※ 引述《cooljeff19 ()》之銘言： : ※ 引述《keeptrying (讓我加入你們的談話)》之銘言： : : 想請問一下各位一個問題: : : a.show for a harmonic oscillator in the v=0 state that <K>=<V> where V is the : : potential energy and K is the kinetic energy .(Hint:one way to proceed is to : : calculate <K> and use the fact that <K>+<V>=E) : : b.do you think that this will also be true for the other energy eigenfunction? : : check it out for v=1. : : ============================================================================= : : 麻煩各位替我解答一下～感謝^_^ : 中間計算過程好繁複阿XDDDD 碰到Quantum harmonic oscillator 通常用ladder operator算會方便許多 許多書本都有介紹這方法 不然看wiki也行 http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator x = \sqrt{hbar/(2mw)} (a^+ + a) p = i\sqrt{hbar mw/2} (a^+ - a) V=kx^2/2 = hbar w/4(a^+ + a)^2 T=p^2/2m = -hbar w/4(a^+ - a)^2 H = hbar w/2(a^+a + aa^+) 對eigenfunction 取期望值時我們有 <V> = hbar w/4 <(a^+ + a)^2> = hbar w/4 <a^+a + aa^+> <K> = -hbar w/4 <(a^+ - a)^2> = hbar w/4 <a^+a + aa^+> 故 <K> = <V> 也可從下列算式驗證 <H> = <V+T> = <V> + <T> = hbar w/2<a^+a + aa^+> = hbar w (n+1/2) 這題也可用牛刀來殺 從(Quantum) viral theorem可知 若V有以下特性 V(ar)=a^n V(r) 則2<T>=n<V> 對harmonic oscillator而言n=2 故<T>=<V> 有興趣的可去驗證氫原子 此時n=-1 故2<T>=-<V> --

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