Harmonic Oscillator Quantum Equation Formula Latex Code and Summary

Harmonic Oscillator Quantum

Tags: #physics #quantum

Equation

$$H=\frac{p^2}{2m}+\frac{1}{2} m\omega^2 x^2= \frac{1}{2} \hbar\omega+\omega A^\dagger A \\ A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} \\ A^\dagger=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x-\frac{ip}{\sqrt{2m\omega}} \\ HAu_E=(E-\hbar\omega)Au_E \\ u_n=\frac{1}{\sqrt{n!}}\left(\frac{A^\dagger}{\sqrt{\hbar}}\right)^nu_0 \\ u_0=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right) \\ E_n=( \frac{1}{2} +n)\hbar\omega$$

Latex Code

                                 H=\frac{p^2}{2m}+\frac{1}{2} m\omega^2 x^2= \frac{1}{2} \hbar\omega+\omega A^\dagger A \\
            A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} \\
            A^\dagger=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x-\frac{ip}{\sqrt{2m\omega}} \\
            HAu_E=(E-\hbar\omega)Au_E \\
            u_n=\frac{1}{\sqrt{n!}}\left(\frac{A^\dagger}{\sqrt{\hbar}}\right)^nu_0 \\
            u_0=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right) \\
            E_n=( \frac{1}{2} +n)\hbar\omega

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Introduction

$H=\frac{p^2}{2m}+\frac{1}{2} m\omega^2 x^2= \frac{1}{2} \hbar\omega+\omega A^\dagger A \\ A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} \\ A^\dagger=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x-\frac{ip}{\sqrt{2m\omega}} \\ HAu_E=(E-\hbar\omega)Au_E \\ u_n=\frac{1}{\sqrt{n!}}\left(\frac{A^\dagger}{\sqrt{\hbar}}\right)^nu_0 \\ u_0=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right) \\ E_n=( \frac{1}{2} +n)\hbar\omega$

Latex Code

            H=\frac{p^2}{2m}+\frac{1}{2} m\omega^2 x^2= \frac{1}{2} \hbar\omega+\omega A^\dagger A \\
            A=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x+\frac{ip}{\sqrt{2m\omega}} \\
            A^\dagger=\sqrt{\mbox{$\frac{1}{2}$}m\omega}x-\frac{ip}{\sqrt{2m\omega}} \\
            HAu_E=(E-\hbar\omega)Au_E \\
            u_n=\frac{1}{\sqrt{n!}}\left(\frac{A^\dagger}{\sqrt{\hbar}}\right)^nu_0 \\
            u_0=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\exp\left(-\frac{m\omega x^2}{2\hbar}\right) \\
            E_n=( \frac{1}{2} +n)\hbar\omega

Explanation

Latex code for the Parity Equation. If the wavefunction is split into even and odd functions, it can be expanded into eigenfunctions of P. I will briefly introduce the notations in this formulation.

$H$ : Hamiltonian
$A$ : Raising ladder operator
$A^\dagger$ : Lowering ladder operator
$u_0$ : Eigenfunction for $Au_0=0$ holds

Discussion

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Hugo Chandler

Hoping to make it through this exam successfully.

2023-02-28 00:00
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Mark Jackson reply to Hugo Chandler

Gooood Luck, Man!

2023-03-06 00:00:00.0
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Brandon Campbell

I hope the odds are in my favor to pass this exam.

2023-04-28 00:00
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Eric Gonzalez reply to Brandon Campbell

Nice~

2023-05-27 00:00:00.0
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Norma Harvey

The goal is clear: pass this exam.

2023-11-01 00:00
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Paula Burton reply to Norma Harvey

Nice~

2023-11-20 00:00:00.0
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