves to be equal to everyone else in knowledge, in expertise, in access to the truth. We don't accept authority. We don't accept expertise. We reject authority. We hate authority. We hate expertise. So the old religions don't fit the new mindset, don't fit the new metaphor, don't fit the modern world. And your religion is emerging. We're in the middle of a massive religious revolution similar to the emergence of Islam in the 7th century. Or more precisely the emergence of Protestantism in the 16th century. It's a massive religious revolution. Bigger than anything in human history because it's everywhere. All previous religions were localized religion. Islam was Middle East. Christianity was Europe, largely. They were all localized. Judaism was, God knows where. I mean it was all localized. This is the first global religion because we live in the age of globalization. All the features of the modern world are features of this religion. It's globalized. It's network.


Meanwhile, the president of the Christian Association of Nigeria, Dr. Daniel Orkin Sunday, urged the federal government and the leadership of the academic staff union of universities to end the seven-month strike. Orkin made the appeal in his address during the interdenominational chat service to mark the 62nd Independence Day at the National Acumenical Center in Abuja.


Hello, this unit looks at conversation, a linguistic interaction between two or more participants that is highly contextualized, goal directed and cooperative. In the virtual session we will discuss the principles of turn-taking which ensure everyone's right to speak. We will discuss the central implications of the so-called cooperative principle, including politeness principles. And we will define and exemplify the notion of implicature. Use the question sheet with its hyperlinks and its video links for guidance if you don't know where to start. The mastery worksheet is an identification task where you have to examine randomly chosen dialogues concerning their conversational maxims involved, certainly manageable after going through the virtual session. Well and the practical provides you with dialogue and conversation data where you can apply the knowledge you have gained in the virtual session. Quite enjoyable, I'm sure.


Couple's fake garage gets discovered, police ask for harshest sentence possible. After noticing that their neighbors were starting to get nosy, they decided to set up a fence around the perimeter of their property. Now that they were away from prying eyes, they thought that they could breathe a sigh of relief. Isolated from the world, they kept working on their project while blasting music to hide what they were doing. But they were unaware of one flaw in their fence, a small gap that anyone could look through. Rita Herzala and Handy Almasri had built something that they felt was special, but everything came crashing down when a nosy neighbor decided to report them to the authorities. Almasri, 39, and his wife Herzala, 37, enjoyed their life in Inderby, United Kingdom. It was a small quaint town where there wasn't much going on. So when everyone found out what drama the couple had stirred up, they were surprised. Their house was made of brick and was close to a big highway. Being the road was a garage that could fit one car. Little did people know what was hiding behind its doors. The couple always had kept a low profile. They both worked as doctors and had spent their fair share of time in hospitals. When they got home from their irregular work hours they never came out of their house. Even though their neighbors never made a ruckus they were rather nosy. They all wanted to know what the couple did with their free time since they almost never came outdoors even on weekends. What could they be doing? They grew more curious once they noticed something strange coming out from inside their garage. It poured high up into the air and then disappeared as abruptly as it came out. The neighbors started talking amongst themselves as to what it could be. It was some kind of smoke or vapor. Maybe dust from cleaning? Perhaps they were constructing something, but why keep it a secret? Some said that the couple bebequeued inside their garage so nobody could see them. Others said that they were conducting weird experiments behind the doors. The couple knew that the neighbors were on to them. They began to grow suspicious so they had to take measures to keep their business a secret from prying eyes. They bought some timber from a local company and once delivered they started working. They built a wooden fence that surrounded their property. Nobody could find out what they were doing, but the neighbors weren't going to give up so easily. Even though before now the neighbors were just engaging in harmless gossip, now they felt that the couple actually had something to hide. Warning bells sounded in their heads as they wondered what they had to hide. They were worried about what their neighbors could be doing and how it could be impacting their community. It was fair to assume it could be harmful since there was smoke dispersing into the air. The neighbors kept talking and soon they grew worried. Once their fence was constructed they thought they could breathe a sigh of relief. So they kept hard at work and listened to music whilst tending to their project. They thought no one was the wiser. Their fence did quite the opposite though, the neighbors were more curious than ever and they were going to try to learn the truth. The couple was oblivious to what would come next. Right in handy arrived home from their jobs at the hospital and before doing anything else, they headed to the garage and continued work on their project. This hidden lifestyle had become something of a habit. He opened the toolbox on the garage floor and pulled out the relevant tools, there wasn't a speck of dust on them with how frequently they were used. With how early it was they knew they wouldn't be breaking any disturbing the peace rules. They knew how to keep things subtle these days. But while the couple worked away in their garage they were unaware of how motivated their neighbors were to find out what their secret was. They were in their daily groove with their music blasting while more than one pair of eyes were watching them through the small gap in their fence. Still unaware, the eyes grew two by two and soon people were looking for their phones and an emergency number to call to put a stop to their hobby. Rita and Handy were none the wiser when a scene was unfolding right outside their home. Neighbors gathered around as police showed up and their dozens. Things were about to go down and the neighbors were glad to see they finally solved the mystery. No one acts that strange if they have nothing to hide. And this couple definitely did. On closer inspection of the garage door, authorities found that it wasn't attached. It was nearly a facade propped up to conceal what the couple had been busy with for the last few months. They had surrounded the garage with fencing to hide it from authorities. The development included the conversion of their garage to habitable accommodation and the erection of fencing, preventing off-street parking at the property, said planning authorities. And when the media got hold of the story, things took a strange turn. Planning permission in the U.K. is taken very seriously. The couple had built another dwelling on their property by converting their garage into a tiny home even though this garage was supposed to be a parking area to prevent on-street parking. One of these alterations had been legally approved. The couple could appeal, but if unsuccessful, they would have to comply or face prosecution. But the false garage wasn't the only alteration they had made. Building developments that could be potentially damaging to the neighborhood or an inconvenience to the public are strictly forbidden in U.K. property development laws. If it is decided that a property is in breach of permissions, the local council will enforce changes. The authorities also discovered other planning breaches. The audacious couple had also built an extra driveway that was positioned to give them direct access to a busy highway. The couple also undertook illegal works within the highway and created unauthorized vehicle access onto the busy B4114 dual carriageway, authorities stated. The message from this case is clear. If you breach planning regulations and ignore us, we will not just go away. Bringing in access onto a busy highway and blocking off valuable off-road parking space was completely unacceptable, Councillor Sheila Scott said. Rita and Handy had made an after-the-fact planning proposal to the district council in a bid to retain the changes to their Enderby property. But the couple's application was refused. They tried again, this time appealing to a higher body, the National Planning Inspectorate for England and Wales. But again, their application was refused. Months later, Rita and Handy took down the illegal fencing. It looked like they were about to finally comply with the law. However, when authorities travelled to their home for the follow-up inspection, they found a large garage door propped up across the front of the garage. The tiny structure was still being used as a tiny home. Neighbours had told officials that an employee of the couple lived in the converted garage, and they had no choice but to initiate legal procedures against the couple. Indeed, a hearing on the matter was held at the nearby Leicester Magistrates Court, but Rita and Handy did not attend the proceedings. And since the report about the couple secretly converting their garage into a tiny dwelling was released, the story has gone viral online. But Rita had her own version of the story, which she offered up to Facebook. And soon, netizens were in a frenzy. Rita says the story contains false information, has been twisted into a more negative light by the media, and is highly discriminatory toward her and her husband. Rita felt that she sadly detected racial prejudice behind a lot of the press reporting. Furthermore, she denied the allegations that she and her husband had converted the garage into a dwelling for the purpose of renting it out, by an employee or anyone else. This was a simple single garage conversion into a small garden annex in the house that we own on the land that we own, and it cannot be called a separate house in any stretch of the imagination. Rita wrote, We had no intention in breaking any laws. Rita then went on to explain that the contractor that the couple had hired to complete the conversion had told them that their modifications were compliant with the law. They are currently embroiled in a dispute with this service provider over converting the structure back to its original condition. Rita posted, A simple council dispute over a tiny garage within our property that we converted for personal use was made into the foreign doctors that hit a house behind a propped up garage door story. It doesn't feel random that this was given particular attention when there are thousands of other council disputes, she wrote, nodding to their ethnicity. Rita also added that the fake garage door had only been propped up against the front of the structure temporarily while they were preparing for installation. She claims that it was not put there to try to hide their tiny second home.  She angrily asked in her post. I suggest some respect to the reader's intelligence at least would have been nice. She added. She concluded her post by addressing the haters directly. I also suggest that before you try to attack a law abiding, hardworking couple who have been misled by the council and builders, get both sides of the story. Looking through the comments that users have left on Rita's post, as well as on articles about the dispute, the issue is clearly a divisive one. But around half of the readers have decided to side with her Zala and her husband. Seriously why can't people's homes be their own too, do, with as they please, asked one commenter, such a shame, I say boo to the neighbors that reported it and boo to the planning. Rita's Facebook post caused a divided reaction online, but the authorities were determined to carry out a successful prosecution. They were each ordered to pay a $1,000 fine, including a legal cost of $1,600 and a $100 victim surcharge and ordered to restore the garage to its original use. The authorities used this as a stark warning to others, the message from this case is clear. If you breach planning regulations and ignore us, we will not just go away.


Okay, so the latest twab is now live and it includes some interesting information regarding the game and features coming over the next month that I must cover. Things like a community challenge, changes to the menagerie and a few other bits and bobs. How's it going guys? My name's D.B. Jen. I'd like to thank you for stopping by and checking out the video. If you do enjoy it, leaving a like really helps out and if you're new around to and enjoy daily Destiny 2 videos be sure to subscribe. So the twab you will find linked within the video description if you do want to read through the entire post. It basically starts off covering news I've brought you guys already. Things like the Directors Cup by Luke Smith, the Articles or 3, Trustsave, Finishers what we learnt about Gamescom, Gamescom Trailer of Shadowkeep, The Season of the Undying Trailer. I've brought you guys all kinds of videos on these covering every speck of these actual videos. Like I said if you do want to go through all your findings linked within the video description. Then we'll talk about the Great Migration which is basically Trustsave and people move through platforms which are platforms and a lot of people move into PC which you can understand. And it also talks about this also which is probably worth noting so I will. As we prepare to transition Destiny 2 to Steam, your last chance to purchase Destiny 2 in expansions or silver on battle.net is September 3rd 2019. Additionally players may pre-order Destiny 2 Shadowkeep on Steam but will not be able to play until October 1st. Obviously. Additionally players may begin to purchase Silver on Steam starting October 1st. With Season of the Opulence coming to a close we wanted to make sure you had ample time to get into the action with your friends on PC and got proper warning before your favourite ever verse items went on Hiatus. Make sure to grab Silver before September 3rd so you don't miss your final chances to snag some community favourites and we see a few emote while an emote, a gold shell and a speck which looks pretty epic. Okay so the interesting stuff. Iron and Valor. Next week begins your final month of Season of Opulence. Gold, Saladin returns with the power enabled combat of the Iron Banner and we see we have Iron Banner which starts on the 27th which is next Tuesday I do believe. Is it next Tuesday or the Tuesday after I'm not sure? Well let me double check back there. 27th yes is next Tuesday pretty cool. Can not wait for that and I just minimise the screen. What have I done? Oh yeah what have I clicked on? Let's go back. Okay we're back. So yes it starts next Tuesday I'm looking forward to that and my last chance is actually getting a Swarm. Pretty epic. Basically that's it. We get another Iron Banner on the 17th of September and we get Mayhem the week after. So yeah Iron Banner starts August 27th and ends September 3rd. Looking forward to that. Increased Valor also returns so if you are hunting for those final research required for the Red Rixie's Brutal Quest this is the perfect time to jump in. Moments of Triumph has also been extended through September 17th so this may help you earn the Crucible Pinnacle weapon that's been holding you back from the MMX1X title. September Community Challenge on Mars. Whether you wanted it or not we've stepped into a war with the Hive on Mars. So let's get to taking out there. Wait a second this isn't what I meant to start with. With a bit of extra time before Shadow Keep ships we thought it would be a good idea to get the community in fighting shape to turn to the moon. Not just you but the entire community. There's a better way to focus your attention on the Hive and to challenge you in the ways of old. As we have a Quality of Life update coming on September 3rd that touches on Mars content. Why not double down and maybe help a few aspiring Wayfarers along the way. Start September 3rd, 10AM PDT and end September 10th, 10AM PDT. Community Objectives. Defeat 175,000 Wave 7 escalation protocol buses. Wow! Defeat 300,000,000 Hive on Mars. Mars Nightfall. Defeat Nuckers 150,000 times. Defeat Zull 100,000 times. Use 300,000 override frequencies. Wow! Rewards. Now this wouldn't be interesting without rewards. If all objectives are completed during the event any player to complete at least one nightfall will be granted a unique emblem. Granted to only those who rose to the Mars challenge. And we can see the emblem right there zooming a little bit. Looks pretty cool. Really does. Actually I really like it. Okay so it also states this also serves as an opportunity for you to further impress Emperor Calus. If the community reaches all the above goals Calus will unlock additional rewards for successful Menagerie completions. Starting when goals are met players will be guaranteed two rewards when slotting their runes and opening the chests. An additional reward will be added each Friday for the remainder of the season. Now if this starts on September 3rd and we actually complete this within the first week or whatever it is because the 10th is the week after yes. That would mean on the Friday the 13th we can get an additional reward besides the two we are already going to be getting or promised by doing actual community challenge. The 20th will get another additional reward so that could be four in total there and then five on the 27th if my calculations are correct or it could be four on the 27th, three on the 20th and two on the 13th. It just depends on the way they actually do this and drop this but it's actually a pretty great incentive in getting on to Mars on September 3rd and killing the heck out of as many hives as you can. I know I'll be farming this. I'll definitely put in in my percentage of escalation, protocol, nightfalls and whatever else is needed. Good luck guardians we'll be watching so that's pretty epic people and that's basically it where we're in this video I want to cover now there's a few other bits and bobs you may want to check out if you have basically linked your account to steam they do cover a few of the issues people have been having with their report and so forth I think all is good now but on that note guys we have come to the end of the video tell me what you think about this epic new community challenge I love this kind of thing I really do and if we actually complete it it'll give them more incentive to drop more like this in the future so yes people make sure you get your ass to Mars on September 3rd and do what's needed but yeah guys tell me what you think about this down below in that comment section hope you enjoyed the video I really do if you did drop a like it really does help me and my channel out if you're new around to enjoy daily destiny 2 videos be sure to subscribe and if you never want to miss a video I already can turn notifications on by hitting that bell button but guys thanks as always for stopping by hopefully you enjoyed the video and hopefully I will see you on that next one