We mentioned the states of well-defined momentum, referred to the playing waves in mynd on the wave number Pe upon momentum upon H bar. We can go back to the problem of too-slit interference and ask ourselves why it is that quantum interference is not observed in Macroscopic objects and assess what experimental set up will be required to see quantum interference with say electrons. I can put some numbers into this experiment. So, as you will recall, we had some gun here that was symmetrically placed with respect to a couple of slits here in an obscuring stream. Screening it fired out particles. Some of the particles got through the holes and they came to here and we said that the amplitude up here would be the sum of the quantum amplitude to arrive at this point would be the sum of the amplitude to go via this root, or to go via this root. These roots, well, the distance from the gun to the two slits by setup is symmetrical, is equal. So any difference in the amplitude that they come there is to do with the change in the amplitude when it goes along this root as against along this root. So let d plus be the distance from the upper slit to this place and d minus the distance to the lower slit, then we can write a formula that d plus by Pythagoras' theorem is going to be l squared plus x minus s squared square root, and correspondingly d minus is going to be the square root of l squared plus x plus s squared, and we make the reasonable conjecture. I mean the right way to think about this is these particles, let's imagine that we've got our gun here has been tuned to emit particles with some well-defined energy. That means that as they go along here, the particles will have some reasonably well-defined momentum because their energy will consist of their kinetic energy. So along here we will have that, that the amplitude is going to be on the order of e to the i p upon h bar times x. So that's a reasonable model for how the amplitude varies with position. That's the new information that we bring to bear on this. So what will be the difference? So the probability to arrive at x, as you recalled, was equal to the amplitude to arrive by the top slot plus the amplitude to arrive at the bottom slot, mod squared, and the argument that this was a plus mod squared plus a minus mod squared plus twice the real part of a plus a minus, and these were about equal, and well, these were the classical probabilities. So these were p plus plus p minus, and then we have this quantum interference term which we want to assess. So what is this quantum interference term? So the interference term is going to be mod a plus mod a minus, neither, not very interesting, and the crucial thing is that this one is going to be e to the i p upon h bar d plus, and the other one's going to be e to the minus i p upon h bar plus, sorry, one of these needs to have a star in it, which means that one of these requires a minus sign. So we're interested, excuse me, in the real part of this, and the real part, so this thing can be written as cos p upon h bar d minus minus d plus, et cetera, et cetera, et cetera. So this is looking like to the probability to go either way through each hole separately times the cosine of p over h bar d plus minus d minus, actually it's easier to do it the other way around, isn't it, minus plus. So what is this difference of distances? This difference of distances from up there, well, let's binomial expand this, and because we can argue that l is going to be big experimentally compared to x, x will be millimeters, l will be a meter or so, so we can binomial expand this and say that this is l, it's about half of l brackets one plus x minus s squared over l squared plus dot dot dot, and the other one is going to be about half l one plus x plus s squared over l squared plus dot dot dot. So when we take the difference of those two, so this is going to be two p plus minus, this probability times the cosine of p upon h bar of the difference is going to be x plus s squared over l minus x minus s squared over two l in fact. So when you take the difference of those two, you will be looking at two x s over l, yes, two x s over l is what that difference will be. So now let's put it, now we need to put in some numbers, right? Suppose we take the energy which is p squared over two m, what do we want to do? We want to, no. So what does this do? This gives us a probability of arrival as a function of x which is going to consist of twice, the sum of these two which are going to be about equal and so about twice this plus twice this thing times this cosine. So it's going to be an oscillating commodity which we'll be doing this and there's some characteristic distance between these minima which, so this we'll call this delta x say, no let's call it big x, I think that's what the, so we call this big x, the distance between the places where it's a minimum because the quantum interference term is cancelling the classical term and this difference is what causes that, the argument of that cosine to become two pi, so we can write that two pi is equal to p over h bar times two s over l times x. In other words, we have a formula for the distance between the minima which is two pi, two pi h bar, but h bar is h Planck's constant over two pi, so that's going to be h over p hl, sorry, yes. And we could also write this, I guess, we could also say that, oh no, let's not bother. So let's put some numbers in, let's say that e, the energy is, so in order to get a big value of x we want to take a big value of l, needless to say we want to take a small value of p and of course a small value of s, I've lost a two, yeah you're quite right, so let's take l to be a metre, let's take s to be, I think, yeah what did I do, I think it's, yeah, a micron, because you want to make it as small as you can, but if you make it much smaller than a micron you'll find it difficult to make the whole, using ordinary materials, and in order to get a small value of p we want to take a small value of the energy, but you can't take too small a value of the energy or your particles will be deflected by stray electromagnetic fields and stuff and it'll be difficult to keep any kind of coherence. So let's take 100 ev say as a sort of convenient low speed, if you plug all this stuff into there, so that gives you, what does it give you, a twentieth of a millimetre or something, I think it's, is it 0.06 millimetres, which is obviously perfectly, perfectly observable, yeah. So this such an interference experiment is possible but hard using electrons. If you do the same thing with bullets, when we're not expecting anything to happen, what could we do? We would, we would take that the velocity for a gun is say 300 metres a second, might be a bit faster these days, I'm not sure, but that's a classical, that's, you know, faster than sound, so that's sort of a reasonable ballpark figure. I suppose we took L to be 1 kilometre, 1000 yards per reasonable shooting distance for a rifle, and if we took out the mass to be 10 grams, put it into the same formula, and we discover that X is some ridiculous figure, 10 to the minus 29 metres. So it's obvious that you cannot observe this interference using anything like a bullet, any kind of macroscopic, any, any kind of macroscopic object, because it's going to be vastly bigger itself than the, than the size of the interference pattern. Obviously an absolutely basic requirement for this experiment to work is that the physical size of your particle has to be smaller than the value of X that you derive out of this, so you haven't a hope of measuring this interference. So that's why classically we don't, we are unaware of this interference term, but I would remind you that in the last lecture we recovered classical results, which would explain why cricket balls move as they do, why satellites and so on move as they do by interpreting, we calculated this, we obtained results which recovered classical physics by decomposing the amplitude to arrive into a sum of contributions from states of different well-defined momentum, and these were all interfering with each other, and the classical physics came back as a result of quantum interference, so this quantum interference on the one hand is something which is very hard to observe with classical objects, on the other hand our entire picture of the classical world, a classical world is only recovered through quantum interference. It's not, it's not some esoteric corner of the subject, but it is hard to, it's hard to have it happen in a controlled way. Okay, so we, yeah, we should just, so we've done, we've done the position representation in just one dimension, everything has been with a one-dimensional motion, motion along X, we obviously need to generalize the position representation to three dimensions because we live in a three-dimensional world for whatever reason, and the generalization is, is nice and trivial, we don't need to worry about it, we have, we now have three position operators, X, Y, Z, also known as XI, all right, and we have, of course, three momentum operators, three more operators, PX, PY, and PZ, also known as PI, and we have that every one of these operators commutes with the other one, so we have that XI, XJ is nothing, and every one of the momentum operators commutes, PI, PJ equals nothing, so it is possible to simultaneously know your X coordinate, your Y coordinate, and your Z coordinate, there's a complete set of eigenstates of well-defined states, states where you know all those strings simultaneously, or you can know all three components of momentum, but you can't know there's not a complete set of states for knowing, and so on, and the only other interesting thing we have to have is XI commuted with PJ is IH bar delta IJ, so it is possible to know the X position and the Y momentum, but it's not possible to know the X position and the X momentum, so most of these operators commute with, well, each operator commutes with five of the, sorry, four of the remaining five operators, but it does not commute with its own momentum, that's what each of these position operators, so that's the generalisation there. What else do we have to say? Well, we used to have a wave function of psi being a function of scalar X, we now, it's trivial, the argument of the wave, we can now label a complete set of states by X, Y and Z, so we can write that there's a, we have states of well-defined position which are labelled by a vector now, vector position X, because there are three, this is an eigenstate of the X operator, it's an eigenstate of the Y operator and an eigenstate of the Z operator, so we need three eigenvalues written inside here to describe what this is, it is, that's at a mathematical level, at a physical level, this is the state of being at the location position vector X, correspondingly our wave functions become functions of X, Y and Z because they become these complex numbers, right, that's still a complex number, this complex number, but it's a function of X and Y and Z, the locations of the particles, similarly we have states of well-defined momentum, UP, we have states, yeah, yeah, UP of X, which is X, P, so now we have, here we have PX, PY and PZ because we have a state of well-defined momentum which is labelled with all three components of momentum, so we have this function of a complex of three, sorry, this complex function of three variables X, Y and Z labelled by the momentum, this is just an identical notation, whereas in single, when we were doing this in one dimension we found that this was E to the I, P over H bar times X not vectors, now that's a vector, that's a vector, and whereas on the bottom we used to have H bar to the one-half, now we have H bar to the three-halves reflecting the fact that there's an X component to this, a Y component to this, and a Z component to this, so this wave function of a state of well-defined momentum has now become a plain wave whose wave surface is a normal to the vector P, and that's what it is, it's easy to check that that stuff works, it's a very straightforward generalisation of what we did before, and I think that's all we have to say, oh you know, not quite, we also want to say what the momentum operator P looks like, so previously we had an X, P, X, P, sorry, it's not what I'm talking about, yeah, X, P, of Psi was minus IH bar was introduced by this formula here, D by the X of X of Psi, all right, that was what we did in one dimension, that generalises in three dimensions very straightforwardly to X, P, Psi, so that's become a vector, that's become a vector because we have to write down what it is for PX, PY and PZ, this is really going to be a shorthand for three formulae and it's going to be minus IH bar gradient operator on the function of three variables, function on space, this one here, all right, the wave function, so this is a vector reflecting the fact that that's a vector, this is just a label which appears on both sides of the equation, that's what this formula generalises to that formula, I don't think we need to be detained about that any longer, before we leave the position representation it's good to do a useful result which falls into our laps now because of what we've already done called the virial theorem, which is a theorem, it's a result in classical physics which you may not have met, I don't know but you in a way should have met, did you cover the virial theorem in classical mechanics anywhere, no anyway, so it's nothing quantum mechanical about the virial theorem, it has a classical counterpart but it's going to fall into our laps because it's got this powerful machinery, so do you recall if we are in a stationary state that is to say a state in which the result of measuring energy is certain then all expectation values for such a state are constants, that's why we call it a stationary state, it's going nowhere, so every expectation value for a stationary state, for a state of well-defined energy is independent of time, so we want to exploit that result, so for a stationary state, this is just recalling what we already had, it was a consequence of Ehrenfest theorem, for a stationary state we have that dbd time of EqE equals 0 for all operators Q, it doesn't matter what observable you stuff in there, as long as the observable is defined in a way that is independent of time, so it's something like position momentum, angular momentum, whatever, it has a vanishing rate of change or with respect to time, it's a constant, so we now apply this result to Q is equal to x dot p, so then we have that 0 is equal to dbdT, sudden moment of doubt, so I want to apply this to x dot p E, and let's divide that by Ehrenfest theorem is x dot p comma h, so Ehrenfest theorem tells us that this rate of change which vanishes is equal to this here, and now let's take, suppose we're dealing with a particle which has kinetic energy and potential energy, so we'll take the Hamiltonian to be of that form, which is a pretty useful form, and stuff it in there, and we're going to have that 0 is equal to E dot p comma p squared over 2m plus v, closed square brackets, so now we need to work out what this commutator is, and this is where a little bit of, so this is where we get a bit of practice in using the three-dimensional generalisation, we obviously have two things to work out, we've got a commutator of x comma p with p squared, so let's work that out, x comma p with p squared, now we write that in components, we write x dot p, sorry x dot p, did I say comma x dot p comma p squared, this x dot can be written as a sum over j, j equals 1 to 3 of x i p i, sorry x j p j, so that's just a way of writing that, and now I have a sum pk, well p squared k, so now I'm summing over k as well, right, p squared is px squared plus py squared plus pz squared, now we can work out this using our rules for a commutator, we had that rule that a comma b, sorry ab comma c was equal to a comma c b plus ab comma c, p, we know that pk commutes with pj that's been written down up there, so that commutator vanishes, that's this one here in some sense, sorry there's this one here in some sense, and so what we're left with is, so we have this double sum, we're going to have xj pk pj, sorry let's squared, squared comma, no no, no comma, so that's what we get, so this has to be commuted with that, that's what I've written down I hope, and then there should in principle be another term this commuting with this but that vanishes because pj commutes with pk for all for all j and k, so we have to work out what this one is now, and we can use the same rule if we're being pedantic, we would say this is x on pk pk, so we would say that this is xj pk pk plus pk the commutator of x and pk, I'm using the same rule and that all has to be multiplied by pj, the same because I'm now writing p squared as pk pk, but this is ih bar, this is ih bar, so these two terms actually contribute the same thing, this becomes two ih bar times delta jk times pk times pj, and I'm sorry I've lost track of the sum sign, here we have a sum sign, we're summing over j and we're summing over k, sum over j and you get nothing because of this delta jk unless j is equal to k, so this becomes pk pk summed over k, but pk pk summed over k is the same thing as p squared, so this is two ih bar p squared, that's what the commutator is of x dot p with p squared, now let's write, now let's do the x dot p commutator with v which is itself a function of x of course, these things all ought to have hats really but one gets difficult to write down enough things, well what we want to do is write this thoroughly in the position representation, in the position representation x dot p is minus ih bar x dot gradient, right that's what this becomes in the position representation on v which becomes a function of x, just so this is in the position representation, so what does that mean, that means ih bar minus ih bar brackets x dot gradient working on v minus v x dot gradient, and this is an operator statement so it's waiting for you to put in the function of your choice of psi on the right, right, there's a virtual function there for it to work on that's what that's the meaning of this v x dot gradient, and this x dot gradient v doesn't mean extra gradient only v, it means of everything that is to the right of it including your psi, so when you use this x dot v on v times alone you'll get a term and then you will still have to use the x dot v on the psi, but the result of using the x dot v on the psi will be killed by here an x dot v on psi, so what this is equal to is minus ih bar x dot gradient of v, that's all that survives this is the action of nabla the gradient operator on the potential itself, the operation of the action of the gradient operator on the wave function that's virtually sitting here is cancelled by this contribution here, so we now have we can put these results back into what we had up there so what we had was naught is equal to yeah is equal to the sum of these of these commutators is equal to e x dot p comma p squared um e plus e x dot p v, that's just summarizing where we stand, this we've discovered to be this is 2 ih bar, this commutator turned out to be p squared so it becomes the expectation oops there should have been over 2m on this shouldn't there because it was the Hamiltonian p squared over 2m uh yeah this p came from the Hamiltonian where it was p squared over 2m this v came from the Hamiltonian where it was just v so we have over 2m uh no let's leave that alone of p squared over 2m e plus um we figured out that this one was uh minus ih bar um so we want to cancel uh what we this is the expectation value of the kinetic energy clearly right p squared over 2m is the kinetic energy this is the expectation value of it um so canceling the ih bar we can say that 2 times the expectation value of the kinetic energy is equal to this stuff that's as far as we can go in general um but consider now very important cases have that v of x is proportional to mod x to the alpha so for example for a simple harmonic oscillator we're about to discuss the potential energy goes like x squared alpha is 2 if we were doing with the coo dealing with the Coulomb interaction the potential energy potential energy goes like 1 over radius so it would be v of r is proportional to 1 over r um well this mod x is r so alpha would be minus 1 so we can say that alpha equals 2 is simple harmonic motion alpha equals minus 1 is Coulomb there are you know you can think of other power laws uh which are relevant in this case so then if we ask yourself what is x dot gradient of um of v well that's going to be so so we'll say that this is equal to some constant a times x to the alpha what is this going to be it's going to be um alpha mod x to the alpha minus 1 times x dot the gradient of mod x and the gradient of mod x is uh uh the gradient of mod x is x the unit vector x so it's at the vector x over mod x so this is equal to sorry this is a we have a x to the alpha minus 1 times x dot x over mod x so here this mod x is going to make this an alpha to the minus 2 but from this x dot x we're going to get a mod x squared so this is going to be and i've lost sorry this was an alpha there was also an a unfortunately yes sorry we need an a and an alpha um this is going to be alpha times a x to the alpha which is alpha times v so if v has a power law dependence on distance from the origin then x dot grad v is simply alpha times v so when we put this result back into that formula that back into this statement here we have that twice the ke expectation value is equal to alpha times the expectation value of the potential energy so that's our kepler formula in the case of simple harmonic motion alpha is 2 and kinetic energy is equal to potential energy in the case of Coulomb interaction where alpha is minus 1 you have that the potential energy is minus twice the kinetic energy which is to say that the particle has lost two units of energy uh radiate in falling in from infinity into a bound orbit it's lost two units of energy uh one unit's been sent off to infinity in radiation or something and one unit is used as kinetic energy uh of its orbit so that's the this is a virial theorem so now we opened a new chapter as it were by talking about harmonic motion the harmonic oscillator is the single most important dynamical system in physics most of field theory most of condensed of quantum field theory most of condensed matter physics is fiddling with uh more or less with harmonic oscillators which uh which are which are decorated in some way so the basic physics is that of the harmonic oscillator and it's it's worth just taking a moment to understand why harmonic oscillators are all over the place the universe physicists a fundamental position of physicists well physicists like to represent the universe is a collection of harmonic oscillators and this is partly because physicists uh may be brighter than some other people but they're still pretty stupid we have a quite a small bag of tricks and a harmonic oscillator is is is a trick that we we have and it's an incredibly useful trick for this reason that if you plot force in some direction versus displacement from a point of equilibrium you will get a curve which does something like this um the force vanishes uh at an equilibrium at a point of equilibrium of a system the force on it obviously vanishes so if you do a plot of force versus distance you'll get a curve something like this passing through zero at the point of equilibrium which i happen to put at the origin of x but you know that's by construction clearly and the general idea is that most of the time you can rep sorry that's meant to go through the through the origin most of the time you can represent this uh to some to a good approximation you can say that f of x uh is about equal to kx plus order of x squared or whatever so to lowest order approximation because f has to vanish at a point of equilibrium in the neighbourhood of the point of equilibrium f is going to be proportional to x and if f is and if we neglect this if this is small then we have harmonic motion for displacements these small displacements around here so this is why harmonic oscillators are ubiquitous a very incredibly an incredibly valuable model we can apply we can use to understand many many systems because many systems for small displacements uh almost all systems for small displacements look like a harmonic oscillator okay so let's agree what the Hamiltonian of this thing should be the Hamiltonian of our harmonic oscillator should be p squared over 2m plus a half kx squared right because if the force is going like that you integrate it up this becomes the potential energy and this is of course the kinetic energy we're familiar with that already it's better though to write this in in a different way to anticipate results that are to come and to write this as p squared plus m omega x squared over 2m so uh and of course omega squared is k over m so it's easy defining omega squared to be k over m it's easy to write this formula like that and that's how I want to write it if you want to reproduce this formula just think about dimensional analysis we want to have p squared over 2m because it's the kinetic energy we're always saying that and here I want something that's proportional to x squared and has the dimensions of momentum and obviously omega x has dimensions of speed so m omega x has dimensions of momentum so that's why you know that enables you to recover that quickly from this and that's the way to go for practical purposes so there's our Hamiltonian and we're trying of course to solve the stationary states are the key to understanding dynamics because they have this trivial time evolution and by by decomposing any initial condition into a sum of of stationary states into a linear superposition of stationary states then evolving the stationary states we find out how any arbitrary initial condition evolves in time so that's why we want these stationary states I've said that before and I'll say that again so we want to find states of well-defined energy this is the problem we want to solve and this is a completely generic situation in physics first of all you think about your physical system on the grounds of physics you write down the Hamiltonian then the next thing you do is you find the damn stationary states because that once you've got those you can do anything you want pretty much so that's what we're trying to solve the the the way to do this is the the proper way to to find these states uh so we need to find the energies that that uh are possible and we need to find the corresponding states and the way to do this is to introduce some new operators let's introduce a which is m omega x plus ip over the square root of two m omega or h bar omega why do I write that down well basically because I know where I'm going but just to give you some sense of direction the general idea here is that we want to factorize that that's the general idea we want to factorize the Hamiltonian into you know it's a quadratic expression it seems kind of reasonable to factorize it uh if these were if these weren't operators because these are operators sorry and in future I'm not going to even attempt to put hats on operators right these are operators despite the absence of hats it's just too difficult to remember to put the hats on and takes too much time and cronups never do but these are operators now if if they but if they weren't operators this and its complex conjugate would factorize that so that's the drift okay let's write down its complex this of course is an operator and it's not an observable it's not a Hermitian operator it's what is its dagger a dagger is Hermitian adjoint is this thing dagger which is itself because x is a Hermitian operator on its own dagger so it's m omega x plus this thing dagger p is its own dagger but i has the the dagger of i the Hermitian adjoint of i is minus i so this is minus ip over of course this on the bottom is a real number so it's its own it's its own complex conjugate so here we have two operators and the general idea is they're going to factorize h or they almost do whatever that's the plan um and this is called annihilation operator and this is a creation operator and we'll we'll the reason they have these names will emerge but it is that if you use this on a state this operator increases the excitation of our harmonic oscillator and this oscillator reduces the excitation of our harmonic oscillator and since in quantum field theory particles are excitations of the vacuum this thing creates a particle because it creates an excitation which is a particle and this thing destroys a particle because it destroys an excitation so what we next do is work out what a a dagger a is because the idea was it that this product would be more or less the Hamiltonian so what exactly is it let's get this right m omega x minus ip m omega x plus ip over two m h bar omega now when we write this out we have the obvious terms we have p squared and we have m omega x squared so let's write those down that's p squared plus m omega x squared all over two m h bar h bar omega two m omega h bar whatever um and um and then we have some additional terms which would class it would cancel in classical algebra but don't now because we have an x we have an m omega x times ip and here we have an m omega x on the sorry have an i minus ip times an m omega x so the additional term is an m omega x m omega i x comma p and it's again over two m h bar omega right so this is the Hamiltonian over h bar omega and this is an ih bar which and the i's make a minus one with this so this is going to be minus a half and everything else will cancel right because we'll we've got an m omega here and we're going to get an h bar from there so the rest cancels so I should have I should have explained sorry I wanted to factorize this and this on the bottom this normalizing factor on the bottom is put in it's not really essential but it's very convenient and it's put in in order to make this dimensionless so just to check that that's true um h bar has dimensions of position times momentum right so it has the dimensions of position times momentum so what we have here is m x sorry m omega x which we've agreed has dimensions of momentum times p which has dimensions of momentum then we take the square root so this on the bottom has dimensions of the whole square root of momentum and therefore cancel the dimensions of what's on the top so it's dimensionless that's the purpose of the that's the purpose of the horrible square root so we find that this product which is dimensionless is equal to the Hamiltonian divided by h bar omega which has the dimensions of energy because h bar also has the dimensions of energy times time omega of course has dimensions of one over time it's a free is the frequency of the oscillator so this is has dimensions of energy minus a half which is obviously dimensionless so we have indeed almost factorized we have the statement now that h can be written as h bar omega which carries the dimensions times a dagger a plus a half we've almost factorized h just there's that there the next thing we want to do is calculate the commutator a dagger comma a yes we've just got time to do this a dagger a of these two operators so of course we will have a one over two m h bar omega as a factor on the bottom because each of these a's brings in its own square root and then we will have the commutator of m omega x minus ip on m omega x plus ip now we we have this breaks down into four commutators in principle there's the commutator of this with this and the commutator of this with this the commutator of this with this obviously vanishes because x commutes with itself and the commutator of this with this is so we're going to have an m omega i times x comma p that's the commutator of this with this and now we have to deal with these with these terms this produces a non-negotiable commutator with that we're going to have minus m omega i times p comma x and then we'll have the commutator of p with self which will vanish if i swap those two over then clearly i changed the sign in front and then this becomes a plus x comma p it becomes this thing all over so that cancels this and this whole caboodle is going to equal i x comma p over h bar because we're going to get a two these two terms are going to add together to make us a two which cancels with this and the m omega is clearly go x comma p is itself equal to i h bar so the i's make a minus one the h bars cancel and this is equal to minus one so these two operators have non vanishing commutator actually equal to minus one yeah well we seem to still have time to to nail this this problem i think so let us suppose we have got a state of a station restate let us now apply the operator a dagger to both sides of this equation right then this is just that i can value it's only a number so i can then write e a dagger e is equal to a dagger h e that's obvious i would like to swap these over so i jolly will do i say this is equal to h a dagger plus a dagger commutator h so this commutator puts in what i'm supposed to have and takes away what i'm not supposed to have but have previously written down but we know what h is in terms that we have that h is equal to there it is h bar omega a dagger a so let's use that so this is h a dagger plus commutator of a dagger and h turns out to be a dagger a plus a half closed brackets h bar omega to carry the dimensions close that close that and stick in our e that we first thought of so all i have done is replace h by an expression we already derived um yep now i have to take the commutator of a dagger with this and with this the commutator of a dagger with the half clearly vanishes because a half is just a number not an operator um the commutator of a dagger with itself van vanishes so so when we do the commutator with this product there should in principle be two terms but only one of them survives and that term is that term is uh this sticks uh this stands idly by while the a dagger works on that and then i have an h bar omega h bar omega close brackets close brackets e but we just worked this thing out and found that it's minus one right a dagger a turned out to be minus one so this is equal to um h a dagger e minus h bar omega a dagger e just to remind you of it mind us what we had on the left what we had on the left was e a dagger e this is just a restatement of what's been at the top so we take this a dagger e and we obviously join it onto that a dagger e and we discover that h on a dagger e is equal to h working on this the cap that you get by using a dagger on e is equal to e plus h bar omega of a dagger working on e what does this tell us it tells us that we have out of a state which had energy e we have constructed a state by multiplying by a dagger which has energy e plus h bar omega so this means that a dagger e is equal to a constant normalising constant not discussed times e plus h bar omega a new stationary state this is an incredibly powerful result because it immediately follows that we have states if we can find a state e we can immediately generate e plus h bar omega by using this a dagger beast and also if we use a dagger on this it follows we're going to get e plus two h bar omega and we're going to get another if we use a dagger on this we're going to get e plus three h bar omega so we're going to get a whole infinite series of states of ever-increasing energy simply by applying a dagger again and again and again so what remains is to find what e what number e is and that we will do first thing tomorrow


Over the last year or so, as I've done more distro reviews, I've started to get more picky when it comes to arch-based distros. Mainly because I feel that in order to be anything good you have to be a little bit special. You can't just be an easy to install arch-based distro because there's 12,000 of those things. You have to do something that is at least a little bit unique. And there really aren't that many out there that have strived to do this. Very few of them do anything beyond just creating a really pretty rice and installing the Calamari's installer. Today I'm going to be taking a look at a distribution called ArchCraft. Now I've been asked to take a look at this several times over the last year or so and I'm finally getting around to it. Now the first thing you should know is that as I said in the last distro review that I did, I'm going to be doing this on hardware. So this is going to be a little bit different. This is my first attempt at a refabricated distro review and it's going to be a little bit rough. So just keep that in mind as we go along that the layout of this review might be different from the ones that come after simply because I'll refine it and get better as I go along. So the first thing we're going to talk about as we usually do in these things is installation. Now the installation you're going to be seeing in B-roll was done in a virtual machine but I do have this on hardware. I'll show you that here in a minute. The installation for ArchCraft is very simple if you've chosen the Calamari's installer. So you have two options when you first go into the ISO. You get an option for the Calamari's installer and you get an option for a terminal base installer. And the biggest issue I have here is that they don't really tell you what the benefits are of one or the other. However, if you were to choose them you would soon figure out exactly what the benefits of one are over the other. So the Calamari's installer, which is what you'll see in the B-roll, is a very simple installer. It will take you through the regular Linux setup tips. It allows you to choose between EXT-4, B-butterafast, and things like that, whether or not you have swap. There's no special pages there to choose what applications are installed. Nothing like that, like you would see with ARCO or GRUIT or something like that. It's just a very straightforward step-to-step Calamari's installer. And it was very fast. Now on hardware I also used the Calamari's installer one time and I will say that this is the first time Calamari's hasn't got hung up scanning all of my hard drives. This was a very fast on hardware installation. Now the other installer, which I didn't try on camera, but I tried on hardware, is a terminal-based N-curses installer. And this is very much an expert installer. It's something that you will only want to use if you know what you're doing. It allows you to tweak the installation of ArchCraft in a way that is very similar to what you'd get if you installed Vanilla Arch. And it gives you options for like build parameters, kernels, things like that. It's very much in-depth and much more complicated than the Calamari's installer. So I highly recommend only using that if you know what you're doing. Now like I said, the biggest issue I have here is that on that screen that pops up at the beginning that gives you an option between the two, it doesn't really tell you which one you should use or what the difference is. So if you are a fairly new Linux user, you may end up choosing wrong. But the good news is it's very easy to stop. If you get confused going through the N-curses installer, just close out of that, open up the Calamari's installer, it's fairly easy. So that's installation. And like I said, on hardware, it was a very quick installation. It was maybe four or five minutes. And usually Calamari is very slow for me, at least at the beginning, because it always scans all my hard drives. It did not do that this time, which I was very impressed with. Now let's go ahead and show you what this looks like installed on hardware. This is what it looks like. And this right here is the open box window manager. So the first thing you should know is that it comes with open box and BSPWM. I will show you the BSPWM version here in just a moment. But this is the open box version. And it is, as you can see, very pretty. This is like a very Nord-like theme. They've done a very good bar up there at the top, which I find very nice. They've done a wonderful job of customizing and adding modules and making it look just spectacular. It has a button up here for the menu, which brings up Rofi. It has a power menu. It does have some extra empty spots here, which I'm assuming probably for battery or Wi-Fi or something like that. I'm actually not sure what those would be. But it does look very, very nice. Now, as I said, this comes with BSPWM and open box. Now, I'm not as familiar with open box as I am with other window managers. I've never been a floating window manager guy. If I'm going to use something floating, I'll just use a desktop environment. So I haven't spent very much time at all in open box, but I will have to say the work that they have done to make open box very, very usable is impressive. So the first thing you will probably understand about open box is that most of the moving around of the operating system is going to be done through either key bindings like a traditional window manager or through this menu that pops up when you right click the desktop. And this is where the version of ArchCraft really shines. They have done a tremendous amount of work to make this menu really nice. Now, this is a custom menu, and I know that because that's the way open box works. When you customize open box, you create this menu on your own. It exists, but you have to do all the entries and stuff on your own.  And that includes not only being able to launch applications through categories, which we'll go through here in a minute, but also being able to take a screenshot, being able to record the desktop common places in your files, recent files, preferences, which is a place where they've done a lot of work to allow you to customize your setup right here from the menu. So you can get to open box settings. You can get to the compositor settings. You can change settings for the monitor right here in the menu. So if you needed to change the position of a monitor, if you needed to change the resolution, you can do it all from the menu right here. This is a very in-depth menu system. It doesn't require you to open up an app or anything to make any of these changes. You can also change the font from here. So if you do this, you can change the font. It brings up a font picker. You can then change to like just say you want to do this hit select. And it would actually change that font right from there. That is spectacular because what that means and that is Polybar up there. What that means is they've created an integration between that font picker and the Polybar configuration file. That is a lot of work. From someone who spent a lot of time customizing Polybar, let me tell you that they've spent a lot of effort to have all those integrations to change fonts, to change like the size of the font. It's really quite awesome. You can also change the terminal color scheme from here. Change the wallpaper here. This brings up nitrogen. So if you wanted to change to like this wallpaper, you just hit apply and it would change the wallpaper. The fact that it uses nitrogen is very nice. And again, that it's integrated. So you select the options from the menu and this pops up and you can change the wallpaper. Just like you would in a desktop environment. If you use like KDE or whatever, you can right click on the desktop and click an option in the menu and then change your wallpaper. And you can do that similarly just like this. You can also change the appearance settings, QT5 settings and Covantum from here. And it gives you links to the power settings, audio settings and the settings manager. Now, if this settings manager looks familiar to you, very good job because this is the XFCE settings manager. And you'll find that a common thread throughout most of ArchCraft is that they've used a lot of XFCE stuff here in order to have integrations with it.  All that stuff is done through XFCE tools or XFCE like tools, which is very nice because it gives you the power of a desktop environment without actually using a desktop environment. You don't have any of the cruft the XFCE has, really. If you can say XFCE has cruft, it really doesn't. But you get all the features of a settings panel and stuff like that, which a window manager wouldn't normally have without any of the other stuff, right? Now, when you first boot in, you get this welcome screen. I've kept this up for a reason because unlike other welcome screens where they're mainly useless, the best thing about this one here is that it gives you the key bindings. So if you scroll down here, you hit this button here, it'll actually scroll all the way down and you can see all of the key bindings. And this is important because when someone else creates a window manager for you, you're going to be dealing with all of their key bindings. And that means they're going to be probably different than the ones that you probably would have chosen yourself. So you'll need to go here in order to see all of the key bindings. Now, there is a way to find them elsewhere. If you right-click in a menu and go to key bindings and then show key bindings in window, you get this window here. So if you don't have the welcome screen up and running right away, you can find them later on. You can also go to the rc.xml file and find them in there if you want to go spelunking into the configuration file itself. In addition, the welcome screen also gives key bindings for the BSPWM version as well. So if you need to get to the key bindings for BSPWM, those are all right here. Now, they are very similar to the open box key bindings, but obviously tailored for BSPWM. That is pretty much it for the welcome screen. Everything else here are tips and then some information about the distro itself. Overall, a very good welcome screen. It doesn't really have links to any applications or anything like that. So that's not really the focus of this distro. But if you are looking for information on how to use the thing that you installed, it has some good information there. Now, let's talk a little bit about installed applications. So the first thing that I want to say out loud is that by default, VIM is installed. So this is an amazing distribution. I mean, we could stop right here and I'd give it all the kudos in the world because no distro or very few distros come with VIM installed by default. It's just kind of a sad fact, unfortunately. I think it should become my default. But if you install Ubuntu, it's not there. If you install Mint, not there. You install vanilla arch, obviously, it's not going to be there. And that's, I mean, that's fine. It's not if it's hard to install. It's just this makes me happy. So VIM is installed by default. That's really all you need to know. That's the only application you need, right? Well, we'll talk about the rest of them, but I just wanted to point that out. Also, if you noticed, they've done some customization for VIM itself. So they have a custom VIM RC. I'm not sure if they have plugins or not. But they do have the status line at the bottom and they use tabs appear at the top. So again, you can see that they've done some work in customizing pretty much every corner of this distro, which is very nice. Now, in terms of installed applications, there's not very many. I was actually assuming because this is a two gigabyte ISO, I was assuming that there's going to be quite a few applications installed for you. But there's actually not. So you get settings, things like the file manager, which I believe is TUNAR, a few other settings, applications or accessories, Kvantum, Nitrogen Plank is the bar at the bottom. It actually comes with two terminal emulators. It comes with a lacquerty and it comes with XFC for terminal comes with X archiver for development. It has genie and meld graphics has GPIC and view noir for multimedia. Two of these things here. I installed. So I'm using OBS and audacity right now. I installed those. So those don't come in preinstalled for network as Firefox is a default web browser, and I'm not sure what it's going to use for male reader. It actually doesn't come with a male reader installed. That's just an XFC e quirk where it has the preferred applications as like web browser and file manager and text editor, right? You can click on those things and it'll bring up whatever you've said is preferred for office. It doesn't. It just has a document viewer. It doesn't come with LibreOffice or anything like that. You'd have to install that. And then for settings, most of the stuff is the XFC stuff that I talked about settings for mouse and keyboard notifications and stuff like that. All that stuff is going to be XFC for related. It's a XFC for back end, if you will. It also comes with G parted H top Ranger time shift, all that stuff installed. So you can use time shift for the butterf s, which if you chose butterf s in the installation, which I did, you can use time shift for the snapshots. So as you can see, it's very light on the installed applications. There's not a ton of stuff here. And I think that that's good.  Now, this is the open box version of Archcraft. And it is, like I said, very pretty. And all the key bindings are fairly intuitive. You change two different workspaces using super in the numbers. You can shoot move between windows using the Vem keys. And super, you can launch several different applications with key bindings as well. All those things that are in the place where I showed you. And then all of the configuration files are here in the open box folder within .config. So if you know how to configure open box, this is where all of your configuration stuff would be, including changing the menu, including changing Polybar and Rofi. All that stuff is right here. The cool thing here is, though, that there are like 20 different themes here for you to choose from. And the ones that I've tried are really nice, just like the default one is very, very pretty. All of these are very well put together. Now, one thing that I didn't notice when I was going through the menu is that you can actually change the themes from the menu, which is not surprising given how thorough this menu system is. But I totally missed this part right here. I skipped right over it. As awesome as I think changing themes is, you'd think I would have noticed this, but I didn't. It's here and you can change them.  And that, again, is another place where it kind of shows you that they put a lot of effort into making this really cool.  Okay, so this is the BS PWM version of Archcraft. And as you can see out of the box, it looks very similar to what the open box version looks like. They've done a good job of keeping cohesion here. Now, the differences between the two window managers are going to be quite obvious if you've ever used them. If you haven't, you'll notice the differences quite quickly because they are quite different. Open box is a floating window manager, whereas BS PWM is by default a tiling window manager. And they just kind of behave differently, but the biggest difference between the two in terms of Archcraft is in the open box, you get a menu system that you get when you right click on the desktop like I showed you before. But when you are in BS PWM, you don't get that. It doesn't have a built in menu system. They could add it if they wanted to. They obviously haven't, but that is going to be the biggest difference. Now, that means you're not going to have the easiest access to all of the stuff that Archcraft comes with, like changing themes. It doesn't mean any of that stuff's not here because you can still get to all the settings and stuff like that just by using Rofi by hitting the super key, but it doesn't have easy access in the menu. Now, the thing that confused me the most is like, how do you change the themes? Because unlike with the open box, there's no scripts here. Well, there are they're not really scripts. They're more like settings for specific things like they have a list of the colors and stuff like that in bash files, but they're not scripts that change the theme like you had an open box. Now, I'm assuming that those scripts do exist somewhere and I just didn't see them because there is a way to change the theme. So if we go back to the welcome screen or the tips and tricks screen, you can see by hitting control Alt and T, we're going to get an option for the themes. So let's just say we want to drag it here. Now, I'm assuming that OBS is going to crash here again. So if that does happen, we'll see a jump cut. Oh, no, we didn't get it. It didn't crash OBS, which is nice. I'm happy about that. So there's the Dracula version. And I have to say I'm not as impressed with that theme as I would normally be because the themes that I looked at in the open box were very well put together and different from each other. Now there were some, obviously they were similar, but this one here is exactly the same as the thing we just had just with Dracula colors. I'm going to see if we're going to change to another one. We'll choose this one. Yeah, I mean, that's a nice theme. I'm not, I'm just not as impressed with them because they don't look as different, I guess. Here's, here's a forest theme that's ever forest. A very nice color scheme. Look at a couple more. Let's see here. Let's look at the, actually go back to the cyberpunk one. That's cool theme and look for another one. Let's do this one. So they have some really cool themes. They're very cool color schemes. I'm very, I'm very interested in scripts that change themes because I'm writing one of my own for I3 and it has been a chore for sure because there are so many different components of the window manager that you need to change. You need to change the GTK theme. You need to change the terminal theme. You need to change the dunce theme. You need to change the poly bar theme. You need to change the window manager theme. All these things, they all go into that script and it is kind of a pain. But so I'm impressed that they've done this and that means that they've gone through and done one rice, one right for another and that takes time and effort. So that's definitely something that has impressed me. I wasn't as impressed with the themes here in BSPWM as I was with OpenBox but maybe that's just because OpenBox is new to me so it just seemed newer to me. Now the last thing that I wanna talk about is issues that I've had in the few hours that I've been using this and really there's only been one and that is that the monitors don't go to sleep. At least in the OpenBox version that I've been using for the most part. I spent most of my time with OpenBox. The screens wouldn't go to sleep. They went to sleep for a moment, came back and they completely forgot the monitor layout. So both monitors stayed on but this monitor wasn't even active. Like you couldn't move windows or anything. It was just like a mirror of this one. I'm not sure if that's an OpenBox problem or an ArchCraft problem. I'm not sure. I didn't do any troubleshooting on it whatsoever but that's just something that I did notice that happened and if that happened all the time it would definitely be a deal breaker for me. I haven't seen it happen again but then my computer hasn't gone to sleep since that happened. So that's definitely something that you might keep an eye out for. In terms of performance, it's been rock solid. It's been very speedy, nothing slowing down or whatever the installation was fast. It used about the same amount of memory as you would expect out of the box as any window manager very little in terms of memory resources. If you're curious about the versions for this it comes with ZSH as the default shell. It comes with kernel 5.18. I can remember this is based on Arch which I probably should have said before but I assumed it was probably, I mean it's called ArchCraft so I'm assuming it was obvious. But so you're gonna have most of the latest and greatest stuff because this is based on Arch. And obviously this is my proof right here that I'm running on actual hardware not on a VM. I'm assuming that I needed to actually point that out. Out of the box it comes with about 968 packages which is fairly low for distribution which is again very nice. And if we look at free-m here which is not going to be very representative to anything because I'm using OBS so it's using like two gigabytes right here. I have OBS and Audacity running. I have a couple terminals on so probably not the most useful piece of information but it's there if you want it. So bottom line, ArchCraft is a very, very nice ArchBase sister. So I wanna go back to what I said at the beginning where I said I'm very picky when it comes to ArchBase sisters nowadays because I want it to feel like they've put effort into making it something other than just Arch. You know what I mean? I want them to have spent time to make it their distro instead of just a Arch distro with a Calamari's installer. And unfortunately the vast majority of ArchBase distros are just that. They are a Calamari's installer with very little effort. Some maybe a theme or something like that maybe a button of applications pre-installed but for the most part very little, very low effort. And I mean, if you're just creating a distribution for yourself, that's fine because maybe that's what you want. But if you're trying to market to other people I want you to have put some effort into it. ArchCraft you can tell is a project that they've put a lot of effort into.  They've created that menu system in OBS which is just fantastic. They've created a whole bunch of polybars. They've done all of that while keeping a very minimal distribution where you can build up your application stack on your own. You can have a ton of stuff there that you don't need and basically they're allowing you to choose what applications you want. I would like to see some gooey way of choosing applications to install along their installers. So something maybe similar to what Arco does but maybe less confusing where they give you an option for some applications to install as the installer goes through. That way when you boot into your system you have all the stuff that you need. You don't have to spend time afterward installing software but that's a very minor nitpick. Isn't really necessary so it's just something that might be nice for some people. Other than that I'm actually very happy with this distribution. Would I run it full time? Yes, if I was still an arch user this would be a very good distribution for me and I would be perfectly happy here. Now the question would become then would this cause me to switch from Fedora to this and the answer to that is no I'm really liking Fedora right now so I don't see myself switching to it full time but I will say this I'm going to steal some of the things so at least I got something out of it I guess. So that is it for this video if you have thoughts on ArchCraft I'm sure there's stuff here that I missed, right? This is just a first look I only spent about two hours with this distribution and there are many different areas of a distribution that it's possible for me to just have missed things so if you have thoughts on this you can leave those in the comment section below if you have ideas on how I can make the first look videos like this a little bit better now that I'm doing them on hardware you can leave those in the comment section below as well if you can follow me on Twitter at the Linuxcast you can follow me on Macedon or Odyssey those links will be in the video description just below the like button if you could hit the like button I'd really appreciate that because it does help the channel so very, very much you can support me on Patreon at patreon.com slash Linuxcast just like all these fine people thanks to everybody who supports me on Patreon and YouTube I truly do appreciate it without you the channel probably wouldn't be where it is today so I truly, truly do appreciate it thanks everybody for your support thanks everybody for watching I'll see you next time


In this video, I'm gonna talk about 13 AI side hustles that anyone can start, and one of them you can make over $13,500 per month with. Bing bot! And in this video, I'm gonna be talking about the AI side hustles that, one, you can actually make money with. Like, it's not impossible to be able to make money with them. And two, ones that you've probably never heard of before. So if you appreciate me doing this type of video, go ahead and let me know by gently tapping that like button. And let's jump into it with number one, which is going to be AI editing for YouTube and TikTok. So two examples of faceless channels that are gaining a ton of views on YouTube and also making a ton of money are going to be the Swedish investor. And they're in the investing niche, which typically makes between $11 and $50 per 1000 views and top cars TV. And they're in the auto niche, which typically makes $10 to $12 per 1000 views. So both of these channels likely make over $10,000 a month. I wouldn't be surprised if the Swedish investor is making more like $50,000 plus dollars a month. Now, some of the skills you're gonna need here are going to be strategic thinking. This is basically things like figuring out your niche, thinking about different topic ideas that you're gonna make videos on and figuring out how you can give value to your audience. And then the second skill you're gonna have to have is prompting skills because you are going to be heavily utilizing AI in just about every aspect of the video creation process. And to succeed in this side hustle, you're going to need this one AI tool and that is called NVIDEO. So NVIDEO is an AI editing tool that automates pretty much everything from producing a video to scripts, to getting stock photos, the voiceover, and heck, even the subtitles. So just a quick run through of the features in NVIDEO, you should sign up first. Let's open your browser, then open NVIDEO.io. The best thing about this is that it's absolutely free, but it's best to use the paid version of it to remove watermarks and have access to millions of royalty free stock footage clips and a human sounding voiceover. If you use a free account, the final video you produce has watermarks as opposed to a clean export you can have with a paid one. These are the features of NVIDEO. There's a workflow slash platform selector. You can choose what types of video you'll be producing. Is it for shorts or a long form video? You have these choices. That's YouTube shorts, YouTube explainer, recent events video and script to video. For script writing, if you want NVIDEO has the script to video feature, which you can utilize if you have a written script. We will only use YouTube shorts for now because we're going to upload this on TikTok. Click it, then you'll see a new panel. It asks you what your video is about. So for example, we're going to make videos about the benefits of a booty band. Let's prompt with this. So five benefits of a booty band. Then for settings, you can select the narrator's voice either male or female. You can add subtitles for this. I'm going to choose the one that highlights every spoken word with yellow. Then you can select what type of stock photos or videos it will use. So we're going to select a female voice with a sexy Southern accent just because our target buyers for our yellow basket product will mostly be women. After you click the generate button, you'll wait a few minutes and you'll have these options. One audience to look and feel three platform. So for audience, we're going to choose fitness enthusiasts for look and feel. I kind of like the minimalist modern and for platform again, for TikTok. So it should be YouTube shorts. At this point, you can still edit the prompt, but for this, let's go and generate the video, wait for a few minutes and watch the magic happen. All right, so for editing on a video, let's watch the generated video list. Looks good considering it only took us a few minutes to do it, although if you don't like certain parts of it, like the background music for example. Two, it's versatile. From squats to lunges, there's no limit to the exercises you can enhance with a booty band. You can use in videos AI to make the changes for you. Simply type out the prompt in the command box. Once you're done, click generate and a new video will be created with the changes you requested like so. Now let's see how it looks. Two, it's versatile. From squats to lunges, there's no limit to the exercises you can enhance with a booty band. Alternatively, you can edit the video yourself. You can click the edit button and then you can replace clips that you don't want using in videos built in stock media library, which contains more than eight million royalty free clips from sources like shutter stock and eye stock. You can also edit the script too. You can change overlays or titles as well. But for now, we're gonna stick with this one. Then step six, exporting. After we settled on something that looks so good and feels so right, now we must export the video. So to do this, you have to click the export video button and then here you'll be presented with a choice. If you have the paid version, you can go for no watermarks and no branding. Otherwise, we're gonna stick with the free version with watermarks and branding. Then choose the resolution of your video. I recommend using 1080p. People are not using Nokia 3600 today, right? Then press continue and voila. We're good to go after it renders and you download it. Now you can literally do the same exact process for YouTube shorts as well. You can also integrate it with other types of AI tools, such as using mid-journey to generate photos. And yeah, it's just a really cool tool. It's made making shorts extremely easy. Now I really do like NVIDIA, but this is a sponsored video and you can get started with NVIDIA AI for free. But if you're serious about video creation and wanna publish videos with a watermark, I highly recommend you should upgrade to a paid plan, which starts as low as $20 per month. You'll need to upgrade to remove the watermarks and gain access to millions of royalty-free stock footage clips and a human-sounding voiceover. So if you looked elsewhere, all of the tools would cost you hundreds of dollars if you source them individually. Plus NVIDIA has the overall most affordable monthly and overall pricing compared to famous traditional video editing tools. So you can compare NVIDIA to Adobe Premiere Pro, for instance, or Filmora, or Animoto, or Final Cut Pro. So make sure to download NVIDIA and you can find the link down in the description as well as the pinned comment below. Now in YouTube, there's two potential ways that you could usually earn money and that's through the YouTube Partner Program and that's basically where you get paid AdSense and that's essentially getting paid whenever an ad shows on one of your videos. And then the second way you could potentially get paid with a faceless channel is going to be affiliate marketing and that's basically where you put an affiliate link for a product or a service down in the description and then during the video, you kinda talk about the product or service and you tell them that, yeah, hey, if you wanna check it out, it's down in the description and if they end up buying it, you get a small percentage of that purchase. So I think that YouTube is the absolute best platform out there when it comes to making money and I do think there's a lot of opportunity when it comes to AI and faceless YouTube channels. So for that reason, I am gonna go ahead and give this one an eight out of 10 opportunity score and the next one on the list is going to be making pitch decks because convincing people to invest in your dreams is harder than it looks, but it pays and this basically involves crafting, engaging presentations to persuade people to invest in your company or idea and there's an example of someone doing this who's a pastor living in rural Arkansas and makes $3,000 a week with his side hustle where he uses chat GBT to make pitch decks for startups. So that's $12,000 a month he's making as a side hustle. It's not even his main job, he's not doing it full-time but typically earnings from a pitch deck are gonna be somewhere between $500 and $1,000 per pitch deck and of course you can make much more money than that once you get your skills to a high quality level and you can make even more money than that once you have some clients and some testimonials and you've done a good job in the past. So the first thing you would wanna do is familiarize yourself with pitch decks. Then you'd wanna study the types of companies that regularly hire people for this type of side hustle. Then you would use chat GBT to generate ideas for pitch decks and then you'd use another AI tool called Tom to create a presentation out of it. Now the secret ingredient to creating pitch decks which is really what separates the successful ones from the unsuccessful ones is going to be the ability to tell a story. So you'd wanna brush up on your storytelling skills as well. Then you would wanna build a portfolio of pitch decks that you've created for different companies and start advertising yourself on freelancing platforms such as Upwork, Fiverr.com or Freelancer. So overall, this is a really good one and as with just about any type of freelancing or any type of business you want to specialize in particular types of companies. So for instance, you might specialize in SaaS B2B companies or you might even specialize even further in cloud based SaaS B2B companies.  So overall, I really like this one. A lot of opportunity here. I'm gonna give it an 8.5 opportunity score. Next is going to be overdubbing videos in other languages. And with overdubbing, you basically use AI tools to translate and dub videos into different languages making them accessible to a global audience. Castile, ooks, masolites. And you can use tools such as wearnova.ai in order to do this. And there's a story of a YouTuber named Chris McCauley who wanted to launch a translation of his videos into Spanish. And he initially turned to translations.com which is a go-to among YouTubers but the company quoted him $4,000 for dubbing 100 minutes of footage. Then he came across Nova AI which is another tool you can use and it's a software that can translate and add subtitles to videos. And Nova AI's pro plan priced at $18 a month allows users to translate 300 minutes of footage into 75 different languages. Now, are these translations going to be absolutely perfect? No, of course they're not. If you know anything about translating languages you know that the translations are not always going to be absolutely perfect.  But this is an easy way to make content accessible to an audience that otherwise would not have any access to it whatsoever. And there might be a few things here and there that are incorrect, but they're probably gonna get the overall picture of what the translation actually means. And these AI translating tools are getting more and more accurate every day. So the first thing you'd wanna do to start the side hustle is to get access to a tool such as Nova AI. Then you would want to choose a particular niche and build a portfolio to show that you have the skills to do this. Then you would want to outreach to different content creators, whether this is filmmakers, YouTubers or businesses that work in different countries and offer your overdubbing services. And you wanna emphasize the value of reaching a wider global audience through multiple languages. You could also post on different freelancing platforms such as Fiverr, Upwork or Freelancer.com and you'd probably wanna start off with some lower priced gigs just to build your reputation. Then you'd wanna handle the projects efficiently, do a good job with them and then maintain your relationship with the client so you get more work in the future. So overall, this one is pretty good. Like I said, these AI tools are getting better and better every day. And of course they're not going to be as good as a professional translator, but as long as you communicate that to the person and they understand exactly what they're getting, there shouldn't be any issues. And I've actually seen some of my friends who do SEO professionally. And basically what SEO is, is they create different blogs and then create articles on those blogs and they try to rank for different keywords that are related to things that people would actually buy in other countries. And many of them are actually using these translation tools to basically copy their entire blog in a bunch of different languages. And like I said before, the translation tools are of course not perfect, but they're still seeing a ton of success doing this. So overall, I'm gonna give this one a seven out of 10 opportunity score. Next is going to be voiceover impressions. It doesn't sound anything. It doesn't sound anything like me. And this is where you either clone your own voice and turn it into an AI generated voice or you use a tool like voice.ai to change your voice into a celebrity impression such as Morgan Freeman. This is incredible true story of how I came to be fired from Pepsi home. And as you can see, there is an entire industry on websites like Fiverr that do exactly this. For instance, Optimus Prime, Leonardo DiCaprio, Kermit the Frog, Borat, Yoda and C3PO and a number of other different impressions. And you can also use a website like 11labs.io to clone your own voice. So there's so many different ways that you could make money doing this. You could start faceless YouTube channels. You could do voiceover impressions. You could have the AI actually read out people's books or you could do freelance voiceover gigs. I hear your breath. Now again, if you are using an AI to do this, it's not gonna be quite as good as a professional voiceover artist. And there may be certain parts of the voiceover that you have to go in and edit, but it's much faster than just doing the voiceover yourself. And as long as you set proper expectations with your clients, this shouldn't be an issue. And voiceovers are getting better every single day. So you could actually start a business here where if you're doing purely AI voiceover recordings, you can just hire other people to do all of the work for you. And the only thing you have to do is market. So the sky's the limit in terms of how much you can make doing this. But I would say if you really give this a good try and you really take it seriously, you could definitely get to at least $3,000 a month with this side hustle. I'm gonna give this one an opportunity score of seven out of 10. The next one is going to be podcast show notes. And this is where you use AI to transform podcast episodes into tidy, digestible summaries faster than you can say the word subscribe. And this is actually mega useful. This is something that I use myself because there are a lot of podcasts that I like to follow, but I don't always have an hour or two to listen to them. And so there are tools out there that you can use where it basically just translates the entire podcast into digestible notes and it shows you exactly when they're talking about these different things. And so you can kind of just skim through the notes on the podcast, see what catches your eye and then just listen to that part. And it's actually becoming more and more common for podcasts to offer this. And you see a bunch of different gigs on websites like Fiverr where they will do exactly this for you. So this person, for instance, will do it for $35. This person will do it for $20 and this person will do it for $35 as well. And you can see they're getting tons and tons of sales. 656 different people bought this and they gave them a five star rating. And what these people don't know is that 99% of the work is done by AI. So you can use tools like Melville or Descript in order to do all of the work for you. And a bonus on this one is if you combine this AI tool with another type of AI tool, which is making shorts, you'll be able to offer a combination of services. So first you would make the podcast show notes and then you would make highlights from the podcast and you would turn those into shorts. So yeah, obviously lots of different opportunities here all the way from just doing little gigs here and there to starting a full blown business. I'm gonna give this one an 8.5 out of 10 opportunity score. And by the way, creating organic content is a part of digital marketing. And digital marketing is one of the most common ways to start making money online, working from home, working remote or doing a side hustle. And I've interviewed a ton of different people on this channel that have been able to get digital marketing jobs, even though they didn't have a college degree or previous experience. And many of them went on to start digital marketing side hustles and even companies. For instance, Fabian went from a factory worker to making $70,000 a year right away with digital marketing skills. And my friend Seth helped all of these people start making money with digital marketing skills. And he actually does have a free masterclass which I'll put down in the description as well as the pinned comment below. So definitely go ahead and check that out if you're interested in digital marketing by clicking down in the description or pinned comment below. Next is going to be resume writing. These are all just blank pieces of white paper. Now this is one that I'm very familiar with because I help people make their resumes all the time. And the thing about resume writing is it's not so much about helping you get a job but not having a good resume will lose you the job. So you basically just need to make sure that it's decent. And there's a story of a 35 year old teacher who brought in 2 million from his resume writing side hustle. And with the power of AI, you can do this much more quickly. Now there's a ton of different AI resume websites out there. In my opinion, the best one is going to be Rezi AI. Rezi AI is great. And you can also combine it with chat GBT just to make things a little bit better. Now typically it's not going to perfectly write anyone's resume right off the bat but it'll get about 80% of it right.  Now I'm gonna be honest with you again like many other types of freelancing or making money online, it's gonna be best if you specialize. Not only will it be best for you but it'll also be best for the client. So you want to pick a particular type of career to write resumes in. And you want to really familiarize yourself with that career and that industry. That's going to be the key for you writing really good resumes using AI. So overall, I think resume writing is a phenomenal opportunity. I've known people who've started seven figure businesses in certain places like finance or management consulting and it was all from writing resumes. So I'm gonna give this one an eight out of 10 opportunity score. By the way, comment below if you like any of these side hustles or there's other side hustles that you'd like me to talk about in the future. I always read the comments and there's a good chance that I will do exactly that. Next is going to be creating stock images. AI is getting so good that it's almost indistinguishable from real life images. And so what you want to do as someone who creates AI generated images is you want to see what type of images people are often searching for. And then see what type of images are already offered on these different platforms. For instance, Adobe stock actually does allow AI generated images on their platform. So if you know the supply and you know the demand then you're gonna be able to see that there are certain gaps in the market. And that's where you want to go ahead and fill those gaps by making a bunch of AI generated images. And on Adobe stock, you'll actually get 33% royalties. So let's say you sell an image for $1 you will get 33 cents out of that. And that probably doesn't sound like much but if you think about the fact that you create an image one time and then it's just on there forever earning you royalties for the rest of your life that can really add up over time.  Now there's many different tools out there that are doing this. The most well-known one of course is mid journey. And there's actually an artwork that was created with AI and everyone knows that it was created with AI that sold for $432,000 at auction. And this was just a portrait by AI program that created it. So there's definitely a lot of money, a lot of opportunities out there for this type of side hustle. I'm gonna give it a seven out of 10 opportunity score. By the way, if you think of anyone else that could get value from this video and you're enjoying it yourself go ahead and share it with them. Maybe it's a friend, a family member or just somebody who needs to make a little bit of extra money on the side. Next let's talk about prompt engineering and I am super excited about this one. So prompt engineering is basically all about crafting specific prompts that you would put into an AI tool such as chat GPT. And in the programming world there's this acronym known as GEIGO. And that stands for garbage in, garbage out.  And it's the same thing in the prompting world. The better you are at prompting AI the better results you're going to get at the end. And this is how you can turn a vague idea into something bizarrely specific and utterly delightful. I wanna put both of them. And I've seen a ton of different articles of people who've turned prompt engineering into a business.  And they typically charge $300 per prompt. And there's even websites that specialize in this like promptbase.com. And this is a collection of over 100,000 different top AI prompts. And this helps you to produce better outputs, save time on API and costs and sell your own prompts. So as a prompt engineer you could make sort of mass market prompts that hundreds or even thousands of different people could use. And they might buy them for $1 to $5 or probably a better use of your time is to go into businesses, figure out what their business needs are and create specific prompts that would help with those. And in these cases you could get paid on the low end $300 to on the high end tens of thousands of dollars every time you do this. So there's a ton of opportunity when it comes to prompt engineering not only in side hustles but you can also get hired as a job. And there's people that got hired as prompt engineers for their job that are making hundreds of thousands of dollars a year. So this one is absolutely hot right now. I'm gonna give it a nine out of 10 opportunity score. Next is gonna be a little bit controversial but just give me a moment to explain. And that is going to be using AI for freelance coding. Now you might be thinking, Shane, there's absolutely no way that AI or chat GPT can do coding. Coding is just too complicated, maybe 10 years from now but it definitely can't do it right now. Well, you're right for most types of coding. However, as Amar Reshi showed there are some types of coding where you can absolutely use AI. So for instance, he used chat GPT and replete to code a snake game that runs in your browser. And he had zero knowledge of JavaScript and he did it in less than 20 minutes. Now more practical uses of doing this would be something like designing a basic website or designing a basic app for someone to just get started with a minimum viable product. So for instance, there's people that offer to create custom desktop applications and they only charge about $30. Now if they were doing all the coding themselves I guarantee you they wouldn't be able to charge $30. The reason they're able to charge that low is because they have an AI doing 95% of the heavy lifting for them. So this one takes a little bit of creativity because you kind of have to know what people are looking for. And you also have to know what is available on the AI market. They can do these different tasks but there is a ton of opportunity here and every day there's new different types of AI programs and AI software that's coming out that make this easier and easier. So definitely look into this one. I think right now I'll give it a 7.5 out of 10 opportunity score but it could be much better in the future. Next is going to be AI graphic design. And there are so many different places that you could do this with. One example would be designing T-shirts. And there's a guy who makes $14,600 a month in passive income and only works one hour a day on his online business and that's exactly what he does. And there are tons of different gigs on Fiverr of people who use AI art and AI graphic design and make a ton of money with it. So for instance, this person charges $30 to create quick custom AI art using mid-journey advanced and they have 506 five-star ratings. That means they made at least $15,000 and you have to take into account that a bunch of people probably didn't rate them so it's probably a lot higher than that and that is just one posting. They might have 10 or 20 different ones. So yeah, lots of opportunity here. The only limit is your creativity and imagination. I'm gonna give this one a 7.5 out of 10 opportunity score. Next is going to be AI copywriting and I know what you're thinking, Shane, there's no way AI can do good copywriting and the truth is again, for most examples you are right about that. However, there's certain types of copywriting that AI can help with tremendously. If you don't believe me, there's an article about a self-made millionaire who said this is how I'd use AI to make thousands of dollars a month in passive income with less than $100 and one of the main things that he talked about was using chat GPT for marketing and copywriting. So for instance, a type of copywriting I've seen a lot of people have success with is you take a long form piece of copy that was written really well already and then you turn it into a series of tweets. So this is a type of ghost Twitter writing that a lot of people do. They find really well-written long form versions of copy. They plug it into chat GPT or something like that and then they turn it into a series of tweets and they make it original and that can be really, really good. There's also tons of people who advertise copywriting on places like Fiverr. Some of them actually admit that it's AI, others say that it isn't but chances are they heavily utilize AI in order to help them write good copy and you can see they charge decent amounts of money. So for instance, this person Chloe charges $100 per copy. This person Kate charges $75 per copy and she's got over 1,000 reviews. Lauren charges $100 per copy and she's got over 1,500 star reviews. So yeah, there is a ton of money in copywriting. There's many YouTubers that hire copywriters to help them write their scripts as well. So there's just a ton of opportunity here and of course you're always gonna have to look over it and make little changes here and there and make sure that it's good. But overall, I'm gonna give this one an 8.5 out of 10 opportunity score. Next is going to be writing tweets for famous creators or businesses. And this is kind of what I touched on just a few minutes ago. Basically, you find really good pieces of content out there that the creator or business owner either wrote themselves or they really like. Then you sort of understand their voice and their opinion about things and then you use AI to break down those pieces of content and make original tweets from them. And this is also known as being a Twitter ghostwriter.  So again, this is one where you definitely want to specialize in a particular niche.  And there's also people on Fiverr who offer to write different tweets for somebody's brand. So yeah, this is a really good one. I've seen people absolutely crushing it with this one lately. I'm gonna give it an eight out of 10 opportunity score. Next, let's talk about a super simple one that literally anyone could do and that's going to be a background removal service. So this is basically where you remove the background behind somebody. So you basically cut it out to where it's just white behind them. And that can be really good for for instance, making YouTube thumbnails. A few years ago, for instance, I paid someone like $500 to remove the background of hundreds of different photos. And then I've been using those photos on my thumbnails ever since. Or in some cases, this might be removing somebody else from a picture. So if someone photo bombed your picture or there's just something really annoying in the picture, you can actually remove that. Now there's many different websites you can do this on. For instance, Canva recently has been experimenting with a lot of AI related stuff. Also, there's a website remove.bg where you can remove backgrounds. And then there's many different websites where you can edit photos. So I don't know how long this one is going to last, but right now it's a relatively good opportunity. There's lots of different gigs on Fiverr selling these types of services. And to be honest with you, removing backgrounds or removing someone from a photo is incredibly hard if you do it manually. It's actually extremely tedious. But if you can do it using AI, it's done almost instantly. So I can see why people are able to make so much money doing this. I'll give this one a seven out of 10 opportunity score. By the way, I had a video that went viral. You can check it out. It's 10 side hustles that you can do to earn money. And you can check that out by clicking right here.