Question

-\frac{\hbar^2}{2mr}\frac{\partial^2}{\partial r^2}(rR(r)) + \frac{\hbar^2l(l+1)}{2mr^2}R(r) - \frac{e^2}{4\pi\epsilon_0r}R(r) = ER(r) R(r) = (1 + br)e^{-ar} a) Find a and b that such the R(r) solution shown above works, you should find 2 independent solutions (plug R(r) into the radical Schrodinger equation) b) Normalized solution in a. c) Find <r> for both solution d) find values of r where radical density is maximal for both solutions, and find a cubic function in r for one of the solutions

          -\frac{\hbar^2}{2mr}\frac{\partial^2}{\partial r^2}(rR(r)) + \frac{\hbar^2l(l+1)}{2mr^2}R(r) - \frac{e^2}{4\pi\epsilon_0r}R(r) = ER(r)

R(r) = (1 + br)e^{-ar}
a) Find a and b that such the R(r) solution shown above works, you should find 2 independent solutions (plug R(r) into the radical Schrodinger equation)
b) Normalized solution in a.
c) Find <r> for both solution
d) find values of r where radical density is maximal for both solutions, and find a cubic function in r for one of the solutions

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-(ħ^2)/(2mr)(∂^2)/(∂r^2)(rR(r)) + (ħ^2l(l+1))/(2mr^2)R(r) - (e^2)/(4πϵ0r)R(r) = ER(r)

R(r) = (1 + br)e^-ar
a) Find a and b that such the R(r) solution shown above works, you should find 2 independent solutions (plug R(r) into the radical Schrodinger equation)
b) Normalized solution in a.
c) Find <r> for both solution
d) find values of r where radical density is maximal for both solutions, and find a cubic function in r for one of the solutions

Added by Josep E.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Stephen Zaffke

Step 1

To find a and b, we need to plug R(r) into the radial Schrodinger equation: h^2 O^2 rR(r) 2mr Or^2 h^2l(l+1) 2mr^2 R(r) = ER(r) 4EOT R(r) Simplifying the equation, we get: h^2 O^2 rR(r) = (ER(r) - h^2l(l+1) 2mr^2) R(r) Now, let's separate the variables by Show more…

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Text: h^2 O^2 rR(r) 2mr Or^2 h^2l(l+1) 2mr^2 R(r) = ER(r) 4EOT R(r) = (1+ br)e-ar a) Find a and b such that the R(r) solution shown above works. You should find 2 independent solutions (plug R(r) into the radial Schrodinger equation). b) Normalize the solution in a. c) Find <r> for both solutions. d) Find the values of r where the radial density is maximal for both solutions, and find a cubic function in r for one of the solutions.

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