Now, let's talk a little bit more about the classical limit of our system. The first topic I wanted to touch upon is conservation of energy, E. The total energy E, is the sum of kinetic energy equal to K. Kinetic energy K equal to M squared over two, and the potential energy U of X. Conservation of energy says that, their sum is constant in time, which gives us Equation 30, shown here. We can also rewrite it as shown in Equation 31, by regrouping terms and taking a square root. This is an ordinary differential equation that we can explicitly integrate. And if we do that, we obtain Equation 32, on this slide. Here, a parameter X nought, is some initial value and the constant in this equation, partially absorbs any changes to this parameter. So, there are a few things we can see about this equation, first it does not give the direct solution as a function X T, rather the method gives an inverse function T, ot which is equal to T of X. Second, an inspection of this equation, shows something that we already know. Namely that, the total energy should be larger or equal to the potential energy, than the potential energy. If this condition is violated, the expression under the square root becomes negative. Let's consider again the example that we already discussed, that has two meta-stable states. In this graph, the energy E of a particle, can be plotted as a horizontal line at the level of this energy. If we go along this line from the left to the right, the point where it intersects with the potential are points where the kinetic energy becomes zero. At this point, the particle turns around. Therefore, those points are called the turning points. Therefore, turning points are simply found by solving Equation 33, shown in this slide. Now, imagine that for a given level E, we have two turning points, X1 and X2. This means that our potential and our energy E are such that, a system can only do a finite motion. Then, these points can be used as limits of integration in above formula, and this produces Equation 34 here. Please note that the common p integral factor here, is twice the same factor in a previous formula, and this is because Equation 34 is written for the full period of oscillations, and this gives us an extra factor of two. Now, a socially useful here to review alternative formulations of classical mechanics, that might be useful to know. The first fundamental concept that we need here, is the concept of action and Lagrange function. The action is simply defined as the time integral over Lagrangian function. In its form, the Lagrangian function, is given by the difference of the kinetic and potential energies. Now, all of classical mechanics can be derived from a single principle called the Hamilton principle. It says that the motion of a mechanical system should be such that, the action along a trajectory is minimized. This is called the Hamilton principle of the least action, if an action S is optimal for a given trajectory, then its variation under small variations of the trajectory itself, should be zero. Well, now that a variation of action can be expressed in terms of variation of the integrant of the action as shown here in Equation 36, if in addition you integrate the second term here by parts, you will obtain the Lagrange equation shown here in Equation 37. Lagrange equations are very general, and in particular, they reproduce the Newtonian Laws of Dynamics. It's also useful to use conservation of energy shown here in Equation 38, and use it to express the momentum P, which is equal to the product M times X dot, in terms of energy E and potential energy U, as shown on the right of this equation. If we substitute this into the definition of action, we obtain equation 39 shown here, this one gives the action as a function of the terminal position of a particle. Now, the reason we talk about the action at lengths here, is that while it plays a somewhat formal role in classical mechanics, it becomes vital in quantum mechanics and statistical mechanics. Indeed, in classical mechanics, you can in principle, proceed just with the Newtonian mechanics and Newtonian mechanics does not use any notion of the action. The Hamiltonian principle is used here in order to derive the Newtonian mechanics from something else, such as variational principle. But at the end, the principal selects one optimal trajectory, and this is a trajectory that is followed by a classical system. On the other hand, when we move from classical mechanics to quantum mechanics, then the action becomes a first-class citizen there. In quantum mechanics, all paths simultaneously contribute to a transition over a quantum particle from one point in space to another. The full probability is a weighted sum of probabilities of transitions along each given pass. The weights of those passes are given by the exponential of the action multiplied by the imaginary unit and divided by the Planck constant H bar. Now, if action S along a trajectory is real, the whole expression in exponential, is purely imaginary and therefore it doesn't change the norm of the exponential. On the other hand, assume that S is imaginary, then in the exponential, we will have a negative number, that is, such trajectory would be exponentially suppressed by its action. On the other hand, if you look at Equation 40, you can see when you can get imaginary actions. For this, you need to have a negative expression under the square root in this equation. But this happens only when energy is less than the potential energy U of X. But we just said that for a classical motion, we should have the opposite relation, E larger than U of X. This means that exponential suppressed contributions are obtained when a particle goes through a classically forbidden region in the X space. This is called quantum tunneling, and this is one of the most fascinating phenomena in quantum physics. We will talk a little bit more about it after we discuss the Fokker-Planck equation in the next video.