quantum mechanics - Why does $Delta x Delta p_x$ increase linearly with $n$ for stationary states?

Harmonic Oscillator

• $\displaystyle \Delta x\Delta p_x = \hbar \left(n+\frac{1}{2}\right)$

Particle in a box

• $\displaystyle \Delta x\Delta p_x = \frac{\hbar}{2} \sqrt{\frac{n^2\pi^2}{3}-2}$

Similarly, the cone potential $V(x)=|x|$ and the exponential potential $V(x)=\exp(|x|)$ have been shown to have $\Delta x\Delta p_x$ grow linearly with $n$.

We notice that for small n the product is of the same order of magnitude as $\hbar$ and for large n it grows linearly with it:

• Is that behavior expected? If yes, then why?


• Is this a general behavior for any stationary states of any system?