Which of the following pairs of operators commute?

The de-Broglie wavelength for a He atom travelling at 1000 m/s (typical speed at room temperature) is

A. 99.7 × 10^-12 m

B. 199.4 × 10^-12 m

C. 199.4 × 10^-18 m

D. 99 × 10^-6 m

The vibrational partition function for a molecule which can be described as a simple harmonic oscillator with fundamental frequency $$\nu $$ is given by

A. $$\exp \left( { - \frac{{h\nu }}{{{K_B}T}}} \right)$$

B. $${\left[ {1 - \exp \left( { - \frac{{h\nu }}{{{K_B}T}}} \right)} \right]^{ - 1}}$$

C. $$\exp \left( { - \frac{{h\nu }}{{{K_B}T}}} \right){\left[ {1 - \exp \left( { - \frac{{h\nu }}{{{K_B}T}}} \right)} \right]^{ - 1}}$$

D. $$\exp \left( { - \frac{{h\nu }}{{2{K_B}T}}} \right){\left[ {1 - \exp \left( { - \frac{{h\nu }}{{{K_B}T}}} \right)} \right]^{ - 1}}$$

The wave function for a quantum mechanical particle in a one-dimensional box of length a is given by $$\psi = A\sin \frac{{\pi x}}{a}.$$   The value of A for a box of length 200 nm is

A. 4 × 10⁴ (nm)²

B. 10$$\sqrt 2 $$ (nm)^1/2

C. $$\sqrt 2 $$ /10 (nm)^-1/2

D. 0.1 (nm)^-1/2

First order perturbation correction $$\Delta \varepsilon _n^{\left( 1 \right)}$$  to energy level $${\varepsilon _n}$$ of a simple harmonic oscillator due to the anharmonicity perturbation $$\gamma {x^3}$$ is given by

A. $$\Delta \varepsilon _n^{\left( 1 \right)} = \gamma $$

B. $$\Delta \varepsilon _n^{\left( 1 \right)} = {\gamma ^2}$$

C. $$\Delta \varepsilon _n^{\left( 1 \right)} = {\gamma ^{ - 1}}$$

D. $$\Delta \varepsilon _n^{\left( 1 \right)} = 0$$