32 lines
1.2 KiB
Markdown
32 lines
1.2 KiB
Markdown
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$$-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi$$
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- Time Independent
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- $\psi$ is the wave function
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Quantum counterpart of Newton's second law in classical mechanics
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$$F=ma$$
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[From](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation)
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Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of a [wave function](https://en.wikipedia.org/wiki/Wave_function), the quantum-mechanical characterization of an isolated physical system.
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[From](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation)
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[Time–Independent Schrödinger Equation](https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_4.pdf)
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[RadialEquation.pdf](https://physics.weber.edu/schroeder/quantum/RadialEquation.pdf)
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## Hamiltonian
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- Operator
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- Total energy of a system
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- Kinetic + Potential energy
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$$\hat{H}=\hat{T}+\hat{V}$$
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- $\hat{V}$
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- Potential Energy
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- $\hat{T}=\frac{\hat{p}\cdot\hat{p}}{2m}=-\frac{\hbar^2}{2m}\nabla^2$
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- Kinetic Energy
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- $\hat{p}=-i\hbar\nabla$
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- Momentum operator
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## Wavefunction Normalisation
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- Adds up to 1 under the curve
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