Greetings, we will begin unit 2 today and this will be on many body theory. Our focus will be on the study of electron correlations and techniques which make use of diagrammatic perturbation theory, the Feynman-Golstone diagrams and so on. So, we will develop these techniques which are of importance in studying atomic collisions and spectroscopy. And I will spend a few minutes at the start of this unit, recapitulating a few points that we did earlier in the previous course on the special topics in atomic physics. In particular, I will invoke the discussion on the Hartree-Fock formalism which is also a many body theory in a certain sense. It is a many electron theory, but then we are going to go well beyond the Hartree-Fock in the discussion in unit 2. So, let me recapitulate a few points from unit 4 of the previous course on atomic physics. And our interest in studying atoms molecules condensed matter of any kind is in studying electron interactions and electron correlations and of course, how the many body system responds to various probes like electromagnetic radiation or other particles like projectiles, electrons, positrons, composite particles and so on. So, I have underlined the words electron interactions and electron correlations to draw attention to the fact that these two terms will have different connotations in our context. And in particular, when I refer to correlations, I will be referring to two kinds of correlations. One are the exchange correlations and the other are Coulomb correlations. So, obviously Coulomb correlations and Coulomb interactions are two different things. Coulomb interaction is the usual 1 over r 1 2, Coulomb correlations is what many body theory is about. So, that is the focus of unit 2. Now, the exchange correlations come from statistics. Exchange correlations they are because of the fact that electrons are Fermi particles, Fermi Dirac particles and the statistics they must observe is the Fermi Dirac statistics and that requires that a many electron wave function must be anti-symmetric. So, there is a certain correlation coming in from statistics. In addition to that, there is a correlation which is the Coulomb correlation which is not included in the Hartree-Fock formalism and that is the one that we are going to talk about. That is a result of what we will refer to as many body correlations or many electron correlations and let me quickly remind you that statistics enters classical mechanics and quantum mechanics in different ways. In classical mechanics it enters just because you are dealing with a large number of particles and there is too much of information not all of it all the details are interesting. So, you do some averaging like in principle in classical mechanics since every single particle is distinguishable and it obeys dynamical laws in which you can predict the motion using the equation of motion. In principle you can follow the trajectory of every particle, but all this is not interesting. What is interesting is what is the average kinetic energy and what it generates is a thermodynamic parameter like the temperature. So, these are average properties and this is how statistics enters classical mechanics. In quantum mechanics statistics enters because of the uncertainty principle. So, laws of nature are intrinsically statistical there is a probabilistic interpretation that quantum mechanics demands even when you are dealing with a single particle or even when you are dealing with vacuum. So, it is nothing to do with the number of particles. Number of particles only makes it more complicated, but having fewer particles having a single particle does not eliminate statistics. And then you have the spin which is an intrinsic angular momentum for elementary particles and this also leads you to a statistical considerations because there are two kinds of particles in nature fermions and bosons and they observe different statistics and they have different spins. So, if you have two identical particles and the signature of quantum mechanics. So, far as many body formalisms are concerned is the fact that these particles are indistinguishable. So, if you have two identical particles one at coordinate q 1 and the other at coordinate q 2. If you interchange these two particles and I is the interchange operator and on the result you carry out the interchange yet again. So, you are carrying out an interchange twice you naturally expect that you will recover the original state. Now, this is the geminal wave function which is a wave function for the twin particles they are completely identical in all respects. And you will recover the same function which means that if you interchange only once you can at the most change the phase of this wave function by e to the i alpha. And e to the 2 i alpha must be unity which is what you will get over here which means that alpha is either 0 or pi and depending on it is 0 or pi you have either both particles or Fermi particles. Now, all particles which have half integer spin are Fermi particles or Fermi Dirac particles and all particles with integer spins are both particles or both Einstein particles. And typically the wave function would be written in terms of the spin quantum number as well and then you will have a one electron state written as n l m l m s for a particular state which is alpha. So, you have got a set of commuting operators.  Now, it is very interesting that you have this connection between statistics the spin of the particles and the sign of the wave function on interchange because on interchange of both particles the wave function remains invariant whereas, on interchange of Fermi particles the wave function of the geminal changes sign. So, this connection is rather interesting and it is nevertheless not so very easy to understand as Tomonaga tells us in the statement that the relation between spin and statistics is apparent, but hard to understand and I would like to quote Feynman one more time over here that you have a rule which can be stated very simply, but the explanation is complicated it lies deep down in relativistic quantum mechanics. So, I will not go into those details because our focus in unit 2 is the discussion on how many body correlations are to be treated and what they are in the first place. So, let us deal with electrons on interchange the electron geminal the 2 electron wave function changes at sign and you consider the wave function which is a function of 2 electrons q 1 and q 2 one at q 1 the other at q 2 to be separable in one particle coordinates. So, the 2 electron wave function is written as a product of 2 1 electron functions, but the product must be an anti symmetric product as we have discussed in details in our previous course anatomic physics on the in the discussion on the Hartree-Fock. So, you have to reconcile 2 properties 2 attributes which can appear to be contradictory the particles are indistinguishable in the sense that you really cannot separate one from the other yet you talk of them as individual elementary particles in which each has an identity of its own as an elementary particle the elementary particle of nature is the electron not the twin it is not the pair of electrons. So, the elementary particle is still the electron, but the 2 electrons are indistinguishable and this is indicated over here that the 2 electron wave function is written as a product of one electron functions which respects the elementary nature of the 2 electrons, but then it is a superposition of these 2 states electron at q 1 in the quantum state 1 and electron at q 2 in the quantum state 2 and the other possibility that the electron at q 1 is in the quantum state 2 and the electron at q 2 is in quantum state 1. So, these 2 situations are not distinguishable.  So, you can see that you can write this as a determinant this is just an equivalent form and this is the 2 by 2 determinant for the 2 electron system this is called as the Slater determinant you can automatically see that the poly exclusion principle and the anti symmetry of the wave function is automatically built into it, because if you interchange q 1 and q 2 the determinant would change its sign and if you have 2 rows to be the same the determinant will vanish that is the poly exclusion principle. Now, basically the single electron wave functions which are called as spin orbitals these are made up of a spin part and the orbital part. So, there is a separation between spin and orbit part over here, but the spin orbit coupling of course will give you a quantum number which is different, because l and s will combine to give you a j and then j m j will be good quantum numbers and not m l m s. So, basically you deal with these spin orbitals in the Slater determinants for an n electron system now we go quickly from the 2 electron system to an n electron system where n is any number it could be 2 or 3 or 4 or whatever and you have an n by n Slater determinant in which the columns are labeled by the q's which are the coordinates and the rows are labeled by the quantum numbers these are the complete set of commuting operators and their Eigen values. So, the quantum number is obtained from the Eigen value of the observable which is measurable and once you have a complete set of commuting observations you get a set and this set is what I call as alpha. So, alpha is not just 1 quantum number in our case it is 4 quantum numbers and together this gives us 1 set. So, each row in this Slater determinant corresponds to a particular set of alpha where alpha contains all the 4 quantum numbers. So, these are the 4 quantum numbers in alpha n, l, m l and m s and you can write this determinant also as you can take all the diagonal elements construct a product of this and then carry out the permutations of these this is how the determinant yes you have a question. The other e subscript is a combination different combinations of n, l, m, m. Exactly right. So, either one or more quantum numbers will be different when you go from one row to the other no two rows will have the same set of quantum numbers that will violate the Pauli exclusion principle and the determinant will vanish, but if even one of them is different then you are ok right. So, that difference is manifested inside the q the position I mean in each row in each row or each. The q's go from one column to the next. So, the first column is q 1, the second is q 2 and the last column is q n. So, in a particular up to q n. Yeah. So, look at all other quantum numbers Yes. So, basically what you are doing is interchanging the positions that is where the indistinguishability of the particles comes in. So, typically this is a measure of the probability amplitude that an electron at coordinate q i is in the quantum state alpha, but in the next row belonging to the same column you will have the probability amplitude that an electron at coordinate q i will be in the quantum state alpha plus 1, the next one right and that cannot be rolled out because these particles are completely indistinguishable. So, you must construct a superposition of these and this must be an anti symmetric superposition. So, that is what this later determinant guarantees. So, you can write this basically you are carrying out these permutations between these coordinates and then you must have the parity of permutation. You will have factorial n permutations possible and then 1 over root factorial n is the normalization as you can get if these individual spin orbitals which are elements of the determinant are individually normalized. So, these things we discussed at some length in our previous course on atomic physics and you may want to refer back to some of those details and I just wanted a very quick recapitulation of that and the many electron Schrodinger equation is h psi equal to e psi for the n number of particles and this is what we called as the catch 22 situation. The reason is it is catch 22 a phrase which is which comes from the novel by Joseph Heller and the main idea in this is the following that if you write the n electron Hamiltonian as a sum of these single particle operators this is the single particle of kinetic energy. This is the kinetic energy of each electron in the field of the nucleus, but this is the electron-electron repulsion and the 1 over r i j which is the electron-electron repulsion. So, this requires for your consideration the distance between these two electrons and this distance can be specified only in terms of distance between the two probability densities because charges these are not classical point charges. So, you need the probability densities which can come only from the wave function. So, you need the wave function even to construct the Hamiltonian and unless you have the Hamiltonian you cannot set up the differential equation at all. This is the catch 22 situation and to break it you follow Hartree's procedure and have approximate numerical solutions in which you obtain self consistent field solutions. So, some of these details we have discussed in our previous course I will not go through it now. This is just a quick reminder that these are the techniques which are used and then of course, you go beyond Hartree you must include the spin and that is the Hartree-Fock formalism. This is a nuclear part this is a single atomic system. So, there is a single nucleus for molecules of course, you have the nuclear-nuclear interaction this is a single nucleus. So, I am developing the formalism in the context of a single atom, but of course, in condensed matter where you have a molecule or a cluster then you have the nuclear-nuclear term as well. So, the contribution from the nucleus is not neglected it is z over r i over here for in molecule where there are more than one nucleus, you can carry out the Born-Oppenheimer approximation or you can do even better that, but that is a matter of detail. So, essentially it is hopeless to look for an exact solution of a many body system. There is no chance that you can get an exact solution, it is not that it is difficult. So, it is absolutely hopeless I think you can understand it only in those terms. So, it is impossible to get an exact solution and this is best stated in remark by Brown in his book where he says that having no body at all is already too many if you are looking for an exact solution. So, you have to look for approximate solutions and even to write down these solutions you have a very huge task because just how much of storage space would you need to write handle these slater determinants. How many terms do you have in the slater determinant? If you just take a small atom like every electron has 3 degrees of freedom. So, it has got 3 n variables and if you write the wave function just on a 10 point grid actually it is more than 3 because you know that there is a spin also. So, just to simplify this I have taken 3 if you take a 10 point grid then you have 10 to the power 3 n numbers to tabulate and you will get a very bad wave function on a grid which has got only 10 points. But even that you have 10 to the power 3 n numbers and what kind of a number is it for n equal to 1 it is already 1000 and then for neon for mercury atom these are huge numbers. So, you need techniques and the occupation number formalism the second quantization methods that we are going to discuss in this unit will tell you very elegant ways of doing this. So, you have to deal with spin as well and we know that self consistent field solutions are possible in the Hartree-Fock formalism and the main strategy of Hartree-Fock is to get an extremum of the expectation value of the n electron Hamiltonian in the slater determinant. This is the single slater determinant mind you and in this you obtain this extremum subject to the condition of constraints of the orthogonality of the single particle wave functions and their normalization as well. So, you can determine the expectation value of the single particle and two particle operators and proceed and this is what we discussed in our previous course anatomic physics. The two electron geminal state which is always nice to have some kind of handle on because all the two electron interactions are expressed in terms of these functions. Now, you can have the anti symmetry of the wave function that when you interchange q 1 and q 2 q 1 is here and q 2 is here when you interchange the first one is q 2 and the second one is q 1 the wave function must change its sign, but this sign must be attributed either to the spin part or to the orbital part and either of the two possibilities will give you an anti symmetric wave function. So, there are two possibilities here that the orbital part is anti symmetric and the spin part is symmetric or the orbital part is symmetric and the spin part is anti symmetric. So, you have two kinds of you know solutions and you must ask which part is symmetric and which part is anti symmetric and depending on whether you are dealing with a symmetric orbital part or a symmetric spin part you get the triplet or the singlet states at this again I will not spend any time discussing this we have had fairly extensive discussion in our previous course in atomic physics. So, you can certainly refer back to that and I will just like to remind you that if you consider the diagonalization of the coulomb interaction in this you know that the triplet state is less punished as Landau and Liffchitz call it has got a lower energy than the singlet. So, now you can of course write the single particle quantum numbers either as n l m l m s or as n l j m j and you can write one in terms of the other by simply carrying out this rotation of the base vectors in the coupled angular momentum space. You have got two sources of angular momentum for the electron one is orbital angular momentum the other is spin angular momentum and you can have their uncoupled spaces or you can have the coupled spaces and you can write both are equally valid you can write the wave functions either in terms of one basis or the other any basis is good all you need is a complete set of basis and some of these details again we have discussed at length in our previous course in atomic physics and some of these video lectures are also available on the internet and you can refer to that if required, but I just want to mention that you can use either of the two designations and depending on the coupling scheme if you are coupling you know if in the j m basis you have these spherical harmonic spiners which are you know which have two rows and one column. So, you can write these wave functions in terms of these spherical harmonic spiners so instead of the m l m s basis. So, here again the details can be found in unit 3 of the previous course in atomic physics, but I want to quickly go over to the many electron formalism which really requires us to go in for second quantization methods. Now, let us take a particular case of many electron system let us take the case of magnesium. Now, magnesium has got 12 electrons the usual configuration is 1 s 2 2 s 2 2 p 6 3 s 2 the 2 p is a spin orbit split because of the relativistic effects in the non relativistic quantum mechanics you will not have the spin at all. In relativistic quantum mechanics or in non relativistic quantum mechanics in which spin is plugged in on an ad hoc basis then also you can do some quantum mechanics with spin then you have j becomes a good quantum number and j is either half or 3 half for l is equal to 1 for 2 p l is equal to 1 s for the electron is half. So, you have two possibilities l plus half and l minus half from angular momentum coupling and the l plus half gives you the j equal to 3 half state in which you can have 4 electrons because this has got a degeneracy which is 4 fold for each j m j can take 4 values. So, for j equal to 3 half m j can take 3 half 1 half minus half and minus 3 half. So, it has got a 4 fold degeneracy and for j equal to half you have got a 2 fold degeneracy for j equal to half m j can be plus half or minus half. So, this is your configuration the superscript S D stands for the slater determinant and this is the slater determinant which I call a slater determinant 1 which I write as a subscript which is an anticipation of the fact that we will have a slater determinant 2, 3, 4, how far can you count? Many right. So, you can have a number of slater determinants let us consider this one. Now, instead of 3 s 2 I have got the first 10 electrons go in the same states, but the 11th and the 12th electron instead of going in 3 s go into 3 p. Now, this is also magnesium atom you have got the same number of neutrons the same number of protons the same nucleus the same number of electrons, but the configuration is different and if you are to write a slater determinant for the first configuration compare it with the slater determinant for the second configuration you obviously, recognize the fact that the 2 slater determinants must be different from each other. In other words each slater determinant corresponds to a particular configuration there is a 1 to 1 correspondence between the given configuration and the slater determinant. There is a 1 to 1 correspondence and you can have many different slater determinants you can have 2 electrons in 3 p 3 half you can have 2 electrons in 3 d 1 half 3 d 3 half 3 d of course, does not have 1 half 3 d has got either 5 half or 3 half. So, you can have what else 4 s 5 p anything you have really infinite number of possible slater determinants and strictly speaking to have a complete basis. You must include all of these and according to the fundamental consideration that we have in the expansion of an arbitrary wave function in a linearly independent complete set of basis. You must have expand the magnesium wave function in terms of this complete set of basis. So, you really have infinite slater determinants yes the coefficients of most of them may go to 0, but the coefficient of many may not go to 0 and how many really you really do need to include depends on the nature of the correlation that is there in the system in the 12 electron system. So, you have got a many body system which contains 12 electrons this is your many body system this is your many electron system and it is a correlated system. Now, if you have more slater determinant what happens to your Hartree-Fock in the Hartree-Fock you did an extremum of the Hamiltonian the expectation value of the Hamiltonian in a single slater determinant right, but now you recognize the fact that you have more than one slater determinant 2, 3, 4 may be more a certain number of slater determinants and all of these slater determinants need to be considered this is what is meant by saying that you must go beyond the Hartree-Fock and what requires you to go beyond the Hartree-Fock is the Coulomb correlation. So, this is the definition of Coulomb correlation Coulomb correlation is what is left out of the Hartree-Fock in the single configuration Hartree-Fock you of course, have the Coulomb interaction all those 2 electron integrals that you did in your previous course they are matrix elements of 1 over R i j that is the Coulomb interaction right.   So, the confinement to a single slater determinant is the single particle model when you go beyond that you begin to include correlations it does not mean that there is no correlation in the Hartree-Fock there is the exchange correlation, but there is no Coulomb correlation sorry because it is coming once again from the 1 over R 1 2, but the 1 over R 1 2 is coupling not only states from a single configuration, but also from this there are so many other configurations and all of these correlations they are coming because of the Coulomb interaction, but then coming from different configurations in the last 2, but they could refer in many others. Now, you have different corpical angular form right that is the I mean I thought that is why we have called Coulomb. No that is just another possible state of the 12 electron system at the reason you have this alternative state as an accessible state for the 12 electron system is because of the Coulomb correlation. They should all have the same angular momentum. So, finally, they must couple to the same j that is a matter of detail, but I am not getting into that at this point due to the anti symmetry of the wave function. It is there in each slated determinant. So, this is one slated determinant which is completely anti symmetric. So, if you interchange any 2 rows or any 2 columns the sign of the determinant will change. So, the sign of the determinant will change under any interchange. So, the exchange correlation which is the anti symmetry of the wave function or the poly exclusion principle that is already built into this and this is a different slated determinant, but the system wave function is a linear superposition of these 2. So, you are then lead to what is called as a multi configuration harsh form. So, now you cannot deal with just a single slated determinant, but a linear superposition of slated determinant. What is it coming from? It is coming because you have to consider the interaction between the first configuration and the second configuration. This is what is called as C i or the configuration interaction. So, this is the configuration interaction and this requires you to go beyond the harshly fork and that is what many body theory is about. So, even the harshly fork in a certain sense is a many body theory. In a certain sense it also has the correlation, but the correlation it has is only the exchange correlation, but the name many body theory is typically reserved for those formalisms in which you go beyond the harshly fork consider the configuration interaction and include the coulomb correlations not just the fermi correlations not just exchange correlations. So, these are what are typically called as many body correlations and I will describe it once again for the magnesium atom. So, if it has this 2 p 6 3 s 2 this is the usual configuration that one talks about this is the default that one that comes to your mind and the slated determinant for this will consist of 12 columns. So, q 1 through q 12. So, as you go from left to right you are going from first column to the second to third until you get to the 12th column and when you go through the rows 1 to the next and so on you go from different quantum states this is n l j m j quantum numbers are what I have used. So, this is the 1 s up state and then the last one will be the 3 s down with m j equal to minus 1. So, that is the 12th row. So, you have got a 12 by 12 determinant this is the determinant wave function and this is corresponding to different single particle states each row is labeled by a set of 4 quantum numbers. All the 4 quantum numbers are the same in a given row they change only from 1 row to the next in at least 1 quantum number. They may change in more than 1 quantum number like if you look at the first row and the last row the first row has got m j equal to half the last row has got m j equal to minus half but that is not the only difference the first row has got n equal to 1 the last row has got n equal to 3. So, they may change in 1 or more quantum numbers but there must be at least 1 quantum number which is different if all of them are the same then you have 2 rows which are equal in a determinant the determinant would vanish. So, this is where you have the slated determinant corresponding to the 2 p 6 3 s 2 configuration but then we agreed that you have not only this configuration but you also have the possibility of the 3 p 2 configuration and not just this there are many more but let us take this as an example. So, if you are to write a slated determinant for this then you will have the first 10 rows will be the same but the 11th and 12th row will be different and the 12th row will now have l equal to 1 here mind you because this is the 3 p state. So, you have got a 3 p number and not a 3 s anymore. So, what you are going to do is you have these different single particle states and you can arrange them in some order you can say that n equal to 1 l equal to 0 j equal to half m j equal to half is my state number 1 n equal to 1 l equal to 0 j equal to half m j equal to minus half is my state number 2 and you start giving numbers to each row. So, the first 10 you have label then 11 and 12 you label according to the 3 s 2. Now, these 2 rows you cannot label any more as 11 and 12 because the label number 11 corresponds to the 3 s up. So, this will be your 13 and 14 what is happening in the slated determinant is that 11 and 12 are empty and 13 and 14 are occupied the first 10 are the same but what was 11 and 12 with the previous configuration is now vacant and you have an occupancy of 2 in 3 p. So, in other words these different slated determinants which correspond to different configurations you can refer to them in terms of occupation numbers of single particle states. What is the occupation number of state number 1? What is the occupation number of state number 2? What is the occupation number of state number 11? What is the occupation number of state number 13? What is the occupation number of state number 20, because you have got an infinite set of slated determinants. So, you can write all of this in terms of these occupation numbers. Now, that is the formalism which corresponds to the occupation number space, and this is where we use second quantization methods, which is what we are going to discuss. Yes, Ankur, you have a question. You will always have 12 by 12, because you are describing a 12 electron system. The 11th label is empty. The 11th label is empty. The 11th row is occupied by what would be the 13th label. The 12th row will be occupied by what is the 14th label. You will always have each determinant for an n electron system will be an n by n determinant. So, it has to have 12 rows and 12 columns, but which 12 rows that set is what we are talking about. And in this set, the labels each label corresponds to a particular one electron state. And the last two labels in this are different from the last two of the previous. So, if you look at this, you have got the 3 p half is the last one. In the previous one, you had this is you had the 3 s. So, the 12th row was occupied by the 3 s quantum numbers, but now the 12th row is occupied by 3 p 1 half. So, you have 12 rows and 12 columns, but which 12 rows is different in different slated determinants. In the previous case, we are denoting by this 2 and the position is by q 11th and q 12th. And in this case, whether we should use some q 13 and q 13. No, no, you are talking about 12 electrons. You are talking about 12 electrons. What is the probability amplitude that a particular electron at coordinate q y is in the state alpha. So, here this is the probability amplitude that the electron at q 1 is in the state. This probability amplitude is different from the probability amplitude that the electron at q 1 is in the 3 s state. So, you have got 12 electrons, they are where they are in the spin orbital space, but the probability amplitude that they are in different quantum states, single particle quantum states is different, which is why the rows are different. The coordinates of course, will be from q 1 to q 12, they are only 12 electrons. It is certainly possible. So, you have the levels 11 and 13. You could have that as a possible configuration. So, you will have infinite configurations. Of course, there are some other constraints because the total angular momentum must be the same and so on. So, those are the details that we can talk about, but basically you will have many different kinds of possibilities. You can have two electron going from first configuration to the other. You can have one going from one state, the second going to a different state. So, here you have what we have done is we have elevated 3 s 2 to 3 p 2. You can elevate 3 s 2 to 3 d 2. You can elevate 3 s 2, one going to n l and the other going to n prime l prime. So, there are all kinds of possibilities and that is what makes many body theory very challenging. And all this is happening because of the electron correlations. So, there are these possibilities and the configuration interaction wave function is now a linear superposition of all these later determinants. And depending on how many configurations need to be considered in this, n can be either a small number like 2 or 3 or 4 or it can be quite large and people do calculations with hundreds, sometimes even with thousands of configurations. Now, it is a mess because each determinant has got factorial 12 terms. We already felt like quitting when we were dealing with a slingle determinant when we thought about a number like factorial 10 just for the neon atom, which is a small atom. Now, you have factorial n terms and so many of them. So, there is a good way of handling this, which is what makes use of the occupation number formalism. This is where the second quantization methods come into the picture. So, the occupation number formalism is sometimes called as the second quantization. Why second? Because the first quantization was doing away with classical dynamical variables q and p position and momentum, which were simultaneously measurable accurately in classical mechanics, recognizing the fact that such a simultaneous accurate measurement is not possible and therefore, replacing them by operators that was your quantization. Now, what we are going to do is that the wave functions that you are talking about will also be treated as operators. In the first quantization methods, the wave function psi was like a scalar function, it was not treated like an operator. Now, in the second quantization method, so it is a technique and because it is quantization of the scalar field, it is sometimes also called as field quantization, but these are different expressions to talk about the same technique. The technique is essentially that of occupation number formalism, also called as second quantization or field quantization, field theory and all of these are related terms depending on different context. They do have different emphasis contained in these alternate expressions and they are inspired by the fact in the context of the fermions that each slater determinant is decided by a particular configuration, which means that which single particle states are occupied. In the first configuration, other than the first 10, these two are occupied and these two are empty, but in this case the 3 s are empty and the 3 p are occupied. So, the occupation numbers are different and you can talk about going from one to the other by destroying two electrons in the first and creating two electrons in the other. So, you can begin to make use of operators, which are called as creation and destruction operators or creation and annihilation operators and the second quantization methods make use of these operators and that is the formalism that we are going to learn in this unit. So, your slater determinant in the first configuration was made up of these states 1, 2, 3 and 4.  So, these are the 12 single particle states. When you talk about the second configuration, the 11th and 12 those two states electrons in those two states are effectively destroyed and you create two in 3 p 1 half. So, in the second configuration you have instead of the 11 and 12, you have 13 and 14 that is what I was referring to. And you can refer to these in terms of single particle operators, creation and destruction operators and these are the ones, which are used to describe the n particle system, essentially you are describing them in terms of a complete set of compatible observables or Eigen values of complete set of commuting operators from Dirac's CSCO and you have n number of identical particles. So, the individuality of the particle that idea is carried over, but so is the indistinguishability. So, is the statistics and so are now also the many body correlations. So, you use the single particle labels, even in the presence of not just interactions, but also in the presence of correlations and I have made a very clear distinction between the term interaction and correlation. So, correlation has got a specific connotation in the context in which we are discussing this. So, you describe a particular slater determinant, a particular configuration by spelling out how many particles are there in state alpha 1, each alpha is a set of 4 quantum numbers, how many particles are there in state alpha 2. Now, this number is either 1 or 0 for Fermi particles, but it can be anything for both particles, because there is no poly exclusion principle for bosons. You can put any number of particles in a one both state, as a matter of fact you can put all of them and get both bosons and condensation. So, there is no such restriction in that, but the basic property of any configuration is of any many electron system is how many particles reside in these single particle states and that number is the occupation number of that state. This will give you one configuration, but then of course, the system wave function will contain many other configurations and you will have a linear superposition of such states. So, this is the idea behind indistinguishability, you take into account the correlations, you recognize the elementary nature of these particles, although they are indistinguishable. So, you reconcile with that and essentially you can get these numbers as Eigen values of the occupation number operators, which are complete set of commuting Hermitian operators. You define these occupation number operators and of course, the description will be different in terms for both particles and for Fermi particles, because if you have created a particle in one vacuum state and if that particle is a Fermion, you cannot create any more in that state, but if it is a Bose particle, the creation operator can operate twice or 10 times or have a Bose Einstein condensation and pump everything into that. So, the properties, the commutation properties of the creation and destruction operators for Fermi and Bose particles will be different. So, the statistics will now be incorporated in the operators. So, you have some sort of an arrangement, this alpha 1, alpha 2, alpha 3 and is some predetermined sequence, you decide what is your state number 11, you decide what is your state number 12, decide what is your state number 13, 14 and then you say that state number 11 and 12 is occupied in this and 13 and 14 is vacant and in the next configuration 11 and 12 are vacant and 13 and 14, but these numbers have to be pre-assigned. So, you have some sort of a predetermined sequence and in this sequence, you give the occupation numbers and then you have a vacuum state in which all of these particles, all of these states are vacant. You can have a single particle state, in which there is only one particle in the state i, so in the i th state n i is equal to 1 and whenever j is not equal to i, this occupation number only for the i th state the occupation number is equal to 1, whenever j is not equal to i, the occupation number is 0. So, this is how you would describe a single particle state and depending on the number of particles, you can write different occupation number states. So, I will conclude my discussion today, I will be happy to take a few questions and we will go from here in the next class. In the meantime, there are some references which I will like to draw your attention to, for Hartree-Fock, we have already given these references in the previous course, which is either Betty and Jackie or Branstad and Joshain, these are the primary sources and for second quantization and occupation number formalism, I will primarily really use Fetran-Valekas book or the book by Reims called many electron theory. Questions? In the magnesium 12 electron system, the alphas will be the set of alphas, you are referring to each configuration will be having 12 ones and all of them. So, each of that is a configuration. Yes, exactly. So, out of the infinite numbers, 12 will be occupied, all the others will be 0, but which 12 is going to be different for each configuration, a particular choice of 12 defines once later determinant. So, there is a one to one correspondence between a configuration and a slated determinant and an occupation number state vector. So, these are state vectors in what is called as the occupation number space, sometimes also called as a Fox space. Maybe a range annihilation and… Right, right. So, in our next class, we will get into these second quantization operators, the creation and destruction operators and the number operator and so on. Alright, thank you.


Hi, and how are you all today? The question says a plane left 30 minutes later than the scheduled time and in order to reach its destination 1500 kilometer away in time it has to increase its speeds by 250 kilometer per hour from its usual speed find its usual speed So let's proceed with the solution here let The usual speed Be equal to x kilometer per hour right now the distance That needs to be traveled is 1500 kilometer so time taken to travel Will be equal to distance upon speed that is 1500 upon x hours right now Time taken to cover 1500 kilometer by increased speed That is x plus 250 kilometer per hour Will be equal to 1500 upon x plus 250 hours right now Further we are given from the question It says that a plane left 30 minutes later Then the scheduled time and it is reaching the destination in time. So that means We can say that X upon sorry 150 upon Sorry again 1500 upon x minus 1500 upon x plus 250 Is equal to 30 minutes that is half an hour right now what we need to do further is we just need to solve This quadratic equation that will be formed Simplified now we'll open the brackets in the denominator also we have x square plus 250 x after opening the bracket is equal to 1 by 2 further implies x square plus 250 x is equal to 250 into 1500 into 2 that is 75 0000 That further implies x square plus 250 x minus 75 0000 is equal to 0 now splitting the middle term we have plus 1000 x minus 750 x minus 75 0000 is equal to 0 so we have x Plus thousand as one of the factors and x minus 750 as one of the factors This implies the value of x is minus thousand or Equal to 750 since distance cannot be Sorry speed cannot be in negative hence The usual speed is Equal to 750 kilometer per hour Right, so this is the required answer to the question. Hope you understood it well enjoyed it to have a nice day


Hi everyone, I'm Joe for jaspyscasebrakes.com. Happy Tuesday, coming at you with our last filler, that tier one double header break, coming up in the next video. But we gotta give away those eight spots. First, we're gonna do a box of Chrome Platinum Adversary Baseball, random team break number eight. That'll be the first dice roll. Second dice roll will be for the giveaway part at the end of the video. Big thanks to this group for making it happen. Appreciate it. All 30 teams are in. Let's roll it. Guys, names and teams four and a one, five times. One, two, three, four, and a one, fifth and final time. Travis going big on this filler. Thank you, thanks everybody for making this happen. Four and a one, five times for the teams. One, two, three, fifth and final time. All right, no trade windows in these filler breaks. Here's the first half right there, second half right over here. And let's just print and rip. So here is the final printout, hot off the presses. There's the anniversary edition right there. And good luck. There might be some surprises in this box. There is an autograph in here, yeah, there is an autograph in here. So let's get a little NLB tonight on in the background. There's only one final today, right? I think the Cubs Cardinals are in a double header. Yeah, Cubs shutting out the Cardinals, two nothing. Albert Pools had a, went one for four with a double. I think there's some fan interference there or something like that, but yeah, he's hitting the ball well. Wonder if he'll play in the second game of the double header. Maybe as DH, he DH this game. You can go up there, swing a bat, right? I'd like to see him run down 700 homers, that'd be awesome. But just not the Cardinals day today. Adam Wainwright takes the loss. And Rucker, Michael Rucker takes the win. Oh, Assad had the start, but Rucker takes the win. Couple more packs here. All right, good luck everybody, here we go. X-Fractor here, Santiago Espinol wave, Aqua wave, shimmer, something. Got to adjust some more note, 12 out of 199 for the Twins. Blue speckle for the Twins, that'll be for Jeremy. We've got Ahmed Rosario, 40 out of 99 for Cleveland. That's gonna be for Travis. Should be down here somewhere. There's the auto. It's Julio Franco, Cleveland, this is for you. Goes to Travis, the Guardian, it's nice. Old school player here, nice. And there's Tariq Scoogle at the end right there. There you go. All right, now let's see who's gonna win those tier one double header spots. Let's gather everybody's names again, one through 30. New dice, new list, top eight after four. One and three, four times, good luck. One, two, three, and once more, fourth and final time. Now from nine on down, Nestor, my apologies. Maybe your name's still up there in the top eight, but for now, nine through 30, thank you. Thanks for giving this a shot. We'll get them next time. Happy times though for the top eight. Jeremy is happy, so is Travis. Martin, Nestor, there you go, Nestor got in. Travis, Chris, Travis, and Nestor, again. There you go, top eight. I'm Joe for Jaspy's CaseBreaks.com. I'll see you in the next video for that double header break. Bye-bye.


Well, at the top of the show, we mentioned we would have some news on Aroldis Chapman. Last night was the sixth consecutive game in which Chapman did not appear. And during Aaron Boone's pregame media session, he had news regarding the left-hander, providing our quote of the day presented by Toyota. Aroldis is going to go on the IL today. It hasn't been a non-pitch the last couple of days. He got an infection in his leg right now. So we were hoping we'd kind of knock it out and be able to take care of it the last couple of days. But it's still significant enough from a tattoo that he recently got that got a pretty bad infection in there. So he was actually at the hotel yesterday, he's still back at the hotel now. So we're treating him with, you know, medicine, antibiotics and things like that. So he's got an infection that he's dealing with that's going to force him onto the IL. On the left leg? I don't know if it's the left or the right. And is he on a no-throw for a while? He's got an infection. So that means how long is that? Do you expect him to even? Yeah, it's at least going to be a few more days. So yeah, the last couple of days he's been unavailable. He had a new tattoo and it got infected. So it's a pretty bad infection. So he did not have to go to the hospital. He saw the doctor here. But we sent him home. Well, what's today? Saturday? So he was here Thursday and then Friday we kept him back at the hotel. Yesterday we kept him back at the hotel. Today he's still at the hotel. They're treating it and addressing it and, you know, antibiotics and things like that and hoping that it's not something too serious. IB? No one as of now. Wouldn't be able to get anyone here today. Obviously with Clay coming off on Monday, you know, we'll just kind of monitor the situation today. If we need something for tomorrow, we'll maybe cross that bridge. Do you expect him back this year? Chappy? Yes. Yeah, I think hopefully he's good in several days. We can backdate this a few days, but we got to get our arms around and get the infection out of there because it's gotten a little bit worse over the last two days. Is there any frustration that he would do something like this during the season and perhaps put himself at risk? Do you burn infected or anything else that might go wrong? Right. Well, I mean, it's hard for me. I don't have tattoos. I don't know. You know, so, I mean, that's a personal choice, like usually a pretty safe thing to deal with. And this is more of an unfortunate situation that, you know, has turned into an infection. So, I mean, my focus is on trying to get him right and healthy.


Sometimes life is like getting 100 metres into a 42 kilometre marathon and accidentally stepping knee-high into a puddle, make it a muddy puddle and then having to complete the rest of the marathon in those shoes, I mean, could you imagine it?   But that's Cooper's knob. I'm almost, yeah, very, very close to being three miles in. I'm going to put my phone away for a decent amount of time. I'll come back with you when I've got some news. But that last mile was 22 minutes 38, so let's keep going.  Welcome back to another video. Today I bring you once again Banks Peninsula. I was parked right here about a week ago, taking you guys overnight in that van around the corner and up the road a wee bit. I had a fantastic time. In fact, I've taken that van overnight again, probably about four or five times. And the date is Wednesday the 5th of January 2022. So guys, I'm off once again to my favorite spot, the sign of the Bell Bird. I'm going to spend another night in the van. That's a given, but what's not a given is what I'm going to get up to tomorrow. So I'm going to try my very best to hike, slash run, slash walk, slash crawl if need be. And look, we've actually got a fellow mountain dweller right there. Anyways, guys, I'm actually going to get burned to a crisp. It's a scorcher. Well, it's not a scorcher, but as far as temperature, but as far as what the sun does to my skin. Although recently, I mean, a lot of people have said I'm looking quite brown, parts of my body anyway. Maybe not my face. I'm not sure. Anyways, guys, I'm super excited to endeavor on a 20 mile hike, slash run, slash walk, slash crawl. Like I said, a 20 mile hike around the summit of this crater along the crater rim walkway. And so I'm going to wake up tomorrow morning, bright and early, about quarter to six when the sun comes up. I'm going to make myself a nice coffee. I'm going to sit down with you guys and we're going to map out a bit of a plan. A 20 mile hike. I don't know how long it's going to take, but it is up and down. There is quite a bit of elevation, not so much in one go, but certainly going up and down at quite an angle. Anyways, guys, this is for fun. This is for entertainment purposes only. It's time to continue along to our parking spot for the night. And I'll see you soon wherever that may be. You know what, guys, I'm going to utilize some nice shade because in our eventual parking spot called the sign of the Bell Bird, there is absolutely no shade. But what I have found here is a nice, well actually, there does seem to be a gate. But I'm just going to chill here for a bit. Well, alright then. That is officially number 2016 Summit Road. So in a place like Christchurch, you can never predict the weather. Even the weatherman doesn't know when it's going to rain. It could literally be raining tonight. In fact, I do think it's going to be raining in the morning. But for now, for now, guys, this is pretty much as good as you're going to get. It's summer's day in Christchurch. I mean, you can't really see me, but hopefully you can see the colour of the sky. Have you got my? No, absolutely not. It's all good. There's not much shade along here, so... No, that's why I'm taking advantage of this. Shade is far. Good idea. Well, that's interesting. I just came across a cyclist who did the exact same thing as me. You know, it stopped in the shade for the reason that there is hardly any shade along this road. So apparently she's going from a place called Beckenham, which is on the Christchurch side, on the flat, probably about two or three k's away from the bottom of the hill. And she's gone all the way from there to the bottom of the hill, you know, cycled all the way up that road that I was on.  And I dare say that's a decent cycle. That's a decent workout. Not something I'm ready to do. But interestingly enough, she said that she can't run, unfortunately, because her knees stuffed. Or at least she can walk, she can hike. She can pretty much do anything that doesn't require... I mean, she can cycle, obviously. She can do anything that doesn't require the leg and the knee to be, you know, having that jarring impact. And so apparently after about 100 metres, her knees or her knee starts playing up. But what she can do, at obviously a pretty decent level, is cycle. So that's really interesting. You know, if I do have knee problems, then it's not all over.  I don't know how old she is. She's probably like 40 or something, South African woman. But you know, to me, that's reassuring. Because if I do have knee problems, that do take me out of running. Cycling wouldn't be too bad. I'm not going to lie. You can definitely access some pretty epic places. And what goes up must come down.  But that'll have to wait. Tomorrow, it's hiking. Tell you what, this is living. This is living. You can't see much, can you? Still not sure. Well, jeez. Right guys, I'm just going to finish off my snack here. Don't you just love Kettle's chips? Are they a New Zealand thing? Australia slash New Zealand? I'm not sure. Anyways guys, here we are in the car park. This will be my final destination. Hang out down the road there for a good three hours actually. It was quite nice. But here we are. We've got three other neighbors. Not sure if they'll be here for the night. Most likely not. I'm not sure where they are. Because there's no one around. Tomorrow morning, the planners get a good rest, get a good rest, wake up about 5.30 in the morning, get my coffee going, get my shoes on, my hat on, my sunblock on, my Garmin smartwatch tracking device on. GPS tracking device. Which is mean because I don't have to use any apps on my phone whilst running anymore. So it doesn't drain my battery as much. So what I can do is just use my smartwatch to track the entire run. That's better. So what I can do now is just use my watch. I could literally go out for a walk or a run or a swim. Doesn't matter what it is. And I could track it directly from my watch. Where are we? I can track it directly from my watch and it's gonna log it automatically without having to have a phone connection. So what I can do now is I can utilize my entire phone battery for music rather than use its GPS capabilities and basically drain the battery because I'm using a tracking app. And I found that out. I was actually really, really lucky. It goes back to when I did my first marathon. I think my phone battery must have been at about 85% when I started the race. Literally. You couldn't get any closer than this. So I had to stop listening to music for the last, like, hour which was torture. But the reason was, and it did come in handy is because my phone battery was dying extremely quickly. And so I just crossed the finish line. I just managed to stop the activity. So I logged the marathon in Strava.  Thank you for allowing me to log that run.  But it just goes to show, you know, using your GPS for an extended period of time, as well as music can and will drain your phone battery. Anyways, guys, learn from my mistakes. I won't do that again. And I won't have to because I've got this. Anyways, the plan is to take off in the morning. 20 miles, which is around about 32 kilometers. And I'm assuming it's probably going to take in the ballpark of around 6 or 7 hours. So that should be fun. Anyways, guys, I'm going to chill out, have a bit more food. Crack a crack a beverage, actually. Because I'm allowed. It's the new year, guys. That's the new year. The festive season. I'm feeling festive. You know, these long summer evenings in New Zealand special. And I'm grateful. And I'm grateful for you guys. Anyways, I'll see you soon. What's up, guys? It's me again. I'm not going to lie. Discipline needs work. Tell you what, this van is so cozy. Seriously.  I'll just roll the other way and just go back to sleep. And then it's 7.30 in the morning. When you're meant to get up at 5.45, it happens. Things like this happen. Seriously, guys, would you want to get up right now? Would you? No, you wouldn't.  So that's what we're going to get started with.  I've got to get my caffeine in. Got to put my shoes on, etc, etc. It's 7.30 in the morning. Give me a break. Let's get going. Right, guys. We're off to a good start. Teeth are done. Coffee's done. The view's fantastic. Although it's not as good as it was yesterday. So I'm glad. I'm glad I got the camera out yesterday at the very least and showed you guys what this place can look like on a really good day. This morning? Not so much. In fact, it's raining.  It certainly won't be looking for any shady spots this morning. It is cold. The rain's coming through, albeit very light. But it is 7.49. I'm going to sit here, enjoy my coffee, enjoy the last 10 minutes of the view and I'm going to set it off for 8 o'clock. Once again, thank you for being here, guys. This video is not for any monetary gain. Basically, it's to set myself a challenge. I do best when I've got challenges to complete and 20 miles, guys, on the crater rim walking track is one of those challenges. So I'll give you a look at what I can see. How's that? So this, my friends, is what I look like from this angle. That's what I look like from that angle. This is what I can see. So, uh, yeah. Oh, cannot wait to drink this. But yeah, that's it, guys. You'll be getting pretty familiar with this place. I know I am. So, the plan is if I can wake up prior to leaving is, um, if I can, uh, the plan is, okay. What is the plan? Sorry to rack my brain there for a second. My plan is to leave all my camera equipment here in the car, lock it up and use my phone as my camera whilst walking. Which would be extremely useful and the picture's pretty damn good, too. So, I'll see you guys on the road. One and all.  I start moving. Fuck, it's cold. They call these windbreakers. Ain't breakin' nothin'. But I have been enjoying my little, uh, hood here when need be. It sits right on the edge of a cap and I absolutely love it. So, we'll do this up. Not sure how I'm gonna get my headphones on now. But that's all right. Packin' the bag.  That would be silly. What we're gonna take in the form of food is one, one square meal bar in there. I've got an apple. I've got a berry smoothie drink. I've got a 1.5 liter of water. We'll get that apple out. Actually, you know what? We're gonna leave that here.  I don't want that. I don't particularly want to be taking enough at all. But I will. All right, so we've got the water. We've got the one square meal bar. One apple. One plum. I'm gonna take some chips. I reckon that's about it. I reckon 1.5 liter of water for now will be okay. Well, actually, let's make it three. Let's make it three. So this is probably, I don't know, ooh, yeah, it's got a bit of weight to it. It's probably gonna be about, I don't know, 12 pounds. Something like that.  But not just metaphorically.  There's no Sherpas around here. We're in Christchurch. Wouldn't mind a Sherpa though. Alright. Yeah. Oh no, it's not too bad. That'll be fine. That'll be absolutely fine. Oh. Can't forget that. Sunblock. In fact, I'd probably be best, believe it or not, just to put a little bit on, which I'll use the camera for, on my face. Oh wow, we've gone heavy. I'm not playing football. I'm going hiking. Jeepers. Jeepers, creepers. That was yeah, a little, a little too liberal. Sunglasses. Good time at 7.58. Put this back on. Drink this. I'm going to need one more of these. Not going to lie. Ah fuck, did it again. Alright my friends, the time has come. It's time to lock the van and head up that way. So I'll see you guys at mile number 1. Let's go. Alright guys, I'm not, I'm not saying I'm going to give this much detail every single mile, but for the first mile, I feel like it's about to tick over just over 18 minutes. There we go. 18.12. I'll take it. This is where we're at. It's well, it looks as if I'm in Kansas. Does it? Or doesn't it? I'm not sure. Anyways guys, here we go. Just over 18 minutes. 18 minutes 12. Mile number 1. I've basically been just taking it relatively easy, keeping my heart rate around about 120. Maximum. I've jogged a couple of the downwards sloped areas and here we go.  I mean, credit to the track builder, because it's pretty obvious, but yeah, not in Kansas anymore. So I'll continue up to the top. We're 1.7Ks now. Sitting at about a solid 12.5 minute pace per kilometer going uphill, which is probably around about 5Ks per hour. So I'll catch you later. I'm thinking we're closing in on mile 2 which should be in 16 minutes and something, possibly 17. Just. 16 minutes 14 seconds. That is the fastest mile so far. All good. What's up guys? We're on our way. 3.6Ks, which is just over 2 miles. The views well, it looks like we're still in Kansas on that side, but on this side it is pretty cloudy but it still looks cool. Anyways, what I wanted to talk to you about was in the distance on top of that hill is a marker and I believe they have markers like these to signal the peak of the mountain or possibly even a mountain range. I'll have to check that, but what we're going to do is head right up the top. We're going to touch that and continue on our way because I think it's out of our way, so to speak. It's called Cooper's Knob and I'll see you at the top. There she is. Careful up here. There's no one to save me. Gotcha. Cooper's Knob. What a sight.  It's pretty windy. I'm not sure if you can hear me, but maybe that's Cooper's Knob. I don't know about you, but the mountain and the distance looks a little bit more like a knob. In fact, there's a few knobs around these parts. I never actually knew how many there were.  I've got a couple of things to look up. I'll be back with you in just a minute. Alright guys, two pieces of information. You learn something new every day and today I've learned two things. Number one, we are 573 meters above sea level, which is over 15, 1600 feet, which is far cool to be fair. But anyways, that's number one. Number two, that marker that we see at the top here is to coincide with maps. And on a map, obviously you see the summit, you see a marker at the summit. If you reach the marker, you're effectively at that point on the map. And that makes a whole lot of sense. So these are these would be extremely useful, especially in places where there's no reception and you're all on your own. Now I am all on my own, but I do have reception thankfully. Makes you want to get out there. It really does. Anyways, 573 meters above sea level and we continue. We are we're in rainforest, or at least it feels like it. Well, we were. But what beautiful surroundings. It's just a pissy. That once again, my choice of shoe isn't very waterproof. But you know, you live and live. I'm really enjoying this. Really, really enjoying this.  Surprisingly. I mean it's early days and obviously I'm taking my time to talk about heart rate. Yeah, you got to be careful. I should put my phone away, but I don't want to. Yeah. Choice of shoe on a dry day. Okay. On a wet day. No. But I complained honestly. I've already thought of this. I've had time nearly being on the road for 55 minutes. The last time I came out I got disheartened because my feet got wet and I turned around. Right. And on my way back I actually forgot about my feet because I was feeling so good.  I'm like I'm not even feeling it. Obviously it's something you just got to get used to. And on things like marathons, ultramarathons, hikes. I'm going to come up against some adversity and it's all about poshing on. And so today my feet are soaked again. But there's no way I'm going to turn back because you've just got to deal with it. And that's a lesson I learnt the last time I did this. I mean I'm no pussy. But I thought do I really have to deal with wet shoes, wet socks? Yeah, sometimes you do. Sometimes life is like getting 100 metres into a 42 kilometre marathon and accidentally stepping knee-high into a puddle. Make it a muddy puddle and then having to complete the rest of the marathon in those shoes. I mean, could you imagine it?   I'm not going to say I'm I'm not going to guarantee anything. It's dependent on food for me. But that's Cooper's knob. I'm almost very very close to being three miles in. I'm going to put my phone away for a decent amount of time. I'll come back with you when I've got some news. But that last mile was 22 minutes 38, so let's keep going. I tell a lie well, this is some news it's a freaking call. This is a forest. This is sick. By the way, if anyone was wondering I'm not going to hold my breath. I'm not going to hold my breath for you guys to even still be here, but for those who are if you're wondering how I'm passing the time for this first part of it I'm listening to our old mate Joe Rogan and it's his latest podcast well, one of his latest ones and it's with Tim Dillon and as a lot of conversations go, not only on Joe Rogan's podcast but in my life too the first thing you talk about or if not the first, definitely the second is the virus and differing opinions and majority versus minority and look it's definitely people and I'm going to admit for the first time basically I'm going to admit to myself, I'm going to admit to you guys it's finally caught up with me the whole situation, COVID not being in control affecting plans certainly affected my plans and something I've been well, a doll that I've had in my head for a long time but seriously oh shit now where do we go down here we're on the road guys I couldn't be completely sure if we're on the right track or not I hope we are I feel like we are if we're not then we're stuck in the labyrinth that's the vibes I get but the cool thing is guys the best thing is the fact that we're up we're up and at it it's only nine in the morning, we've already been hiking for an hour we've seen so many things we've learnt two new facts life is good right, life is good the only thing missing is sharing this with someone but that'll come in time anyways, back to my plans being affected by COVID the whole thing's finally caught up with me I'm feeling frustrated, depressed stuck at the mercy of certain select governments being New Zealand and the United States so that is what it is guys anyways, I'll see you I'll see you a bit later on because something like that could happen I could have just hung myself there that wouldn't have been good alright guys so it was the correct route that's the fence, it continues all the way up to the top of there the path is pretty much non-existent if I just woke up from a coma not that that's a laughing matter but let's say it's a self-induced alcohol, coma toast state and I woke up in the middle of this tear field I wouldn't know how the hell I got here there's literally hardly any path I'm going to turn around now we're at about three and a half miles I'm going to turn around now, this is the end of this path I'm going to continue trying to I'm no track maker put it that way but we've made it a fair way that's actually, I haven't really looked back since leaving and these are the moments aren't they where you turn around and you admire just how far you've come in life and from Cooper's knob anyways guys look I am turning around should I? jeez, you know what if it wasn't for you guys being here I would certainly be turning around but for some reason I want to get up to the top come with me come with me, let's see what the track's like maybe it just hasn't been walked on for a while I tend to think that the top of it's closed that's what I was thinking maybe that's just wishful thinking negative thinking though can't be doing that so we will continue thanks to your guys support so thanks in advance alright, that's about enough fuck getting to the top there's been about 3 or 4 spots that I literally have not known where to go I am indeed making my own path now and if I don't stop soon I may bite off a little more than I could chew just jokes anyway time to go back down my own path we'll call it Jacob's path there's already Jacob's ladder Jacob's path I had Jacob's beach when I was a kid that was pretty cool but yeah, we're going to make our way all the way along that fence line along the ridge line and not go up to Cooper's we're going to stop about there and continue all the way down to the car and on now what guys this may be obvious this may be obvious but what I've just done could be one of the stupidest things I've done in a while I could have just stuck to the side here I could have just stuck to the fence line the entire time I've literally gone this way that way like this roundabout couple of circles and then finally made my way up here by the looks of things I could quite easily just hit up the fence line and finally make it to the top you see that that is a metaphor if I ever did see one for perspective being down there looking up I couldn't see that for some reason I just simply couldn't see that didn't make sense to do that for me but now being up here looking back knowing what I went through and from this perspective the path is clear the route is obvious whatever this metaphor means for you guys take it put it in your pocket and save it for a rainy day can you guys can you see anything what do you reckon worth it is it I don't know yes it is every time I put the camera away oh man no there's a massive prickle bush where am I going it's time guys it is time what's up guys we're taking a break the first break of the hike we covered exactly 11km it's been up and down as you guys know I'm just having a wee snack I had my plum I'm going to have my apple now get some water on board and then we'll continue the other way right my friends we continue the journey 11.1km into it of an eventual 32 which seems like a daunting task but there's only one thing to do and that's to do what this guy is doing and get into it so yeah let's go let's do it that's where I went before this is where I'm going now oh shit oh good oh can't be front there he goes a man in his natural habitat if I ever did see it anyways guys I'm going to follow him believe it or not I've actually reached the exact spot that I pulled up at yesterday afternoon it was right there and there was a car parked here for a wee walk and obviously they've gone for a walk up there so not only is it the spot that I parked at yesterday but it's also it's also my glimpse of a wild animal and there she is she he I couldn't even be sure but they are sheep New Zealand's well known for its sheep farming lamb wool and all the rest of it anyway I'm just trying to work out which is the way to go and I'm thinking it's over here so it was four and a half K's from the sign of the bell bird to the sign of the kiwi so we must have done around about three kilometres let's keep going so we're crossing Diaz Pass Road which is the road that goes over the hill and it's the road that I took to get up here and that building up there is called the sign of the kiwi and what we're going to do is cross the road and continue along the crater rim walkway for another five kilometres then I'm going to turn around come back and by the time I get back to the car it should be around about 32 kilometres or in other words 20 miles what's up guys I couldn't lie to you that's not what this channel is about I decided to turn back 20 miles is indeed biting off just a little bit more than well not my lungs but certainly my legs can handle can't handle 20 miles on the trails certainly not these ones they're pretty rugged man I'm not sure what kind of racing especially the route that we did first I'm not sure what kind of difficulty there is or even if there is one there's got to be one but surely that's like at least intermediate couple of shaky moments couple of sketchy steps on the wrong thing at the wrong time almost fell haven't fallen touch wood but I think we're about like 200 metres away from the car and we've just walked over 20 kilometres, 4 hours, 20 km you know my my legs are feeling it I brought my massage gun which I've just bought I brought it with me and I bought it online so I'm going to get that out and give it a go, especially my calves my hamstrings and my glutes and my quads everything so thank you once again the cloud has gone somewhat so if I turn you round here we can again see the harbour so I'll see you back at the car right guys, we're back on the good camera I like this better even if the picture is not as sharp I just like the aspect anyways, in other news I'm absolutely starving it's um well actually put that down there I just stopped my watch, I forgot to stop it when I got back which pissed me off but all in all 4 hours 15 minutes 20.47 km averaging 12 and a half minute which is pretty damn slow but we are hiking and what's that, 2,273 calories jeepers that's not bad by the way don't go past a chip sandwich seriously, get your favourite flavour chip butter your bread up nice and thick and give it a go yourself could be because I just burnt two and a half thousand calories and I'm extremely hungry or it could be because it tastes delicious either way, that's lunch and just like that with the power of YouTube and the power of editing and also the power of the internet because if you or I didn't have the internet you wouldn't be watching this, we're back it's the next day, today's Friday the 7th yesterday was Thursday the 6th and we're back on dry land guys we're back in my my little YouTube studio editing the video I've probably put in about 3 or 4 hours already you wouldn't know that, you wouldn't know would you even when there's very low levels of editing on a video it still takes time because you've got to watch the clips back, I mean yeah you've got to make sure you don't embarrass yourself too much I just wanted to use this opportunity to close out the video to let you guys know how much I appreciate your support on the channel to let you guys know that I did indeed hike slash jog at times a total of almost 13 miles so it was 20 20.47 kilometers in a total time of 4 hours and 15 minutes but a total moving time of 3 hours and 20 minutes so 20.47k is almost 30 miles in 3 hours 23 minutes going if I didn't stop if I didn't stop and I pushed myself I reckon I could probably do 20 miles in just under 5 hours so that might be a challenge that might be our next challenge to go equipped to go prepared with both footwear, weather water was huge, water was important I probably got through about 2 litres of water on my travels, I took 3 so now I know, I was getting a dry mouth probably within about 45-50 minutes of starting and I was like shit I need to stop I'm going to take a wee rest, get my water out have a drink and you do need fuel you need fuel in the form of food, drink and mindset and I had all 3 so for any of you statheads out there you'll know how much I enjoy my stats so I'm going to run through some with you we had a total elevation change a total ascent of 1,033 metres that's over a kilometre of climbing over 3,000 feet pretty happy with that 2,273 calories burnt which is I'm going to say probably around about the amount that I burnt in my marathon late 2020 so it was a decent amount of calories really blown away and it's these times where you're sort of set out on a run or a walk or whatever and you're not thinking about calories you're literally not thinking about it you're just thinking about getting from point A to point B when you get back and you look at how many calories you've actually burnt due to your heart rate being spiked at certain times it's amazing absolutely incredible so over 2,000 calories burnt 20.5Ks, my fastest mile for anyone wondering was we had a fair few in the 15s but my fastest mile was mile 6 and it was done in 14 minutes 33 seconds mile 6 my fastest kilometre interestingly enough was 8 minutes and 18 seconds which you know is I mean it's jogging it's jogging lightly in my opinion a cruising pace for me is about 6 minutes per K on the flat you know being at 8 minutes obviously hiking going up and down it's not bad, we had 1 2, 3, 4, 5, 6 6 of the 20 kilometres were done within the 8 to 9 minute range which means that you know I'm not it's faster than a walk and when you are going up and down you do have over a thousand metres of climbing it's not bad, I mean I'm happy, how could you not be happy you've just done 20 kilometres along an extinct volcano so it was fun moving time was officially 3 hours 22 minutes 56 seconds an average pace whilst moving of just under 10 minutes per kilometre average cadence 131 steps per minute and look if anyone does want to check out my Strava please do because the cool thing with a Garmin smartwatch a Garmin GPS smartwatch is that it not only uploads the activity to the Garmin app but I've also connected it to my Strava app so all I have to do is press start and stop on here and automatically like I mentioned at the start of this video the data gets transferred over to those 2 apps and you know you can have a look at it I mean I'm not going to lie I love looking at stats I really do and it's something you can always look forward to have at the end of an activity doesn't matter where it is doesn't matter how long it is as long as you put effort into it you're going to enjoy coming back looking at your phone and looking back on your stats you're going to enjoy it you're going to get out what you put in so if you put effort into something this is the way that I get out what I put in and it's through reading my statistics and obviously don't don't get too into the stats because at the end of the day what's going to happen if you go out there and you do a 4 hour activity and the data is corrupted what's going to happen then it's not all about the stats I wish I could have gone out there without having to feel like I needed to sort of track what I was doing but that's not me, it's not the way I do things and it's never going to be so thank you whether I do a marathon next, a half marathon an ultra marathon or a hike who knows whatever it is if it's a physical activity I'll be bringing it to the channel thanks for watching guys once again thank you for the support a whole lot of fun walking around my backyard at the end of the day I'm doing this and I've mentioned this in previous videos I want to be more grateful for my surroundings I want to be more in touch with nature and with the scenery we've got around here because we are incredibly lucky to have it and it's something that until I moved away for a period of years and since come back I was never going to appreciate but I do now so thanks guys and I'll see you in the next video peace out


Everybody forced here up in a box of 2022 five-star baseball hobby for Chad see here. We go chalice see what you got Like maybe you have you either have to keep trying or just straight up But at the same time like That's a fumble Alright when I said Jake burger rookie purple auto numbered 47 of 50 Purple burger Chad And behind that is a Matt Chapman base auto Nice Chapman nice burger appreciate the order Chad Scars nice and protected to get him out to you shortly


I don't have a TV and I don't watch news. I don't watch CNN or Fox News or any of that garbage. I'm not interested in that whole my team versus your team mentality. I don't like all the constant fighting and, you know, that's why I don't hang out on places like Twitter and Facebook, you know, none of that stuff interests me. I think it's a complete waste of my time to get into political arguments with people. Quite frankly, I just don't care. I don't care about Trump, Biden, any of them. You know, I'm much more interested in political movements that actually matter. And that's not to say that, you know, some of the stuff that goes on in U.S. politics, it doesn't matter. But I'm interested in things that really make a difference in the lives of billions of people on the planet. And for me, you know, the political movement that I'm most interested in is the free software movement. And one of the largest companies that supports the free and open source software movements is Mozilla, that is the company behind the Firefox web browser and the Thunderbird email client. Recently, Mozilla put out a statement saying that they want to silence certain voices on the internet. Now, think about that for a second. This company that supports free and open source software and one of the cornerstones of free and open source software is that, hey, the software is free. You have the freedom to do anything you want to do with the software. And that means anybody can use it. They can use it for any purpose. It doesn't matter. It doesn't matter if they're doing something that I think is evil or unethical. It doesn't matter. They have that freedom. Everybody has the freedom. Well, according to Mozilla, they don't want a free internet anymore. They have put out a statement saying that they want to basically censor the internet in some way. They want the ability to have the internet basically silence certain voices, people that put out unpopular opinions or unpopular political stances. They want to silence those voices and have the internet amplify the voices that they say are factual. And this is extremely scary. You know, this is like George Orwell, 1984 kind of stuff. But even beyond that, it is just completely antithetical to the free software movement and the open source software movement. So a couple of days ago, Mozilla put out this blog over at Mozilla.org and they said, we need more de-platforming. There is no question that social media played a role in the siege and the takeover of the US Capitol on January 6. And again, I don't have a TV. I actually don't know really what happened at the Capitol. I know some people stormed the US Capitol building and there were some deaths involved. But I really don't know much about that situation. So I'm a little bit out of the loop on the news. But apparently Trump tweeted some stuff afterwards and then Twitter banned him, which is, of course, Twitter is right to do that because, you know, it's it's not a public place. Right. When you're on Twitter, you're on Twitter servers. That's basically Twitter's house. You know, when you're in somebody else's house, you got to play by their rules. If you don't, they can kick you out. Even if you do play by their rules, it's still their house. They just decide, hey, I no longer want to deal with you. I mean, there's really nothing unusual about that. Twitter has the right, unfortunately, to ban anybody. That's why I like decentralized, federated social network platforms like Mastodon. And that's because you can't really get banned on Mastodon. You can get banned from the Mastodon server that you happen to be on. But you're still not banned from Mastodon. You can go join another Mastodon server after that, maybe one that has more relaxed rules. Or you can always create your own Mastodon server. With Twitter, you really are just at the mercy of that one site and their one set of rules. But according to Mozilla, Twitter banning people and Facebook banning people isn't enough. They are saying that Twitter and Facebook are actually responsible for these people that I guess Mozilla disagrees with, even having a platform to speak on to begin with. Quote, as reprehensible as the actions of Donald Trump are, the rampant use of the internet to foment violence and hate and reinforce white supremacy is about more than any one personality. Donald Trump is certainly not the first politician to exploit the architecture of the internet in this way. And he won't be the last. We need solutions that don't start after untold damage has been done. So they are basically saying people are using the internet to do things that we don't necessarily agree with. Can we stop it before it even happens rather than, you know, after somebody does something and then we ban them from, you know, the social network? Well, they've already said what they had to say. Can we stop them before they actually get a chance to even say something? That's that's crazy. I got this. This is crazy that this is coming from a company that makes free and open source software, right? This is and they make a web browser. They make an email client like they should be all in on a free internet, a free web. And this is so not that this has nothing to do with internet freedom. This is all about censorship. It is almost communist in its tone. And of course, this was from the Mozilla blog, but there's a ton of news articles about the situation and Mozilla, they keep speaking on this situation. They're really doubling down on this, how they want to change the internet and they want to silence certain voices and quote, amplify certain voices. They want people that are verified as speaking factually to actually have a bigger voice and they're asking for, you know, companies like Facebook and Twitter. We should all get together. We should all make sure that we're sharing our algorithms. And we know what the algorithms are doing to promote certain voices and demoting certain voices or in some cases, silencing certain voices all together, so we should be sharing algorithms to make sure the right people get promoted on social media and Mozilla on their official Twitter account has been making some just crazy outlandish statements like this tweet here. It's quote, this week we saw the culmination of a four year disinformation campaign orchestrated by the president. We have to acknowledge how the internet was misused to get here and we have to change it. So Donald Trump, he's been misusing the internet for the last four years. You're just now saying something about it four years later, but he was misusing the internet. So we have to change the internet. That's ridiculous, right?  And I don't know why Mozilla and all these statements, too, they keep up bringing fomenting violence and hate and reinforcing, quote, white supremacy.    But I'm pretty sure Donald Trump was a Democrat before he decided to run for president. Like he's not really a conservative. Then nobody would call him an ultra conservative kind of person. You know, he certainly changed political views to fit within the Republican Party so he could run for president. There's problems with racism. There's problems with racism all over the world, not just white and black, you know, here in the U.S. But it's a global problem, you know, with a bunch of races and ethnicities and religions, but here in the U.S. Was there a real racism when I was a kid in the 70s and the 80s? Yes. Did it get better? Sure. By the time I was in high school in the 90s, you know, in college in the late 90s, it was kind of normal, you know, so a lot more interracial dating.  You know, it was it's things certainly started to change. And that was 30 years ago, right? Now, we're in 2020.  Not what people are doing on Twitter, right? Yeah.  It's mainly a liberal platform. They censor a lot of conservative voices. I mean, we all know this. This is common knowledge. Why would white supremacists be wanting to do anything on Twitter? Anyway, nothing Mozilla says in these statements make any sense, right? This is all for a posturing for the media, you know, no longer is Mozilla worried about creating free and open source software because I think Mozilla realizes that they're never going to become mainstream as far as Firefox becoming a mainstream browser that can compete with things like Google Chrome or even things like Microsoft Edge, Apple Safari. It's because nobody wants to use Mozilla Firefox. The only people that use Mozilla Firefox, quite frankly, the only reason they have any kind of support is because of those of us that feel strongly about free and open source software. We are the ones keeping Mozilla afloat. All of these Linux distributions, these hundreds of Linux distributions out there that ship Mozilla Firefox out of the box on the distributions out of principle, they're keeping Mozilla Firefox afloat, right? It's us that are keeping you afloat. But now you are deciding, hey, we don't care about the free and open source software movements anymore. We don't care about free speech. We don't care about a free and open Internet. What we care about is posturing to make ourselves look good. We have to appear woke and like we're, you know, hip to the latest political wins. So maybe people will give us a shot because maybe some of these ultra liberal people that support socialism and communism and, you know, are really invested in all this racial fighting that's going on. Maybe they'll give Mozilla Firefox a chance because they posted this really woke blog post. And if that's what Mozilla is trying to do here, I understand. From a business perspective, but from a principled perspective, as far as free and open source software, for somebody that just cares about the software, you know, I can't really support Mozilla on this. Right. At some point, I just have to check out. And Mozilla is a very large corporation, so they don't always do things that I personally would do as far as from a free software perspective. You know, they've had problems with certain decisions they've made in the past, as far as plugins and extensions, certain policies that they enact and things like that. But they've never done anything that just was completely and totally antithetical to the free software movement until today. And the idea of censoring the Internet is completely antithetical to the free software movement. And as of now, I don't think I can use Firefox anymore. And the great thing about this, though, is there are a ton of good free and open source web browsers out there. I already have Brave installed in my system. It's a Chromium based browser, but it's free and open source. And it's a fantastic browser and it works. I'll probably just start using Brave or Kube browser, Kube browser. Also Chromium based. It's keyboard driven. It uses a lot of them navigation style key bindings. And I really like Kube browser. So I don't need Firefox. I've been a Firefox user for 20 years. I've been a Firefox user since day one when they started, you know, from the transition from Netscape navigator over to Mozilla Firefox, because I've always wanted to support this company because I thought they were one of the champions of open source software. But for whatever reason, they're going in a different direction now that I just can't go.


Let us now look at thermodynamic interactions. The scheme of thermodynamics is to consider the behavior of two systems. Let us call it system A. Let us call this system system B. You may call this system A as system, system B as surroundings but let us continue with this nomenclature system A and system B. What these two systems do is they transfer energy. They have a transaction of energy transfer between them and because of this energy transfer, system A undergoes a change of state for a process from an initial state say A1 to a final state say A2 and system B also undergoes a process and its state changes from an initial state B1 to its final state B2. These energy transfers require that two systems are involved and second one, the energy we have not exactly defined what is meant by energy but let us use our feeling from our study of physics is transported from one system to the other across the boundary that separates them. So we can consider a boundary separating the two and what transaction takes place must be across this boundary. These energy transfers are known as interactions and the whole scheme of thermodynamics is to study the interactions, the type of interactions, the quantification of interactions and relate them to change in the state of system A and the change in the state of system B. In this course we are going to look at two types of interactions and thermodynamics allows only two types of thermodynamics interactions. The first one is the work type of interaction and this interaction we will define but this interaction is a primitive in the sense that it is defined in other branches of physics. Right from our high school days we know what is work and how work is defined and how work is major but thermodynamics defines another type of interaction which is the heat type of interaction and this is defined by thermodynamics. Our next major topic would be a study of the work interaction. Thank you.


Over time, these larger objects continue to grow by accumulating matter from the disk. We find that in each region, each orbit, each distance from the Sun, everything coalesces into one massive object. These larger objects sweep out the remaining debris in their orbits. This is a defining characteristic for planets. All the little deviations averaged out as the smaller particles with varying elliptical orbits combined. This explains orbits being nearly circular and all in the same plane. But the actual process is very chaotic. Not as simple and straightforward as this illustration. The process of accumulating matter in the disk into larger objects came to an end when the Sun ignited as a main sequence star and its strong solar winds blew away any remaining loose material. Computer model estimates for how long this process takes range from 100 million to 200 million years. Here's a computer simulation created by Caltech that illustrates this chaos. Planets interact with the rotating disk and lose momentum moving their orbits closer to the Sun, or gain momentum increasing their orbital distance from the Sun. Changing orbits create collisions between planets, moons form and collide with each other and with planets, comets and asteroids form and smash into everything, but out of this chaos we get our current order.