The Schrödinger Equation
Tags: #physics #quantumEquation
$$-\dfrac{\hbar^{2}}{2m}\bigtriangledown ^{2} \psi +U\psi=E\psi = i\hbar \dfrac{\partial \psi}{\partial t} \\ H=p^2/2m+U, H\psi=E\psi \\ \psi(x,t)=\left(\sum+\int dE\right)c(E)u_E(x)\exp\left(-\frac{iEt}{\hbar}\right) \\ \displaystyle J=\frac{\hbar}{2im}(\psi^*\nabla\psi-\psi\nabla\psi^*) \\ \displaystyle\frac{\partial P(x,t)}{\partial t}=-\nabla J(x,t)$$Latex Code
-\dfrac{\hbar^{2}}{2m}\bigtriangledown ^{2} \psi +U\psi=E\psi = i\hbar \dfrac{\partial \psi}{\partial t} \\
H=p^2/2m+U, H\psi=E\psi \\
\psi(x,t)=\left(\sum+\int dE\right)c(E)u_E(x)\exp\left(-\frac{iEt}{\hbar}\right) \\
\displaystyle J=\frac{\hbar}{2im}(\psi^*\nabla\psi-\psi\nabla\psi^*) \\
\displaystyle\frac{\partial P(x,t)}{\partial t}=-\nabla J(x,t)
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Introduction
Equation
Latex Code
-\dfrac{\hbar^{2}}{2m}\bigtriangledown ^{2} \psi +U\psi=E\psi = i\hbar \dfrac{\partial \psi}{\partial t} \\
H=p^2/2m+U, H\psi=E\psi \\
\psi(x,t)=\left(\sum+\int dE\right)c(E)u_E(x)\exp\left(-\frac{iEt}{\hbar}\right) \\
\displaystyle J=\frac{\hbar}{2im}(\psi^*\nabla\psi-\psi\nabla\psi^*) \\
\displaystyle\frac{\partial P(x,t)}{\partial t}=-\nabla J(x,t)
Explanation
Latex code for the Schrödinger Equation. I will briefly introduce the notations in this formulation.
: The momentum operator
: The position operator
: The energy operator
: Mass
: Potential Energy
: Total Energy
: The current density
: Conservation law
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