Re: [題目] 三題量子力學問題關於creation & annih …

[xgcj (ㄨGc丁)]

先幫你做幾題好了

剩下自己做喔^^
        +
If f(a,a )is a function which can be expanded in a power series
          +
of a and a ,then show that
           +        +
(a)[a,f(a,a )]=∂f/∂a
         +
(b)[f(a,a ),a]=∂f/∂a
           +                +
(c)exp[-αa a]f(a,a )exp[αa a]=f(a*exp[α],a*exp[-α])

-----
     + n     + n-1
[a,(a ) ]=n(a )------(1)
  n   +        n-1
[a , a  ]=n(a )------(2)

                              2
exp[iA] B exp[-iA]=B+i[A,B]+(i)/(2!)[A,[A,B]]
       n
+...+(i)/(n!)[A,[A,...[A,B]...]B]------(3)

              m   + n
let f=Σg_mn a  (a )



(a)
        +               m   + n
[a,f(a,a )]==Σg_mn [a,a  (a ) ]
         m     + n
=Σg_mn a [a,(a ) ]
         m       + n-1
=Σg_mn a (n)(a )
       +
==∂f/∂a

(b)
      +   +           m  + n  +
[f(a,a ),a ]=Σg_mn [a (a ) ,a ]
         m    +   + n
=Σg_mn[a ,(a )](a )

=∂f/∂a

(c)

        +                +
exp[-αa a]f(a,a )exp[αa a]
                +   m  + n       +
=Σg_mn exp[-αa a]a (a ) exp[αa a]

                  m  + n
=Σg_mn exp[-αN]a (a ) exp[αN]
                  m                   +
=Σg_mn exp[-αN]a exp[αN]exp[-αN](a )exp[αN]

---

exp[iA] B exp[-iA]=B+i[A,B]+(i)/(2!)[A,[A,B]]
       n
+...+(i)/(n!)[A,[A,...[A,B]...]B]------(3)
            n         +n
let A=iN B=a   or B=(a )

and then using

[N,a]=-a
    +   +
[N,a ]=a

then you get the answer

第二題

你用第一題用道的等式就oK^^
----
show that
           +
[a,exp[-αa a]]=(exp[-])

----
第三題

你用exp的定義去展開
第二條式子

有計算上問題可以再提出來問版友喔!

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→ nightkid:作者抱歉(_ _)
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◆ From: 140.113.66.194
※ 編輯: xgcj            來自: 140.113.66.194       (03/04 21:09)
推 murai111:請問一下第二題一開始怎麼下手比較好
推 murai111:第三題還是不太懂...唉 真的是量力新手