The Schrödinger Equation

Tags: #physics #quantum

Equation

$$-\dfrac{\hbar^{2}}{2m}\bigtriangledown ^{2} \psi +U\psi=E\psi = i\hbar \dfrac{\partial \psi}{\partial t} \\ H=p^2/2m+U, H\psi=E\psi \\ \psi(x,t)=\left(\sum+\int dE\right)c(E)u_E(x)\exp\left(-\frac{iEt}{\hbar}\right) \\ \displaystyle J=\frac{\hbar}{2im}(\psi^*\nabla\psi-\psi\nabla\psi^*) \\ \displaystyle\frac{\partial P(x,t)}{\partial t}=-\nabla J(x,t)$$

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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