bei48482_FM

Quantum Theory of the Hydrogen Atom 213

Probability of Finding the Electron

The probability densityof the electron at the point r, , is proportional to ^2 , but

the actual probabilityof finding it in the infinitesimal volume element dVthere is ^2 dV.

In spherical polar coordinates (Fig. 6.9),

dV(dr) (r d) (rsin d)

Volume element r^2 sin dr dd (6.24)

5 a 0 10 a 0 15 a 0

R

(nl

r)

R

(nl

r)

Rnl

(r)

3 s

3 p 3 d

2 s

2 p

1 s

r

Figure 6.8The variation with distance from the nucleus of the radial part of the electron wave function

in hydrogen for various quantum states. The quantity a 0  4  0 ^2 me^2 0.053 nm is the radius of

the first Bohr orbit.

Figure 6.9Volume element dVin spherical polar coordinates.

dV = r^2 sin θ dr dθ dφ

r sin θ

x r sin θ dφ

y

z

dr r dθ

dθ

dφ

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