2. For the ground (1s) state of H atom, the radius of the orbit predicted by the

Question

2. For the ground (1s) state of H atom, the radius of the orbit predicted by the Bohr’s theory is equal to the most probable electron-nucleus distance from the wave function-based quantum theory. (a) Check if you can give more credit to the Bohr’s theory, i.e. if there are other nl states (2s, etc. - consider only those with n = 2,3) with the same property: their Bohr-orbit radius rB equals the most probable electron-nucleus distance rmp for this state. Show this by calculation: - Plot the radial distribution function Rnl2 r2; - Obtain the most probable electron-nucleus distance for each possible l value; - Compare it with the orbit radius predicted by the Bohr’s theory for this state and conclude for which nl state (if any) the results of the two theories are same. (b) For those nl states found in (a) to have rB = rmp, calculate the probabilities for the electron to be inside and outside the corresponding Bohr orbit. How do these probabilities compare with those for the 1s state (already obtained in Lectures), and is the probability “outside” still larger than “inside”?

Explanation / Answer

Radial distributewave functions  Rnl2 r2 for the hydrogen like atom

n=1 , l=0 , specific notation=1s ,   Rnl2 r2= 2(z/a)3/2.e-zr/a

n=2, l=0, specific notation =2s, 1/root2(z/a)3/2 (1-zr/2a)e-zr/2a

n=2, l=1 specific notation =2p 1/root24(z/a)5/2re-zr/2a

n=3 l=0 specific notation =3s 2/3root3(z/a)3/2(1-2zr/3a+2z2r2/27a2)e-Zr/3a

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