The Schrödinger Equation Equation Formula Latex Code and Summary

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

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

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)

Explanation

Latex code for the Schrödinger Equation. I will briefly introduce the notations in this formulation.

$p_{\rm op}=-i\hbar\nabla$ : The momentum operator
$x_{\rm op}=i\hbar\nabla_p$ : The position operator
$E_{\rm op}=i\hbar\partial/\partial t$ : The energy operator
$m$ : Mass
$U$ : Potential Energy
$E$ : Total Energy
$J$ : The current density
$\displaystyle\frac{\partial P(x,t)}{\partial t}=-\nabla J(x,t)$ : Conservation law

Discussion

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Julia Butler

Hoping to make it through this exam successfully.

2023-04-26 00:00
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Joel Harvey reply to Julia Butler

Best Wishes.

2023-04-29 00:00:00.0
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Vance Shepard

My mind is set on passing this test.

2023-03-29 00:00
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Theresa Rivera reply to Vance Shepard

Nice~

2023-04-23 00:00:00.0
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Elliot Stone

Here's to hoping I pass this exam with ease.

2023-11-22 00:00
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Jacqueline Snyder reply to Elliot Stone

You can make it...

2023-11-24 00:00:00.0
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