Question

SMA #8: Bohr and Schrödinger Models of Hydrogen Here we investigate the relationship between the Schrödinger and Bohr models of hydrogen-like atoms, following our work in class on both 9 1. Using the appropriate Schrödinger wavefunctions, compute the most probable electron-proton radii (i.e., distances) for 1s, 2p, and 3d states. Do these agree with the corresponding Bohr radii? Hint #1: Remember to maximize the "radial distribution function" P(r) = [rR(r)], i.e., to include the radial Jacobian factor (r2) in your calculations. Hint #2: You do NOT need to properly normalize the wavefunctions to compute most-probable locations 2. Now normalize the 1s, 2p, and 3d radial wavefunctions, R(r). (deleted from problem set) Hint #3: You don't need to consider the angular functions when normalizing radial functions. 3. Compute <r> for 1s, 2p, and 3d states. The value <r>is = (3/2)ao where ao is the Bohr radius Does this agree with the corresponding most probable radius? If so, why? And if not, why not?

Answer 1

Multiple questions as per guideline only First question will be answerd :

1. For Hydrogen like atom , most probable radii for a electron in a orbital and proton can be computed using non-normalized wave-functions. We find this  radius of the hydrogen-like atom  by solving dP/dr = 0. If there are several maxima, then we choose the one corresponding to the greatest amplitude.

1s : Bohr radius is a₀/Z ; our calculated value for most probable radii matches with it.

1 most probable electron-foroton sadii Is at most probable radii 08 a Pir) = 0 : Pers & 182 RE 1 Pers for is = 42 822228/9 we willl ull non-normalized wave function, Pros (is) = g2 = 220/90 depo) = (2x - 2272) €235/40 =0 ar Imp. (15) a 90/2)

2p : The most probable distance for the 2p is 4a₀/Z ,it in in agreement with the simple Bohr model.

P(r) = Ir²R²I = r⁴ e ^−2Zr/2a₀ = r⁴ e ^−Zr/a₀

We use, (dP/dr) = r³ e^-Zr/a₀ ( 4 − (Zr / a₀ ) = 0

or  (4 − (Zr / a₀ ) = 0   

or 4 = Zr / a₀

or r_2p (m.p.) = 4a₀/Z

3d : The most probable radii for the 3d is 9a₀/Z ,it in in agreement with the simple Bohr model.

P(r) = Ir²R²I = r⁶ e ^−2Zr/3a₀

We use, (dP/dr) = r⁵ e^-2Zr/3a₀ ( 6 − (2Zr / 3 a₀ )) = 0

( 6 − (2Zr / 3 a₀ )) = 0 or 6 = 2Zr / 3 a₀

r_3d (m.p.) = 9a₀/Z