91影视

The harmonic oscillator is an essential concept of quantum mechanics. The oscillator oscillates about an equilibrium point, and the system's total energy is always constant. The potential that can represent the system is harmonic oscillator potential.

a)

Since,

$$\begin{gathered} 尉=\sqrt{m蝇/魔x} \\ 伪={\left(m蝇/蟺魔\right)}^{1/4} \end{gathered}$$

The wave function is:

$$\begin{gathered} 蠄\left(x,0\right)=A{\left(1-2尉\right)}^2{e}^{-{尉}^2/2} \\ 蠄\left(x,0\right)=A{\left(1-4尉\right)}^2{e}^{-{尉}^2/2} \end{gathered}$$

鈥︹赌︹赌︹赌︹赌�(1)

Now,

${蠄}_0\left(x\right)=伪e^{-{尉}^2/2}$

localid="1658127827170" $$\begin{gathered} {蠄}_1\left(x\right)=\sqrt{2}伪尉e^{-{尉}^2/2} \\ {蠄}_1\left(x\right)=\frac{伪}{\sqrt{2}}\left(2{尉}^2-1\right)e^{-{尉}^2/2} \\ So, \\ 蠄\left(x,0\right)=c_0{蠄}_0+c_1{蠄}_1+c_2{蠄}_2 \end{gathered}$$

Substitute values in the above expression, and we get,$蠄\left(x,0\right)=伪\left(c_0+\sqrt{2}尉c_1+\sqrt{2}{尉}^2{c}_2-\frac{1}{\sqrt{2}}{c}_2\right)e^{-{尉}^2/2}$ 鈥︹赌︹赌︹赌︹赌�..(2)

Comparing equations 1 and 2 as:

$$\begin{gathered} 伪\sqrt{2}{c}_2=4A \\ {c}_2=2\sqrt{2}A/伪 \\ 伪\sqrt{2}{c}_1=-4A \\ {c}_1=-2\sqrt{2}A/伪 \\ 伪\left(c_0=\frac{c_2}{\sqrt{2}}\right)=A \\ {c}_0=\left(\frac{A}{伪}\right)+\frac{c_2}{\sqrt{2}} \end{gathered}$$

$$\begin{gathered} \left(1+2\right)\frac{A}{伪}=\frac{3A}{伪} \\ 1={\left|c_0\right|}^2+{\left|c_1\right|}^2+{\left|c_2\right|}^2 \end{gathered}$$

Substitute values in the above expression, and we get,

$$\begin{gathered} 1=\left(8+8+9\right){\left(\frac{A}{伪}\right)}^2 \\ A=\frac{伪}{5} \end{gathered}$$

Therefore, the values will be:

$c_0=\frac{3}{5}$

Substitute values in the above expression, and we get,

role="math" localid="1658127161820"

Thus, the expectation value of energy can be written as: .

$$\begin{gathered} 蠄\left(x,t\right)=\frac{3}{5}{蠄}_0{e}^{-i蝇t/2}-\frac{2\sqrt{2}}{5}{蠄}_1{e}^{-3i蝇t/2}+\frac{2\sqrt{2}}{5}{蠄}_1{e}^{-5i蝇t/2} \\ 蠄\left(x,t\right)=e^{-i蝇t/2}\left[\frac{3}{5}{蠄}_0-\frac{2\sqrt{2}}{5}{蠄}_1{e}^{-i蝇t/2}+\frac{2\sqrt{2}}{5}{蠄}_2{e}^{-2i蝇t/2}\right] \end{gathered}$$

Since for the lowest value, we need,

$$\begin{gathered} e^{-i蝇T}=-1 \\ {e}^{-2i蝇T}=1 \\ 蝇T=蟺 \\ T=\frac{蟺}{蝇} \end{gathered}$$

The smallest possible value of T can be written as: $T=\frac{蟺}{蝇}$.