Welcome to GussianMath's module on Quantum Mechanics.
We start by briefly discussing the postulates of quantum mechanics - the minimum set of assumptions from which we'll build the theory upon.
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Our first official lesson on multivariable calculus. We start by examining the double integral, how we use the limiting process and apply it to two variables.
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Assuming that potentials don't change with time, we deal with the time-independent Schrödinger Equation, which is more manageable. This lesson involves writing its most general solutions.
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We see how, through a simple procedure, we define the limits of a type I region R.
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Up to now, we should have realized that we can either integrate dx dy or integrate dy dx. This lesson looks at reversing the order.
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Using the Double integral to find the volume of a cylinder with a flat bottom and slanted top.
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In order to display quantum effects, we are going to consider potentials which varied considerably over small distances. Quantitatively, this is represented by a 'Square' potential.
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We visit an age-old problem of finding the volume of a sphere but this time curiously using polar double integrals.
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Using the Double integral to find the volume of a tetrahedron bounded by a plane and the coordinate planes.
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Knowing that we need to solve a 2nd-order differential equation in the time-independent Schrödinger Equation, we shall write the solutions here, based on which form the equation takes.
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A short video explaining the Gradient Vector Field, a difficult part in understing vector Calculus. Hope you enjoy it.
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The 5th postulate of quantum mechanics. It says that the dynamics of a system is regulated by the Time-dependent Schrödinger equation. Not easy to solve.
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We learn how to calculate the double integral over nonrectangular regions.
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Advanced Differential Calculus - The Total Differential
Previously for partial derivatives, ?x and ?y were considered separately. Now we look at the effect of changing x and y together. We discover a new term called the total differential which is
dz = a?x + b?y
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Still sticking in the polar coordinates systems, this lessons focuses on evaluating the double integral.
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We see how, through a simple procedure, we define the limits of a type II region R.
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Advanced Differential Calculus - Theorem of the Total Differential
In fact, there is a quick way in calculating the total differential of a function IF it exist. Here, we see how we get to the result:
a = ?z / ?x, b = ?z / ?y
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To study the dynamic properties of a single particle, we 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.
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.
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Getting Bernoulli's Equation by CAREFULLY considering each term.
Gaussian Math Fluid Mechanics module, situable for those studying it as an undergraduate module.
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We derive the magnitude and direction of a hydrostatic force on a plane surface.
Gaussian Math Fluid Mechanics module, situable for those studying it as an undergraduate module.
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Before we head to evaluating double integrals, we need to be familiar with a somewhat new technique of integrating, how we integrate a function in two variable with respect to one variable only.
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![QM0.4: \"Square\" Potentials](http://img.youtube.com/vi/R08_h1IZJTM/2.jpg "QM0.4: \\"Square\\" Potentials")
![Bernoulli\](http://img.youtube.com/vi/rXl0oYtCDZg/2.jpg "Bernoulli\")
