Question

The value of \( \frac{1}{r} \) in the \( \psi_{1,0,0}(r,\theta,\phi) \) state of a hydrogen atom is (\( a_0 \) is Bohr radius):

• A. \( a_0 \)
• B. \( \frac{1}{a_0} \)
• C. \( \frac{1}{2a_0} \)
• D. \( \frac{1}{r} \)

Answer 1

The Correct Option is B

The value of \( \frac{1}{r} \) in the \( \psi_{1,0,0}(r,\theta,\phi) \) state of a hydrogen atom is:
 

\[ \psi_{1,0,0}(r,\theta,\phi) = \frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0} \] In this equation, the wave function \( \psi_{1,0,0}(r,\theta,\phi) \) describes the ground state (1s state) of the hydrogen atom. The Bohr radius \( a_0 \) is a fundamental constant and represents the most probable distance of the electron from the nucleus in the ground state.

Correct Answer: 

\( \frac{1}{a_0} \)

To solve for \( \frac{1}{r} \), observe the exponential form of the wave function. The term \( e^{-r/a_0} \) represents a decaying function with respect to \( r \), and this function will approach 0 as \( r \) becomes very large. The Bohr radius \( a_0 \) is the scale factor that defines the characteristic size of the hydrogen atom. Therefore, the inverse of the radial coordinate \( r \) in this state is given by \( \frac{1}{a_0} \), which is consistent with the ground state of the hydrogen atom.