0:08

Now, classical mechanics is extremely successful

Â when applied to macroscopic objects that is objects that are very, very large.

Â Now, what happens when the objects become very small.

Â Now, a few key experimental observations that emerged in the early

Â 1900s demonstrated the inadequacy of classical mechanics in treating

Â microscopic phenomena.

Â One of the most famous examples is the Photoelectric Effect

Â which was explained by Albert Einstein and one that got him the Nobel Prize.

Â Now, for the electric effect, is the phenomenon of

Â emission of electrons from an atomic surface when subjected to light.

Â Now, if we assumed that the light was simply a wave,

Â then the energy contained in one of those waves would depend only on it's amplitude

Â that is the intensity of the light.

Â Other factors like the frequency should make no difference to the experiment.

Â However, this picture was grossly wrong, as the electron emission

Â was found to occur at a threshold frequency, not intensity.

Â 1:16

For the maximum kinetic energy of the emitted electrons from the metallic

Â surface was found to depend on the frequency of the incident light.

Â Now, these inconsistencies are the result by the introduction

Â of a fundamentally radical new idea, an idea of Quantum Mechanics.

Â The essence of Quantum Mechanics is that physical processes

Â are not predetermined in a mathematically exact sense.

Â Now relating this back to what we had discussed earlier in classical mechanics,

Â the particle motion is not restricted to a single part

Â as would be predicted by classical mechanics.

Â Instead, all possible parts have a probability of occurring.

Â 2:05

Now, we can define the probability of going from two to one

Â in term of a total amplitude K,

Â such that the probability P is giving us the square of the amplitude K.

Â Now, using the previously define quantity S, that is, the action for

Â a particular path.

Â The total amplitude can be considered as sum of contributions from each and

Â every path connecting 1 and 2.

Â Now, the actual contribution of each path can be determined

Â in terms of the action using the following relation.

Â 2:41

Note that all objects are really quantum mechanical in nature, that is they

Â traverse along paths with probabilities dictate by the action S of each path.

Â Now, comes the puzzling question of why is it that microscopic objects

Â travel along only one path?

Â Well, microscopic objects have comparably large masses and

Â have actions which are much larger compare to the quanta of action

Â which is given by the blanks constant.

Â Now therefore, microscopic objects posse only one dominant path

Â which determines their behavior and this part corresponds to the classical part

Â as determined by the principle of least action.

Â While such a formulation mostly merges into metonial mechanics for

Â macroscopic physical objects this has far reaching implications

Â on the interpretation of microscopic physical processes.

Â As discussed before the amplitude is related to the probability

Â of going from one to two.

Â Now, to find the probability of locating a particle at a location Q at a time T

Â we define the wave backed Sie which depends on the location Q and time T.

Â Now, this gives the time dependent probability distribution, P(q, t),

Â defined as the square of the wave bucket Psi (q, t).

Â Using the condition that probability must be Markovian, that is,

Â the system has no memory property, we can define the following equation.

Â 4:32

The governing equation for the wave packet, is what is famously known

Â as the starting of equation that forms the basis of quantum mechanical calculations.

Â Now, using R as the position vector,

Â the same equation can be expressed in three dimensions in the following way.

Â 5:18

This gives us the famous equation H Psi = E Psi which is known as

Â the time independent equation.

Â Now, let's revisit the two examples that we did in classical mechanics,

Â from the perspective of quantum mechanics.

Â Now, the first example, popularly called in Quantum Mechanics a particle in a box.

Â Let us once again consider that a particle is moving in one dimension along X axis,

Â and we have a box of length l from X equals 0 and X equals L.

Â Now, the external potential is assigned to the 0 inside the box.

Â Now, how do we make sure the particle stays inside the box,

Â well we make the potential energy infinite outside the box on both ends, so

Â that the particle cannot escape the box.

Â 6:43

Now, to satisfy the boundary conditions, we must require that C2 is 0.

Â Now, the remaining solution has infinite possibilities

Â as the Sinusoid function is 0 for n pi when n equals 1, 2, 3, etc.

Â Now, this results in the condition.

Â 7:21

The second constant can be found using a condition that a total probability

Â of finding the particle must add up to 1.

Â This is properly known as the normalization condition.

Â Now, this gives us the solution for the way packet,

Â for the nth quantum state of the particle, which is given by.

Â 7:42

Nex, we look at the Quantum Mechanical Analog of a particle in a harmonic

Â potential well.

Â Where do we find a harmonic potential well in an atomic setting?

Â Now, classically we found that the spring

Â is a good example of a harmonic potential well,

Â the atomic version of a spring is the vibration modes of a diatomic molecule.

Â For instance, if we have an oxygen molecule,

Â we can imagine a spring between the two oxygen atoms making up the molecule.

Â Now, let's try to analyze such a case.

Â Now, let's set up the problem.

Â Let's consider a diatomic molecule with atomic masses, m1 and m2.

Â The covalent bond between the two atoms can be modeled as a harmonic spring,

Â with a spring constant, k.

Â Now, in this case, the Hamiltonian operator consists of both

Â the Kinetic Energy part and the Potential Energy part.

Â Let's now define x to be the distance of separation between the two atoms.

Â We can now write down the equation for the wave packet as.

Â 8:47

The solution to this equation is a bit more complex.

Â However, similar to the particle in a box,

Â we find that we have an infinite number of solutions with discrete energy levels.

Â Now, it turns that the energy levels are equally spaced and

Â like the case of the particle in the box.

Â 9:09

And note, that this equation is valid for

Â the case of n equals 0 unlike the particle in the box.

Â This gives what is known as the zero point energy of vibration of the molecule.

Â Now, as in the case of classical mechanics,

Â the characteristic frequency new,

Â plays an important role in determining the solutions of this equation.

Â Now, to summarize, in this module, we learned about Classical mechanics and

Â contrasted it with Quantum mechanics.

Â Classical mechanics is an excellent description for macroscopic objects.

Â While Quantum mechanics is necessary for describing microscopic objects.

Â In Classical mechanics, we can have a continuous spectrum of energy.

Â While in Quantum mechanics, it turns out that they come in discrete packets.

Â We discussed two key examples of a particle that's sitting in a box,

Â and in a harmonic potential well.

Â In the quantum mechanical picture, the particle in a box, results in

Â a quantization of the energy levels that scales as a square of a quantum number N.

Â On the other hand, for the case of a harmonic potential well

Â we found that the energy levels are equi-spaced.

Â Now, these two model problems form the cornerstone of modeling many practical

Â systems and will be used over and over again in this course.

Â