Question

The radial part of Hydrogen atom wavefunction depends on which quantum number
• \(n\) only
• \(n\) and \(l\)
• \(l\) and \(m\)
• \(n\), \(l\) and \(m\)

• A. \(n\) only
• B. \(n\) and \(l\)
• C. \(l\) and \(m\)
• D. \(n\), \(l\) and \(m\)

Hint:

Hydrogen atom wavefunction: \[ \psi=R(r)\,Y(\theta,\phi) \]
• Radial part depends on \(n,l\)
• Angular part depends on \(l,m\)

Answer 1

The Correct Option is B

Concept: The total wavefunction of the hydrogen atom can be separated into: \[ \boxed{ \psi(r,\theta,\phi)=R(r)\,Y(\theta,\phi) } \] where:
• \(R(r)\) = radial part
• \(Y(\theta,\phi)\) = angular part Different quantum numbers affect different parts of the wavefunction.

Step 1: Understand the role of quantum numbers. Hydrogen atom wavefunction depends on three quantum numbers:
• Principal quantum number \(n\)
• Azimuthal quantum number \(l\)
• Magnetic quantum number \(m\)

Step 2: Identify dependence of radial part. The radial wavefunction: \[ R(r) \] depends on:
• Principal quantum number \(n\)
• Orbital quantum number \(l\) Thus: \[ \boxed{ R(r)=R_{nl}(r) } \]

Step 3: Understand why magnetic quantum number is excluded. Magnetic quantum number \(m\):
• Determines spatial orientation.
• Affects angular part only. Thus: \[ m \] does not affect radial dependence. Hence: \[ \boxed{ \text{Radial part depends only on }n\text{ and }l } \]

Step 4: Analyze options carefully.
• \(n\) only \(\rightarrow\) incomplete
• \(n\) and \(l\) \(\rightarrow\) correct
• \(l\) and \(m\) \(\rightarrow\) incorrect
• \(n,l,m\) \(\rightarrow\) incorrect

Step 5: Choose the correct answer. Thus: \[ \boxed{ \text{Radial part depends on }n\text{ and }l } \] Hence the correct option is: \[ \boxed{(2)} \] Additional Understanding: The angular part: \[ Y(\theta,\phi) \] depends on: \[ l \text{ and } m \] while radial distribution depends on: \[ n \text{ and } l \] Final Conclusion: The radial wavefunction depends on: \[ \boxed{ n \text{ and } l } \] Hence, the correct answer is: \[ \boxed{(2)} \]