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Quantum Mechanics
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HYDROGENMOLECULE ION
-e
+e
+e
r1
r2
R
HYDROGENMOLECULE ION
𝐻 = −ℏ2
2𝑚𝛻2 −
𝑒2
4𝜋𝜖0(1
𝑟1+
1
𝑟2)
𝜓 = 𝐴 𝜓0 𝑟1 − 𝜓0 𝑟2
𝜓0 𝑟 =1
𝜋𝑎3𝑒− 𝑟 𝑎
1 = 𝜓−2𝑑3𝑟
1 = 𝐴 2 𝜓0(𝑟12𝑑3𝑟 + 𝜓0(𝑟2
2𝑑3𝑟 − 2 𝜓0(𝑟1 𝜓0(𝑟2 𝑑3𝑟
1 1𝜓0
I
𝐴 2 =1
2(1 − 𝐼
𝐼 = 𝜓0(𝑟1 𝜓0(𝑟2 𝑑3𝑟
=1
𝜋𝑎3 𝑒−(𝑟1+𝑟2 𝑎 𝑑3𝑟
𝒓𝟐 = 𝒓𝟐 + 𝑹𝟐 − 𝟐𝒓𝑹𝒄𝒐𝒔𝜽
𝒓𝟏 = 𝒓
𝑹
𝐼 =1
𝜋𝑎3 𝑒− 𝑟 𝑎𝑒− 𝑟2+𝑅2−2𝑟𝑅𝑐𝑜𝑠𝜃 𝑎 𝑟2𝑠𝑖𝑛𝜃𝑑𝑟𝑑𝜃𝑑𝜙
𝜙 2𝜋
𝜃 𝑦 = 𝑟2 + 𝑅2 − 2𝑟𝑅𝑐𝑜𝑠𝜃. 𝑑 𝑦2 = 2𝑦𝑑𝑦 = 2𝑟𝑅𝑠𝑖𝑛𝜃𝑑𝜃
0
𝜋
𝑒− 𝑟2+𝑅2−2𝑟𝑅𝑐𝑜𝑠𝜃 𝑎 𝑠𝑖𝑛𝜃𝑑𝜃
𝜃 = 𝜋
𝜃 = 0 𝑦 = 𝑟 − 𝑅
𝑦 = 𝑟 + 𝑅=
1
𝑟𝑅 𝑟−𝑅
𝑟+𝑅
𝑒− 𝑦 𝑎 𝑦𝑑𝑦
= −𝑎
𝑟𝑅𝑒−
𝑟−𝑅𝑎 𝑟 + 𝑅 + 𝑎 − 𝑒−
𝑟−𝑅𝑎 ( 𝑟 − 𝑅 + 𝑎
0
∞
𝑒−𝑘𝑥𝑥𝑛 𝑑𝑥 =𝑛!
𝑘𝑛
𝐼 =2
𝑎2𝑅−𝑒− 𝑅 𝑎
0
∞
𝑟 + 𝑅 + 𝑎 𝑒−2𝑟𝑎 𝑟𝑑𝑟 + 𝑒−
𝑅𝑎
0
𝑅
𝑅 − 𝑟 + 𝑎 𝑟𝑑𝑟 + 𝑒 𝑅 𝑎 𝑅
∞
𝑟 − 𝑅 + 𝑎 𝑒−2 𝑟 𝑎𝑟𝑑𝑟
𝐼 = 𝑒− 𝑅 𝑎 1 +𝑅
𝑎+1
3
𝑅
𝑎
2
𝐻 = −ℏ2
2𝑚𝛻2 −
𝑒2
4𝜋𝜖0(1
𝑟1+
1
𝑟2)
ö 𝐻𝜓 = 𝐸𝜓
−ℏ2
2𝑚𝛻2 −
𝑒2
4𝜋𝜖0𝑟1,2𝜓0 𝑟1,2 = 𝐸1𝜓0(𝑟1,2)
𝐻𝜓− = 𝐴 −ℏ2
2𝑚𝛻2 −
𝑒2
4𝜋𝜖0(1
𝑟1+1
𝑟2𝜓0 𝑟1 − 𝜓0 𝑟2 = 𝐸1𝜓 − 𝐴(
𝑒2
4𝜋𝜖0 𝜓0 𝑟1𝑟1
−𝜓0 𝑟2𝑟2
𝜓0(𝑟1,2
𝐻 = 𝐸1 − 𝑎 𝐴 2(𝑒2
4𝜋𝜖0 𝜓0 𝑟1
1
𝑟2𝜓0 𝑟1 − 𝜓0 𝑟1
1
𝑟1𝜓0 𝑟2
𝐷 = 𝜓0 𝑟1𝑎
𝑟2𝜓0 𝑟1
𝑋 = 𝜓0 𝑟1𝑎
𝑟1𝜓0 𝑟2
𝐷 =𝑎
𝑅− (1 +
𝑎
𝑅 𝑒−2 𝑅 𝑎
𝑋 = (1 +𝑅
𝑎 𝑒− 𝑅 𝑎
𝐻 = 1 + 2𝐷 − 𝑋
1 − 𝐼𝐸1
𝐼 = 𝑒− 𝑅 𝑎 1 +𝑅
𝑎+1
3
𝑅
𝑎
2
𝑉𝑝𝑝 =𝑒2
4𝜋𝜖0
1
𝑅= −
2𝑎
𝑅𝐸1
𝐸𝑡𝑜𝑡 = 𝐻 + 𝑉𝑝𝑝
𝐸𝑡𝑜𝑡 = 1 + 2𝐷 − 𝑋
1 − 𝐼−2𝑎
𝑅𝐸1
𝐹 𝑥 =𝐸𝑡𝑜𝑡−𝐸1
𝐹 𝑥 = −1
𝐸11 + 2
𝐷 − 𝑋
1 − 𝐼−2𝑎
𝑅𝐸1 =
2𝑎
𝑅− 1 − 2
𝐷 − 𝑋
1 − 𝐼
𝐹 𝑥 = −1 +2
𝑥− 2
1𝑥 − 1 +
1𝑥 𝑒−2𝑥 − (1 + 𝑥 𝑒−𝑥
1 − 𝑒−𝑥(1 + 𝑥 +𝑥2
3
= −1 +2
𝑥
1𝑥 − 1 + 𝑥 +
𝑥2
3 𝑒−𝑥 − 1 + 𝑥 + 1 𝑒−2𝑥 + (𝑥 + 𝑥2 𝑒−𝑥
1 − 𝑒−𝑥(1 + 𝑥 +𝑥2
3
𝐹 𝑥 = −1 +2
𝑥
2𝑥2
3− 1 𝑒−𝑥 + (1 + 𝑥 𝑒−2𝑥
1 − 𝑒−𝑥(1 + 𝑥 +𝑥2
3
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 1 2 3 4 5 6 7 8 9
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