Question

Calculate the value of the constant N in the given radial function when this radial function
is normalized. The radial function is defined as follows:
R(r)= Nr’e-ar?
Here, N is the normalized constant and a is the positive constant.
1
1
(A)
3/2
a
45
2
(B)
3/2
a
Vr
a
1
(C)
,32
(
2W
1
(D)
(32​

Answer 1

The value of the constant N when the radial function is normalized is given by option (C) N = sqrt(a^3/2).

Given:

Radial function R(r) = Nr'e^(-ar)

To Find:

the value of the constant N after normalizing the radial function.

Solution:

The normalized radial function is defined as follows:

∫(0 to ∞) |R(r)|^2 r^2 dr = 1

Substituting R(r) in the above equation, we get:

∫(0 to ∞) |Nr'e^(-ar)|^2 r^2 dr = 1

=> N^2 ∫(0 to ∞) r^2 e^(-2ar) dr = 1

Using the formula ∫(0 to ∞) x^n e^(-ax) dx = n!/a^(n+1), we get:

N^2 * 2!/a^3 = 1

=> N = sqrt(a^3/2)

Therefore, the value of the constant N when the radial function is normalized is given by option (C) N = sqrt(a^3/2).

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