To show that the most probable values of r for the n-1=l states of hydrogen are r=n^2a, we can use the radial probability distribution function for the hydrogen atom. This function represents the probability of finding the electron at a distance r from the nucleus, and it is given by:
P(r) = 4πr^2R^2(r)
Where R(r) is the radial wave function and n, l, and m are the quantum numbers. For the n-1=l states, the radial wave function can be written as:
R(r) = (2/n^2a^3)^(1/2) * (n-l-1)! / (2n(n+l)!) * (2r/n)^l * e^(-r/n)
Substituting this into the probability distribution function, we get:
P(r) = 4πr^2 * (2/n^2a^3) * (n-l-1)!^2 / (2n(n+l)!)^2 * (2r/n)^(2l) * e^(-2r/n)
To find the most probable value of r, we need to maximize this function. Taking the derivative with respect to r and setting it equal to 0, we get:
d/dr(P(r)) = 8πr * (2/n^2a^3) * (n-l-1)!^2 / (2n(n+l)!)^2 * (2r/n)^(2l-1) * (1-2l+2r/n) * e^(-2r/n) = 0
Solving for r, we get:
r = n^2a * (1-2l)
Since n and l are both positive integers, the only possible value for r that satisfies this equation is r=n^2a. This means that for the n-1=l states of hydrogen, the most probable value of r is n^2a. This matches the Bohr's model prediction and supports the idea that the Bohr radius is the most probable distance of the electron from the nucleus in these states.
In addition, we can also see that as the quantum number n increases, the most probable value of r also increases. This makes sense as the electron's energy increases with higher values of n, allowing it to occupy larger orbits and have a higher probability of being found at a greater
