Simple Harmonic Oscillator

Let us consider the problem of a particle that is subjected to a linear restoring force F = -kΔx, where Δx is the magnitude of the displacement of the particle from its equilibrium position and k is the force constant. This is an important scenario to consider as the forces between atoms in a solid can be approximated by this …

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A Particle In A Well Of Finite Height

Consider now a particle in a well, but now the walls are of finite height (instead of infinitely high), which we call a potential well. If we again choose U = 0 for 0 < x < L, then U will have a finite value for x = 0 and x = L. If the total energy E of the …

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Sense of scale – How small are we?

Look around the room you are in now. What is the biggest thing there? Is it your bed? Now, look outside the window. What is the biggest thing you can see? Is it a skyscraper? The bed looks tiny when compared to a skyscraper. Now, what about our planet – Earth or Mars or even the Universe? How does each …

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Powerful Recoils

In the movies, those toy-guns that the actors are carrying hardly have any recoil. But in the real world, the recoils is much more powerful than you can imagine. If you hold the firearm in an improper position, the firearm will do some very nasty things to you or the people around you. A rifle: A handgun: Back To Interesting …

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The Particle In A Box Revisited

Let us now try to solve the Schrodinger equation for a particle in a one-dimensional box of width L. Consider the walls to be infinitely high, hence U(x) = ∞ for x = 0 and x = L. Since potential energy is constant within the box, it is convenient to choose U(x) = 0 as its value. Hence for 0 …

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Time-Independent Schrodinger Equation

The wave function for de Broglie waves must satisfy an equation developed by Schrodinger. If we simply consider a particle moving in one dimension (e.g. along the x-axis), we can write Schrodinger equation as $\frac{d^{2} \psi}{dx^{2}} = – \frac{2m}{h^{2}} \left(E – U \right) \psi$ where E is the total energy and U is the potential energy. This is the famous …

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Normalization Of The Wave Function

The wave function ψ is not an observable quantity. It manifests itself only on the statistical distribution of particle detection. In one-dimension, the quantity $| \psi |^{2} \, dx$ represents the probability of finding the particle associated with the wave function ψ(x) in the interval dx at some position x. i.e. $\text{Probability} = | \psi |^{2} \, dx$ Thus, ψ2 …

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