0:34

What goes in that motion?

Â What are the principles that allow us to tell how the electron would behave.

Â Now quantum mechanics is the foundation that allows us to answer all

Â these questions.

Â By the end of this module you should be able to develop a Lagrangian formulation

Â for classical mechanics, and apply quantum mechanics and use that to

Â understand two model systems, a particle in a box, and a harmonic oscillator.

Â 1:06

The motion of microscopic objects is dictated by classical mechanics,

Â also known as Newtonian mechanics as it's based on Newton's Laws of Motion.

Â Now an important aspect of Newtonian mechanics is that it allows, for

Â a continuous spectrum of energy, and a continuous spatial distribution of matter.

Â Now, there are three main laws that govern Newtonian mechanics.

Â The first law tells us that when viewed from an inertial reference frame,

Â an object at rest tends to stay at rest, and an object in uniform motion

Â tends to stay in uniform motion unless acted upon by a net external force.

Â Now what does the second law tell us.

Â The second law tells us when you apply a force.

Â This says that the net force f on an object equals the time rate of

Â change of momentum leading to the famous equation F=ma.

Â The third law tells us that every action corresponds to an equal and

Â opposite reaction.

Â 2:07

Now, to cause these laws into a mathematical formulation

Â of various formulations exist,

Â one that is particularly useful is known as the Lagrange in formulation.

Â This formalism is based on the principle of stationary action.

Â The Lagrange L of a particle is defined as the difference

Â between its kinetic energy T and the potential energy V.

Â Now, in generalized coordinates, the energy is given by.

Â 2:37

Now, in this equation, q stands for the generalized coordinate of space.

Â Now, from this, we can define a quantity, called the action, S.

Â Now, this is simply defined as the integral of the Lagrangian between two

Â give an instance of time, say t1 and t2.

Â 2:56

Now what does the principle of stationary or least action tell us?

Â It states that the classical part taken by the system,

Â between times t1 and t2, is the one for which the action is stationary.

Â That is, the change to is zero.

Â Now in other words, we're attempting to take the path of least action.

Â It seems like a useful principle to have for life as well.

Â Now mathematically for the small change del q,

Â this principle states that the differential change in action is zero.

Â 3:47

Now, we can write down a Taylor series expansion for

Â the Lagrangian and this leads to the emergence of two new terms.

Â One that depends on the derivative of the Lagrangian with respect to position q.

Â And one that depends on the derivative of the Lagrangian with respect to momentum,

Â q dot.

Â Now, what does the principle of least action tell us?

Â This says that the sum of these two terms must be zero.

Â A little bit of calculus using integration by parts

Â that leads to the Lagrangian Equation of Motion.

Â Now this states that regardless of the bot, the time rate of change

Â of the differential of the Lagrangian with the respect to the velocity,

Â is balanced by the derivative of the Lagrangian with respect to position.

Â We can substitute the definition of the Lagrangian,

Â which gives the familiar equation of motion from Newtonian mechanics,

Â that is, f equals ma, with the force being given

Â as the derivative of the potential energy with respect to position.

Â 4:50

Now, we used two examples to bring out

Â the distinct difference between classical mechanics and quantum mechanics.

Â The first example will be a free particle.

Â And the second one will be a particle that is in a harmonic potential well.

Â Now, let's look at the first example of a free particle.

Â Now, let's consider that the free particle is allowed to move in one dimension,

Â that is along the x axis.

Â Now, this expedience is no external potential.

Â 5:20

Therefore, its equation of motion is simply the deceleration is 0.

Â Now, we can integrate this equation twice,

Â leading to two unknown constants, C1 and C2.

Â 5:32

Now, how do we determine these constants?

Â Well, we need some initial condition.

Â That is, we some information at the starting time of the experiment

Â say the time is set at t equals zero.

Â Now let's assume that at time T equals zero the particle is at a reference

Â position say X equals zero further let's assume that at time

Â T equals zero the particle is moving with the velocity V that is X dot is V.

Â And you note that in this notation dot here, refers to a time derivative.

Â Now, this results in the equation of motion

Â given us the position x is given us velocity times the time.

Â Now, what does this tell us, the equation of motion tells us that we can

Â precisely locate where the particle will be at any given instant of time.

Â It's a very classical phenomenon.

Â 6:26

There is no restriction in the value that the velocity V can take.

Â So therefore, the energy of the system can have a continuous spectrum.

Â That is, it can take any value.

Â Now, let's move onto a second example.

Â Now, let's assume that there is a potential, and

Â the potential of l is a harmonic potential field.

Â Now, a harmonic potential field is one

Â where the external potential goes as the square of the displacement x.

Â Now, what is a good example of a harmonic potential field?

Â Now if you have a spring and try to stretch it the energy from it's

Â vesposition is well described by a harmonic potential well.

Â Now what is the motion of the particle in this case?

Â Once again, we can derive the equation of motion from the La Grangian equation.

Â Now here it turns out that the solution of the equation

Â of motion consists of two sinusoids with a phase difference of pi over two.

Â 7:26

Now we can invoke the same initial conditions that we used in the previous

Â example and this leads to the equation of motion where the position x

Â scales at the sinusoid frequency times the time.

Â Where new is a characteristic frequency associated with the system.

Â In this case the characteristic frequency

Â scales along with the constant k and the mass of the system m.

Â Like the Lagrangian formulation that we learned another formulation called

Â the Hamiltonian formulation of classical mechanics describes the equation of

Â motion, albeit using a different quantity, H, called the Hamiltonian.

Â Now the Hamiltonian is defined as the sum of the kinetic and the potential energy.

Â Now, for the second example that we studied the Hamiltonian of the particle,

Â it turns out that it would be independent of time.

Â Now, from these two examples,

Â we can infer some key conclusions about classical mechanics.

Â Classical mechanics predicts that the particle motion

Â is completely deterministic.

Â That is, the conditions of a particle at any given instant of time

Â will chart out its future trajectory.

Â The Lagrangian formulation teaches us that the particle traverses along a path

Â such that its action S is an extremum, that is, a minimum.

Â Now we discussed two examples.

Â The first example taught us that a free particle one that does not have

Â any influence of any external potential, will maintain a constant velocity,

Â as proposed by Newton's First Law of Motion.

Â Second example told us that the motion of a particle

Â in a time independent potential field, such as the harmonic potential well,

Â would be governed by a constraint that the total energy is a constant.

Â