"Harmonic oscillator in quantum mechanics"의 두 판 사이의 차이

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9번째 줄: 9번째 줄:
 
<h5>harmonic oscillator in classical mechanics</h5>
 
<h5>harmonic oscillator in classical mechanics</h5>
  
* 고전역학에서의 조화진동자
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* 고전역학에서의 조화진동자([http://statphys.springnote.com/pages/5695329 고전역학에서의 가적분성] 항목 참조)
  
* [http://statphys.springnote.com/pages/5695329 고전역학에서의 가적분성] 항목 참조
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* 질량 m, frequency <math>\omega</math> 인 조화진동자<br>
 
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*  해밀토니안<br><math>H(p,q)=\frac{p^2}{2m}+\frac{m}{2}\omega^{2}q^2</math><br>
 
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*  해밀턴 방정식<br><math>\dot{q}=\partial H/\partial p=\frac{p}{m}</math><br><math>\dot{p}=-\partial H/\partial q=-m\omega^{2}q</math><br>
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*  운동방정식<br><math>\ddot{x}=-\omega^{2} x</math> 즉 <math>\ddot{x}+\omega^{2} x=0</math><br>
  
 
 
 
 
35번째 줄: 36번째 줄:
 
<h5>energy</h5>
 
<h5>energy</h5>
  
According to quantum mechanics, a harmonic oscillator that vibrates with frequency <math>\omega</math> can have energy <math>1/2\omega, (1 +1/2)\omega, (2 +1/2)\omega,(3 +1/2)\omega,\cdots</math> in units where Planck’s constant equals 1. 
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a harmonic oscillator that vibrates with frequency <math>\omega</math> can have energy <math>1/2\omega, (1 +1/2)\omega, (2 +1/2)\omega,(3 +1/2)\omega,\cdots</math> in units where Planck’s constant equals 1. <br>
  
 
The lowest energy is not zero! It’s <math>\omega/2</math>. This is called the ground state energy of the oscillator.
 
The lowest energy is not zero! It’s <math>\omega/2</math>. This is called the ground state energy of the oscillator.

2010년 9월 22일 (수) 17:14 판

introduction

 

 

 

harmonic oscillator in classical mechanics
  • 질량 m, frequency \(\omega\) 인 조화진동자
  • 해밀토니안
    \(H(p,q)=\frac{p^2}{2m}+\frac{m}{2}\omega^{2}q^2\)
  • 해밀턴 방정식
    \(\dot{q}=\partial H/\partial p=\frac{p}{m}\)
    \(\dot{p}=-\partial H/\partial q=-m\omega^{2}q\)
  • 운동방정식
    \(\ddot{x}=-\omega^{2} x\) 즉 \(\ddot{x}+\omega^{2} x=0\)

 

 

creation and annhilation operators

\(a =\sqrt{m\omega \over 2\hbar} \left(x + {i \over m \omega} p \right)\)

\(a^{\dagger} =\sqrt{m \omega \over 2\hbar} \left( x - {i \over m \omega} p \right)\)

\(H = \hbar \omega \left(a^{\dagger}a + 1/2\right)\)

\(\left[a , a^{\dagger} \right] = 1\)

 

 

energy
  • a harmonic oscillator that vibrates with frequency \(\omega\) can have energy \(1/2\omega, (1 +1/2)\omega, (2 +1/2)\omega,(3 +1/2)\omega,\cdots\) in units where Planck’s constant equals 1. 

The lowest energy is not zero! It’s \(\omega/2\). This is called the ground state energy of the oscillator.

 

 

Schrodinger equation
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path integral formulation

 

 

 

 

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