Quantum Mechanics: Theory of Physical Problems

In this video lesson, Physics Instructor Donny Lee starts by briefly discussing the postulates of quantum mechanics and the minimum set of assumptions from which we will build the theory upon.

In this video lesson, Physics Instructor Donny Lee talks about the 5th postulate of quantum mechanics. It says that the dynamics of a system is regulated by the Time-dependent Schrödinger equation.

Assuming that potentials don't change with time, Physics Instructor Donny Lee will deal with the time-independent Schrödinger Equation, which is more manageable. This lesson involves writing its most general solutions.

In order to display quantum effects, Physics Instructor Donny Lee will consider potentials which vary considerably over small distances. Quantitatively, this is represented by a 'Square' potential.

To study the dynamic properties of a single particle, Physics Instructor Donny Lee will consider a potential V(x) general enough to allow for the illustration of all the required features.



First, we deal with bound states which occur whenever the particle cannot move to infinity such as the infinite square well potential and the harmonic oscillator.

To study the dynamic properties of a single particle, Physics Instructor Donny Lee will consider a potential V(x) general enough to allow for the illustration of all the required features.



Second, we deal with unbound states which occur when the motion of the particle is not confined such as the free particle.

Knowing that we need to solve a 2nd-order differential equation in the time-independent Schrödinger Equation, Physics Instructor Donny Lee will write the solutions here on this video, based on which form the equation takes.

Before setting out to solve physical problems, we discuss a general method, relating solutions with bound and unbound states, that we will undertake when dealing with these problems.

Continuing from our development of this general method, we take a closer look at the solutions and see how we get them based on the energy levels of the particle.

A somewhat important lesson concerning these continuity and boundary conditions which can be used at points of potential discontinuity. We employ them throughout all the physical problems.

After getting the wave solutions to the Schrödinger equation, we can now discuss their physical interpretations, as in all of physics, it is very important to know what an equation means. I have used a somewhat amusing analogy here but I hope it aids in the understanding.

Once, we get the solutions to the Schrödinger equation for time-independent potentials, we can use these stationary states to form a wave packet via the superposition principle.

In quantum mechanics, we need a mathematical scheme that can embody both particle and wave features of a particle. Following up from a short talk of stationary states, we finish up with a description of the process of getting a wave packet - a localized plane wave which can represent a particle.