Find , for the nth stationary state of the harmonic oscillator, using the method of Example $2.5$. Check that the uncertainty principle is satisfied.

Beyond conventional turning points, the stationary states (states of definite energy) have nonzero values. A classical oscillator is least likely to be discovered in the ground state at the location of the minimum of the potential well, where a quantum oscillator is most likely to be found.

The value of $⟨x⟩$and $⟨\rho ⟩$will be

So,

$⟨x⟩=\sqrt{\frac{\xcancel{\square }}{2m\omega }}\int {\psi }_n^{*}\left(a_{-}+a_{-}\right){\psi }_{n}dx$

But, $a{\psi }_n=\sqrt{n+1}{\psi }_{n+1},a{\psi }_n-\sqrt{n}{\psi }_{n-1}$

$⟨x⟩=\sqrt{\frac{\xcancel{\square }}{2m\omega }}\left[\sqrt{n+1}\int {\psi }_n^{*}{\psi }_{n1}dx+\sqrt{n}\int {\psi }_n^{*}{\psi }_{n-1}dx\right]$

$=0$

And the value of$⟨p⟩=m\frac{d⟨x⟩}{dt}=0$

According to the value of $x$, the value of

$x^2=\frac{\xcancel{\square }}{2m\omega }{\left(a_{+}+a^2\right)}^2$

$=\frac{\xcancel{\xcancel{\square }}}{2m\omega }\left({a_{+}}^2+a_{+}{a}_{-}+a_{-}{a}_{+}+{a_{-}}^2\right)$

$=\frac{\xcancel{\square }}{2m\omega }(2n+1)$

$=\left(n+\frac{1}{2}\right)m\xcancel{\square }\omega$

Hence, the value is

Again, the value of,

$p^2=-\frac{m\xcancel{\square }\omega }{2}{\left(a_{-}-a_{-}\right)}^2$

$=-\frac{m\xcancel{\square }\omega }{2}\left(a_{-}^2-a_{-}{a}_{-}-a_{-}{a}_{+}+a_{-}^2\right)$

So,

$=\frac{m\xcancel{\square }\omega }{2}(2n+1)$

$=\left(n+\frac{1}{2}\right)m\xcancel{\square }\omega$

Hence,

The value of $⟨T⟩$will be evaluated.

So,

$⟨T⟩=\left(\frac{p^2}{2m}\right)$

$=\frac{1}{2}\left(n+\frac{1}{2}\right)\xcancel{\square }\omega$

Thus, the value of$⟨T⟩=\frac{1}{2}\left(n+\frac{1}{2}\right)\xcancel{\square }\omega$

And to check the satisfaction of uncertainty principal,

$=\sqrt{n+\frac{1}{2}\sqrt{\frac{\xcancel{\square }}{m\omega }}}$

$=\sqrt{n+\frac{1}{2}}\sqrt{m\xcancel{\square }\omega }$

${\sigma }_x{\sigma }_p=\left(n+\frac{1}{2}\right)\xcancel{\square }\ge \frac{\xcancel{\square }}{2}$

Thus, the uncertainty principal is satisfied.