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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: $\text{}F=\mathrm{kx}$ , where k is a positive constant (elastic constant). The potential energy is given by the relation $V\left(x\right)=\frac{1}{2}{\mathrm{kx}}^{2}$. 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 $\text{}F=\mathrm{kx}$, i.e. subjected to the harmonic potential $V\left(x\right)=\frac{1}{2}{\mathrm{kx}}^{2}$, where k is a positive constant (elastic constant).  
${\phi}_{0}\left(x\right)$  The wave function (solution of the Schrödinger equation) corresponding to energy ${E}_{0}$  
${\phi}_{1}\left(x\right)$  The wave function (solution of the Schrödinger equation) corresponding to energy ${E}_{1}$  
${\phi}_{2}\left(x\right)$  The wave function (solution of the Schrödinger equation) corresponding to energy ${E}_{2}$  
${\phi}_{0}\left(x\right){}^{2}$  The square modulus of the wave function gives the probability density of finding the particle at position x  
${\phi}_{1}\left(x\right){}^{2}$  The square modulus of the wave function gives the probability density of finding the particle at position x  
${\phi}_{2}\left(x\right){}^{2}$  The square modulus of the wave function gives the probability density of finding the particle at position x  
${c}_{0}\left(p\right){}^{2}$ 
Probability density for momentum p for the particle in the state with energy
${E}_{0}$.
${c}_{0}\left(p\right)$ is defined by the following formula: ${c}_{0}\left(p\right)={\int}_{\infty}^{+\infty}{e}^{\frac{\text{}i\text{}}{\hslash}p\text{}x}{\phi}_{0}\left(x\right)dx$ 

${c}_{1}\left(p\right){}^{2}$ 
Probability density for momentum p for the particle in the state with energy
${E}_{1}$.
${c}_{1}\left(p\right)$ is defined by the following formula: ${c}_{1}\left(p\right)={\int}_{\infty}^{+\infty}{e}^{\frac{\text{}i\text{}}{\hslash}p\text{}x}{\phi}_{1}\left(x\right)dx$ 

${c}_{2}\left(p\right){}^{2}$ 
Probability density for momentum p for the particle in the state with energy
${E}_{2}$.
${c}_{2}\left(p\right)$ is defined by the following formula: ${c}_{2}\left(p\right)={\int}_{\infty}^{+\infty}{e}^{\frac{\text{}i\text{}}{\hslash}p\text{}x}{\phi}_{2}\left(x\right)dx$ 

All ${\phi}_{\mathrm{n}}\left(x\right)$  The three solutions of the Schrödinger equation (wave functions) in a single graph, highlighting the corresponding energy value ${E}_{\mathrm{n}}$  
All ${\phi}_{\mathrm{n}}\left(x\right){}^{2}$  The three probability density for position x in a single graph, highlighting the corresponding energy value ${E}_{\mathrm{n}}$  
All ${c}_{\mathrm{n}}\left(p\right){}^{2}$  The three probability density for momentum p in a single graph, highlighting the corresponding energy value ${E}_{\mathrm{n}}$ 