[MUSIC] Now let's roll a marble down a hill. How long will the marble take to roll down the hill? What governs it's motion? Now classical mechanics is the machinery that allows us to answer all these questions. Now, let's change the question from a marble to an electron. Now, let's assume that the electron is rolling down some sort of an atomic hill. What goes in that motion? What are the principles that allow us to tell how the electron would behave. Now quantum mechanics is the foundation that allows us to answer all these questions. By the end of this module you should be able to develop a Lagrangian formulation for classical mechanics, and apply quantum mechanics and use that to understand two model systems, a particle in a box, and a harmonic oscillator. The motion of microscopic objects is dictated by classical mechanics, also known as Newtonian mechanics as it's based on Newton's Laws of Motion. Now an important aspect of Newtonian mechanics is that it allows, for a continuous spectrum of energy, and a continuous spatial distribution of matter. Now, there are three main laws that govern Newtonian mechanics. The first law tells us that when viewed from an inertial reference frame, an object at rest tends to stay at rest, and an object in uniform motion tends to stay in uniform motion unless acted upon by a net external force. Now what does the second law tell us. The second law tells us when you apply a force. This says that the net force f on an object equals the time rate of change of momentum leading to the famous equation F=ma. The third law tells us that every action corresponds to an equal and opposite reaction. Now, to cause these laws into a mathematical formulation of various formulations exist, one that is particularly useful is known as the Lagrange in formulation. This formalism is based on the principle of stationary action. The Lagrange L of a particle is defined as the difference between its kinetic energy T and the potential energy V. Now, in generalized coordinates, the energy is given by. Now, in this equation, q stands for the generalized coordinate of space. Now, from this, we can define a quantity, called the action, S. Now, this is simply defined as the integral of the Lagrangian between two give an instance of time, say t1 and t2. Now what does the principle of stationary or least action tell us? It states that the classical part taken by the system, between times t1 and t2, is the one for which the action is stationary. That is, the change to is zero. Now in other words, we're attempting to take the path of least action. It seems like a useful principle to have for life as well. Now mathematically for the small change del q, this principle states that the differential change in action is zero. Now remember, the end points are fixed at q1 and q2 therefore, the perturbation has the condition that del q1 and del q2 are both zero. Now, we can write down a Taylor series expansion for the Lagrangian and this leads to the emergence of two new terms. One that depends on the derivative of the Lagrangian with respect to position q. And one that depends on the derivative of the Lagrangian with respect to momentum, q dot. Now, what does the principle of least action tell us? This says that the sum of these two terms must be zero. A little bit of calculus using integration by parts that leads to the Lagrangian Equation of Motion. Now this states that regardless of the bot, the time rate of change of the differential of the Lagrangian with the respect to the velocity, is balanced by the derivative of the Lagrangian with respect to position. We can substitute the definition of the Lagrangian, which gives the familiar equation of motion from Newtonian mechanics, that is, f equals ma, with the force being given as the derivative of the potential energy with respect to position. Now, we used two examples to bring out the distinct difference between classical mechanics and quantum mechanics. The first example will be a free particle. And the second one will be a particle that is in a harmonic potential well. Now, let's look at the first example of a free particle. Now, let's consider that the free particle is allowed to move in one dimension, that is along the x axis. Now, this expedience is no external potential. Therefore, its equation of motion is simply the deceleration is 0. Now, we can integrate this equation twice, leading to two unknown constants, C1 and C2. Now, how do we determine these constants? Well, we need some initial condition. That is, we some information at the starting time of the experiment say the time is set at t equals zero. Now let's assume that at time T equals zero the particle is at a reference position say X equals zero further let's assume that at time T equals zero the particle is moving with the velocity V that is X dot is V. And you note that in this notation dot here, refers to a time derivative. Now, this results in the equation of motion given us the position x is given us velocity times the time. Now, what does this tell us, the equation of motion tells us that we can precisely locate where the particle will be at any given instant of time. It's a very classical phenomenon. There is no restriction in the value that the velocity V can take. So therefore, the energy of the system can have a continuous spectrum. That is, it can take any value. Now, let's move onto a second example. Now, let's assume that there is a potential, and the potential of l is a harmonic potential field. Now, a harmonic potential field is one where the external potential goes as the square of the displacement x. Now, what is a good example of a harmonic potential field? Now if you have a spring and try to stretch it the energy from it's vesposition is well described by a harmonic potential well. Now what is the motion of the particle in this case? Once again, we can derive the equation of motion from the La Grangian equation. Now here it turns out that the solution of the equation of motion consists of two sinusoids with a phase difference of pi over two. Now we can invoke the same initial conditions that we used in the previous example and this leads to the equation of motion where the position x scales at the sinusoid frequency times the time. Where new is a characteristic frequency associated with the system. In this case the characteristic frequency scales along with the constant k and the mass of the system m. Like the Lagrangian formulation that we learned another formulation called the Hamiltonian formulation of classical mechanics describes the equation of motion, albeit using a different quantity, H, called the Hamiltonian. Now the Hamiltonian is defined as the sum of the kinetic and the potential energy. Now, for the second example that we studied the Hamiltonian of the particle, it turns out that it would be independent of time. Now, from these two examples, we can infer some key conclusions about classical mechanics. Classical mechanics predicts that the particle motion is completely deterministic. That is, the conditions of a particle at any given instant of time will chart out its future trajectory. The Lagrangian formulation teaches us that the particle traverses along a path such that its action S is an extremum, that is, a minimum. Now we discussed two examples. The first example taught us that a free particle one that does not have any influence of any external potential, will maintain a constant velocity, as proposed by Newton's First Law of Motion. Second example told us that the motion of a particle in a time independent potential field, such as the harmonic potential well, would be governed by a constraint that the total energy is a constant.
