My Quantum Mechanics Home \ Harmonic Oscillator
A harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x, that is: , where k is a positive constant (elastic constant). The potential energy is given by the relation . The Schrödinger equation can be solved analytically without taking approximations. For more details open the notebook below. | ||
Open the notebook in Wolfram Cloud | The Wolfram Mathematica notebook contains the solutions of the Schrödinger equation and allows you to produce all the graphs shown on this page. | |
Motion of harmonic oscillator HD video Snapshots |
Motion of a particle subjected to force , i.e. subjected to the harmonic potential , where k is a positive constant (elastic constant). | |
The wave function (solution of the Schrödinger equation) corresponding to energy | ||
The wave function (solution of the Schrödinger equation) corresponding to energy | ||
The wave function (solution of the Schrödinger equation) corresponding to energy | ||
The square modulus of the wave function gives the probability density of finding the particle at position x | ||
The square modulus of the wave function gives the probability density of finding the particle at position x | ||
The square modulus of the wave function gives the probability density of finding the particle at position x | ||
Probability density for momentum p for the particle in the state with energy
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is defined by the following formula: |
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Probability density for momentum p for the particle in the state with energy
.
is defined by the following formula: |
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Probability density for momentum p for the particle in the state with energy
.
is defined by the following formula: |
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All | The three solutions of the Schrödinger equation (wave functions) in a single graph, highlighting the corresponding energy value | |
All | The three probability density for position x in a single graph, highlighting the corresponding energy value | |
All | The three probability density for momentum p in a single graph, highlighting the corresponding energy value |