Question

Column I	Column II
(P) Associated Legendre polynomials	(III) Angular part of H atom   ($Y_\ell^m(\theta,\phi)$)
(Q) Hermite polynomials	(I) Harmonic oscillator   ($\psi_n(x)\propto e^{-x^2/2}H_n(x)$)
(R) Associated Laguerre polynomials	(IV) Radial part of H atom   ($R_{n\ell}(r)\propto e^{-r/na_0}L_{n-\ell-1}^{2\ell+1}\!\left(\tfrac{2r}{na_0}\right)$)
(S) Trigonometric functions	(II) Particle in a box model   ($\psi_n(x)=\sin\!\left(\tfrac{n\pi x}{L}\right)$)

• A. P→III, Q→I, R→IV, S→II
• B. P→III, Q→IV, R→II, S→I
• C. P→IV, Q→I, R→III, S→II
• D. P→II, Q→III, R→IV, S→I

Hint:

Hydrogen atom: Angular \(\Rightarrow\) Legendre, Radial \(\Rightarrow\) Laguerre.
Harmonic oscillator: Hermite polynomials.
Particle in a box: Sine and cosine (trigonometric).

Answer 1

The Correct Option is A

Step 1: Recall the forms of special functions in quantum mechanics.
- The angular part of the hydrogen atom wavefunction (spherical harmonics) involves \emph{Associated Legendre polynomials}.
- The harmonic oscillator solutions are expressed in terms of \emph{Hermite polynomials}.
- The radial part of the hydrogen atom wavefunction involves \emph{Associated Laguerre polynomials}.
- The particle in a box has solutions in terms of \emph{trigonometric functions} (sine and cosine).
Step 2: Match each entry.
- \(P \to III\): Associated Legendre polynomials → angular part of H atom.
- \(Q \to I\): Hermite polynomials → harmonic oscillator.
- \(R \to IV\): Associated Laguerre polynomials → radial part of H atom.
- \(S \to II\): Trigonometric functions → particle in a box.
\[ \boxed{\text{P→III, Q→I, R→IV, S→II = Option (A)}} \]