Re: [問題] 量子化學
※ 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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