[量化]Harmonic Oscillator的波函數運算
在練習波函數的運算時碰到一些問題如下說明:
V=0, Ψ(x)=(a/π)^(1/4)‧exp[(-ax^2)/2]
檢驗total probability是否等於1
∞ ∞
S Ψ(x)*‧Ψ(x)dx = S (a/π)^(0.5)‧exp(-ax^2)dx
-∞ -∞
∞
= 2(a/π)^(0.5) S exp(-ax^2)dx (*)
0
(此時用X even power公式)
∞
S X^2n‧exp(-aX^2)dx = {[(2n)!‧π^(0.5)]/[2^(n+1)‧n!‧a^(n+0.5)]}
0
∞
n=0, S exp(-ax^2)dx = [π^(0.5)/2‧a^(0.5)] 解出 (*) 答案等於1
0
我試著帶入V=1的情況，結果也正確。
接著問題來了我帶入V=2算出來的答案卻不等於1，雖然找過許多同學檢查但還是一樣怪
我列出算式希望有能手幫個忙!!
V=2, Ψ(x)=(a/4π)^(1/4)‧[(2ax^2)-1]‧exp[(-ax^2)/2]
∞ ∞
S Ψ(x)*‧Ψ(x)dx = S (a/4π)^(0.5)‧[(2ax^2)-1]^2‧[exp(-ax^2)]dx
-∞ -∞
∞
= 2(a/4π)^(0.5) S [4a^2‧x^4－4ax^2＋1]‧[exp(-ax^2)]dx
0
∞ ∞
= 2(a/4π)^(0.5)‧{(4a^2)S x^4‧[exp(-ax^2)]dx－(4a)S x^2‧[exp(-ax^2)]dx
0 0
∞
＋ S [exp(-ax^2)]dx } (分別是n=2、n=1、n=0的even power)
0
= (a/π)^(0.5)‧{6(π/a)^(0.5)-2(π/a)^(0.5)+0.5(π/a)^(0.5)} = 9/2
符號以及數字有點雜亂，敬請見諒!!
也謝謝您看到這裡!!
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◆ From: 125.225.104.101
※ 編輯: Chopper 來自: 125.225.104.101 (06/14 22:09)
※ 編輯: Chopper 來自: 125.225.104.101 (06/14 22:52)
推
06/14 23:02, , 1F
06/14 23:02, 1F
→
06/14 23:03, , 2F
06/14 23:03, 2F
→
06/14 23:03, , 3F
06/14 23:03, 3F
推
06/14 23:12, , 4F
06/14 23:12, 4F
→
06/14 23:20, , 5F
06/14 23:20, 5F