Q102 · CSIR NET Chemistry, December 2013
Paper: CSIR NET December 2013 · Subject: Physical Chemistry · Chapter: Quantum Chemistry · Topic: Harmonic Oscillator · Marks: 2 · Difficulty: Medium
Consider a two-dimensional harmonic oscillator with potential energy $V(x,y)=\frac{1}{2}k_x x^{2}+\frac{1}{2}k_y y^{2}$ If $\psi_{nx}(x)$ and $\psi_{ny}(y)$ are the eigensolutions and $E_{nx}$ and $E_{ny}$ are the eigenvalues of harmonic oscillator problem in x and y direction with potential $\frac{1}{2}k_x x^{2}$ and $\frac{1}{2}k_y y^{2}$ respectively, the wave function and eigenvalues of the above two-dimensional harmonic oscillator problem are
(a)
\[ \begin{array}{l} \psi_{nx,ny}=\psi_{nx}(x)+\psi_{ny}(y) \\ E_{nx,ny}=E_{nx}+E_{ny} \end{array} \]
(b)
\[ \begin{array}{l} \psi_{nx,ny}=\psi_{nx}(x)\cdot\psi_{ny}(y) \\ E_{nx,ny}=E_{nx}\cdot E_{ny} \end{array} \]
(c)
\[ \begin{array}{l} \psi_{nx,ny}=\psi_{nx}(x)\cdot\psi_{ny}(y) \\ E_{nx,ny}=E_{nx}+E_{ny} \end{array} \]
(d)
\[ \begin{array}{l} \psi_{nx,ny}=\psi_{nx}(x)\cdot\psi_{ny}(y) \\ E_{nx,ny}=E_{nx}\cdot E_{ny} \end{array} \]
Answer
Answer: C ✓ checked by 4AB · confidence high
The source book printed no answer; this one was worked out and checked — see the explanation.
Explanation
For a separable Hamiltonian H = Hx + Hy, the wavefunction is the product ψnx(x)·ψny(y) and the energy is the sum Enx + Eny (c).
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