Question

2. A classic pair of complementary operators are £ and Pz. In class we showed these operators obeyed the Heisenberg relationship for the harmonic oscillator. Show they also obey this relationship for the particle in the box. Note: your answer will be a fiunction of the quantum number n. Show the relationship holds for n - I and any larger value ofn.

Answer 1

Energy of a particle in a box (1D infinite potential) is:

$E_n = \frac{n^2h^2}{8mL^2} = \frac{n^2\hbar^2\pi^2}{2mL^2}$

but $E_n = \frac{p_n^2}{2m}$

so, $\frac{p_n^2}{2m} = \frac{n^2\hbar^2\pi^2}{2mL^2}$

=> $p_n^2 = \frac{n^2\hbar^2\pi^2}{L^2}$

=> $p_nL = n\hbar \pi$

since $n\hbar \pi \geq \frac{\hbar}{2}$ for every n [n = 1,2,3,4,.....] and 0 < x < L

therefore, $\Delta x \Delta p \leq \frac{\hbar}{2}$ is satisfied for every n.