Now, classical mechanics is extremely successful when applied to macroscopic objects that is objects that are very, very large. Now, what happens when the objects become very small. Now, a few key experimental observations that emerged in the early 1900s demonstrated the inadequacy of classical mechanics in treating microscopic phenomena. One of the most famous examples is the Photoelectric Effect which was explained by Albert Einstein and one that got him the Nobel Prize. Now, for the electric effect, is the phenomenon of emission of electrons from an atomic surface when subjected to light. Now, if we assumed that the light was simply a wave, then the energy contained in one of those waves would depend only on it's amplitude that is the intensity of the light. Other factors like the frequency should make no difference to the experiment. However, this picture was grossly wrong, as the electron emission was found to occur at a threshold frequency, not intensity. For the maximum kinetic energy of the emitted electrons from the metallic surface was found to depend on the frequency of the incident light. Now, these inconsistencies are the result by the introduction of a fundamentally radical new idea, an idea of Quantum Mechanics. The essence of Quantum Mechanics is that physical processes are not predetermined in a mathematically exact sense. Now relating this back to what we had discussed earlier in classical mechanics, the particle motion is not restricted to a single part as would be predicted by classical mechanics. Instead, all possible parts have a probability of occurring. Now, we can define the probability of going from two to one in term of a total amplitude K, such that the probability P is giving us the square of the amplitude K. Now, using the previously define quantity S, that is, the action for a particular path. The total amplitude can be considered as sum of contributions from each and every path connecting 1 and 2. Now, the actual contribution of each path can be determined in terms of the action using the following relation. Note that all objects are really quantum mechanical in nature, that is they traverse along paths with probabilities dictate by the action S of each path. Now, comes the puzzling question of why is it that microscopic objects travel along only one path? Well, microscopic objects have comparably large masses and have actions which are much larger compare to the quanta of action which is given by the blanks constant. Now therefore, microscopic objects posse only one dominant path which determines their behavior and this part corresponds to the classical part as determined by the principle of least action. While such a formulation mostly merges into metonial mechanics for macroscopic physical objects this has far reaching implications on the interpretation of microscopic physical processes. As discussed before the amplitude is related to the probability of going from one to two. Now, to find the probability of locating a particle at a location Q at a time T we define the wave backed Sie which depends on the location Q and time T. Now, this gives the time dependent probability distribution, P(q, t), defined as the square of the wave bucket Psi (q, t). Using the condition that probability must be Markovian, that is, the system has no memory property, we can define the following equation. Now, this property is used to find a diffusion equation for the wave pack inside. The governing equation for the wave packet, is what is famously known as the starting of equation that forms the basis of quantum mechanical calculations. Now, using R as the position vector, the same equation can be expressed in three dimensions in the following way. Now, in order to predict the expectation value of the energy, we note that we can define the Hamiltonian operator. As the sum of the potential energy in the Kinetic Energy operator. Note that each operator in quantum mechanics has an expectation value. Turns out that the energy is the expectation value of the Hamiltonian operator. This gives us the famous equation H Psi = E Psi which is known as the time independent equation. Now, let's revisit the two examples that we did in classical mechanics, from the perspective of quantum mechanics. Now, the first example, popularly called in Quantum Mechanics a particle in a box. Let us once again consider that a particle is moving in one dimension along X axis, and we have a box of length l from X equals 0 and X equals L. Now, the external potential is assigned to the 0 inside the box. Now, how do we make sure the particle stays inside the box, well we make the potential energy infinite outside the box on both ends, so that the particle cannot escape the box. Now, inside the box, the only Hamiltonian that arises is the Kinetic Energy. The equation for the particle inside the box is simply given by a wave equation. Now, this equation has the boundary conditions that the wave function sign must vanish at x equals 0 and x equals l. That is to say, the probability of finding a particle at the edges of the box is z0. The general solution of the wave equation is given as. Now, to satisfy the boundary conditions, we must require that C2 is 0. Now, the remaining solution has infinite possibilities as the Sinusoid function is 0 for n pi when n equals 1, 2, 3, etc. Now, this results in the condition. This results in a rather interesting conclusion that the energy is quantized, that is, the energy of a particle in a box cannot take arbitrary values. It can only take discrete energy values. The second constant can be found using a condition that a total probability of finding the particle must add up to 1. This is properly known as the normalization condition. Now, this gives us the solution for the way packet, for the nth quantum state of the particle, which is given by. Nex, we look at the Quantum Mechanical Analog of a particle in a harmonic potential well. Where do we find a harmonic potential well in an atomic setting? Now, classically we found that the spring is a good example of a harmonic potential well, the atomic version of a spring is the vibration modes of a diatomic molecule. For instance, if we have an oxygen molecule, we can imagine a spring between the two oxygen atoms making up the molecule. Now, let's try to analyze such a case. Now, let's set up the problem. Let's consider a diatomic molecule with atomic masses, m1 and m2. The covalent bond between the two atoms can be modeled as a harmonic spring, with a spring constant, k. Now, in this case, the Hamiltonian operator consists of both the Kinetic Energy part and the Potential Energy part. Let's now define x to be the distance of separation between the two atoms. We can now write down the equation for the wave packet as. The solution to this equation is a bit more complex. However, similar to the particle in a box, we find that we have an infinite number of solutions with discrete energy levels. Now, it turns that the energy levels are equally spaced and like the case of the particle in the box. And note, that this equation is valid for the case of n equals 0 unlike the particle in the box. This gives what is known as the zero point energy of vibration of the molecule. Now, as in the case of classical mechanics, the characteristic frequency new, plays an important role in determining the solutions of this equation. Now, to summarize, in this module, we learned about Classical mechanics and contrasted it with Quantum mechanics. Classical mechanics is an excellent description for macroscopic objects. While Quantum mechanics is necessary for describing microscopic objects. In Classical mechanics, we can have a continuous spectrum of energy. While in Quantum mechanics, it turns out that they come in discrete packets. We discussed two key examples of a particle that's sitting in a box, and in a harmonic potential well. In the quantum mechanical picture, the particle in a box, results in a quantization of the energy levels that scales as a square of a quantum number N. On the other hand, for the case of a harmonic potential well we found that the energy levels are equi-spaced. Now, these two model problems form the cornerstone of modeling many practical systems and will be used over and over again in this course.
