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Show that the following hydrogenic wavefunctions are mutualy orthogonala.) 1s and 2s orbitalsb.) 2p_x and 2p_y

Question

Show that the following hydrogenic wavefunctions are mutualy orthogonal

a.) 1s and 2s orbitals

b.) 2p_x and 2p_y

 

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Step 1

Two different functions (ψi, ψj) are said to be orthogonal when the integral of ψi * ψj is zero.

dt 0 for i
j
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dt 0 for i j

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Step 2

(a)

The wave-functions of 1s and 2s is,

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Step 3

The wave-function of 1s and 2s depends only on r and hence, it is integrated...

3/2
Z
Zr
k
Zr
-Zr/ao 2-
1
2a02 dr
| V10,0 * 420.0dr =
4/27t
e
_
3Zr
2r2 Zr
ao
2ao dr
4/27
3Zr
3Zr
k
Z
re 2a0 dr
r'e
2ao dr
4 2T
0
32a
3
32а?
=0
k
0
42T 27 Z
27 Z3
V10,0* 20.0dr = 0 and hence is orthogonal
8 c
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3/2 Z Zr k Zr -Zr/ao 2- 1 2a02 dr | V10,0 * 420.0dr = 4/27t e _ 3Zr 2r2 Zr ao 2ao dr 4/27 3Zr 3Zr k Z re 2a0 dr r'e 2ao dr 4 2T 0 32a 3 32а? =0 k 0 42T 27 Z 27 Z3 V10,0* 20.0dr = 0 and hence is orthogonal 8 c

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