We must now determine how to calculate matrix elements of the form: (OnzOm). One suggested approach is to again use a recursive method. Given that o, = x"exp(-ax²), find dn in terms of øn-1 and øn+1. That is, evaluate: dộn d -x"e dr -az? Cnn-1"-le-az? + Dn.n+12"+1e-az² (6) dx dộn = Cn,n-1Øn-1 + Dn,n+1Øn+1• da (7) 1. Cnn-1 = 2. Dn.n+1 = From the above expression we can then determine the second derivative in terms of the original set of ø, by using this equation twice. That is, prove that: don dr2 dộn-1 = Cn,n-1 dx dộn+1 + Dn,n+1 dx (8) don da? = Cn,n-1 · Cn-1,n–20n-2 (9) +Cn,n-1· Dn-1,nøn +Dn,n+1 • Cn+1,n$n +Dnn+1• Dn+1,n+2Øn+2 = Pn.n-20n-2Qn,nÓn + Rn,n+2$n+2 (10) da2 n Cn,n-1 Dnn+1 Pn,n-2 Qn,n Rn,n+2 1 2 3 4 TABLE II. Using the above equations complete the table
We must now determine how to calculate matrix elements of the form: (OnzOm). One suggested approach is to again use a recursive method. Given that o, = x"exp(-ax²), find dn in terms of øn-1 and øn+1. That is, evaluate: dộn d -x"e dr -az? Cnn-1"-le-az? + Dn.n+12"+1e-az² (6) dx dộn = Cn,n-1Øn-1 + Dn,n+1Øn+1• da (7) 1. Cnn-1 = 2. Dn.n+1 = From the above expression we can then determine the second derivative in terms of the original set of ø, by using this equation twice. That is, prove that: don dr2 dộn-1 = Cn,n-1 dx dộn+1 + Dn,n+1 dx (8) don da? = Cn,n-1 · Cn-1,n–20n-2 (9) +Cn,n-1· Dn-1,nøn +Dn,n+1 • Cn+1,n$n +Dnn+1• Dn+1,n+2Øn+2 = Pn.n-20n-2Qn,nÓn + Rn,n+2$n+2 (10) da2 n Cn,n-1 Dnn+1 Pn,n-2 Qn,n Rn,n+2 1 2 3 4 TABLE II. Using the above equations complete the table
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Transcribed Image Text:We must now determine how to calculate matrix elements of the form: (ØnzOm). One
suggested approach is to again use a recursive method. Given that o, = x"exp(-ax²), find
n in terms of øn-1 and øn+1. That is, evaluate:
dr
døn
d
Cn.n=1x"-1e-az²
+ Dnn+12"+1e-ar²
(6)
dr
dx
dộn
= Cn,n-1Øn-1 + Dn,n+1$n+1•
da
(7)
1. Cn.n-1 =
2. Dn.n+1 =
From the above expression we can then determine the second derivative
in terms of
the original set of øn by using this equation twice. That is, prove that:
don
dx²
dộn-1
= Cn,n-1 dx
don+1
+ Dn,n+1
dx
(8)
don
Cn.n-1· Cn-1,n-2ºn-2
(9)
dx?
+Cn,n-1 · Dn-1,nÓn
+Dn,n+1 • Cn+1,n$n
+Dnn+1• Dn+1.n+2Øn+2
= Pn.n-2ºn-2Qn,nØn + Rn,n+2Øn+2
(10)
dx?
n Cn,n-1 Dn,n+1 Pn,n-2 Qn,n|Rn,n+2|
1
2
3
4
5
TABLE II. Using the above equations complete the table
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