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| − | = Introduction =
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| − | The quantum world is governed by the [https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Schrödinger equation]
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| − | \[{\displaystyle {\hat {H}}|\psi (t)\rangle =i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle } \]
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| − | where $\hat H$ is the [https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics) Hamiltonian], $|\psi (t)\rangle$ is the [https://en.wikipedia.org/wiki/Wave_function quantum state function] and $\hbar$ is the reduced [https://en.wikipedia.org/wiki/Planck_constant Planck constant].
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| − | The Hamiltonian consists of kinetic energy $\hat T$ and potential energy $\hat V$. As in classical mechanics, potential energy is a function of time and space, whereas the kinetic energy differs from the classical world and is calculated as
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| − | \[\hat T = - \frac{\hbar^2}{2m} \nabla^2 .\]
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| − | The final version of the single particle Schrödinger equation can be written as
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| − | \[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) \psi(t, \mathbf r) = i\hbar {\frac {\partial }{\partial t}}\psi(t, \mathbf r) \]
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| − | Quantum state function is a complex function, so it is usually split into the real part and imaginary part
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| − | \[ u, v \in C(\mathbb R)\colon \psi = u + i v , \]
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| − | which for a real $V$ yields a system of two real equations
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| − | \[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) u(t, \mathbf r) = -\hbar {\frac {\partial }{\partial t}} v(t, \mathbf r) , \]
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| − | \[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) v(t, \mathbf r) = \hbar {\frac {\partial }{\partial t}} u(t, \mathbf r) , \]
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| − | which may be easier to handle.
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| − | = Harmonic oscilator =
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| − | By selecting the potential $V(t, \mathbf r)$ and the initial state $\psi(0, \mathbf r)$ we get a unique solution for time propagation of the quantum state function. Probably the most used and well known example is the quantum harmonic oscilator, where we select a quadratic potential
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| − | \[V(t, \mathbf r) = V(\mathbf r) = \frac{1}{2} m \omega^2 r^2 , \]
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| − | where $m$ is the mass of the particle and $\omega$ is the angular frequency of the oscilator.
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| − | The 1D harmonic oscilator has known eigenstate solutions
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| − | \[\psi _{n}(x)={\frac {1}{\sqrt {2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad n=0,1,2,\ldots .\]
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| − | where the functions $H_n$ are the physicists' [https://en.wikipedia.org/wiki/Hermite_polynomials Hermite polynomials]. Time propagation of eigenstates is described with
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| − | \[\psi_n(t, x) = \mathrm e ^ {-i (n+0.5) \omega t} \psi_n(x)\]
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| − | = Particle in a box =
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| − | A theoretical one dimensional potential
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| − | \[\displaystyle V(x)={\begin{cases}0,&0<x<L,\\\infty ,&{\text{otherwise,}}\end{cases}}\]
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| − | is known as an infinite potential well. Its time independent eigenfunctions are
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| − | \[\sqrt{\frac{2}{L}}\psi_n(x) = \sin\left(k_n x \right), \qquad n = 1,2,3,...\]
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| − | where $k_n = \frac{\pi n}{L}$. With a time dependency similar to Harmonic oscilator
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| − | \[\psi_n(t, x) = \mathrm e ^ {-i \omega_n t} \psi_n(x),\]
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| − | where $\omega_n$ and $k_n$ are connected through dispersion relation through energy $E_n$
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| − | \[{\displaystyle E_{n}=\hbar \omega _{n}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2mL^{2}}}={\frac {\hbar ^{2} k_n^2}{2m}}}.\]
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