91影视

The Heisenberg uncertainty principle relating an uncertainty in energy $\mathbf{鈭�}\mathit{E}$with an uncertainty in time $\mathbf{鈭�}\mathit{t}$are related as

$\mathbf{鈭�}\mathit{E}\mathbf{鈭�}\mathit{t}\mathbf{鈮�}\frac{\sout{\mathbf{h}}}{\mathbf{2}}$

Here, $\sout{\mathbf{h}}$is reduced Planck's constant.

Energy E of a photon with wavelength $\mathit{位}$is given by

$\mathit{E}\mathbf{=}\frac{\mathbf{h}\mathbf{c}}{\mathbf{位}}$

Here, h is Planck's constant, and c is speed of light in vacuum.

(a)

As given Wavelength$\mathrm{位}=656\mathrm{nm}=(656\mathrm{nm})\left(\frac{{10}^{-9}\mathrm{m}}{\mathrm{nm}}\right)=656脳{10}^{-9}\mathrm{m}$

The uncertainty in time $鈭�\mathrm{t}={10}^{-\mathrm{s}}\mathrm{s}$

All values, $\left({10}^{-8}\mathrm{s}\right)$for the uncertainty in time $鈭�t$, and $1.0546脳{10}^{-34}\mathrm{J}.\mathrm{s}\mathrm{for}h$ is taken in the equation uncertainty in energy $鈭�E$

$$\begin{gathered} 鈭�\mathrm{E}鈭�\mathrm{t}鈮�{鈭�}^{\text{'}}\mathrm{E} \\ 鈭�\mathrm{E}鈮�\frac{\sout{\mathrm{h}}}{2鈭�\mathrm{t}} \\ 鈭�\mathrm{E}鈮�\frac{(1.0546脳{10}^{-34}\mathrm{J}.\mathrm{s})}{2({10}^{-8}\mathrm{s})} \\ 鈭�\mathrm{E}鈮�5.273脳{10}^{-27}\mathrm{J} \end{gathered}$$

Therefore, uncertainty in energy for that quantum level $鈭�E鈮�5.27脳{10}^{-27}\mathrm{J}$.

(b)

The photon鈥檚 energy comprises of$位$

$E=\frac{hc}{位}$

Take the derivation on both sides

$$\begin{gathered} dE=\frac{hc}{位}d位 \\ -\frac{{位}^2}{hc}dE=d位 \end{gathered}$$

As we are only interested in the magnitude of the change in wavelength, the absolute value

$d位=\frac{{位}^2}{hc}dE$

Take $6.63脳{10}^{-34}\mathrm{J}.\mathrm{s}\mathrm{for}\mathrm{h},3脳{10}^8\mathrm{m}/\mathrm{s}\mathrm{for}c\mathrm{and}5.27脳{10}^{-27}\mathrm{J}\mathrm{for}dE$in the above equation to solve for

$$\begin{gathered} d位=\frac{{位}^2}{hc}dE \\ d位=\left[\frac{{(6.56脳{10}^{-7}\mathrm{m})}^2}{(6.63脳{10}^{-34}\mathrm{J}.\mathrm{s})(3脳{10}^8\mathrm{m}/\mathrm{s})}\right]\left(5.27脳{10}^{-27}\mathrm{J}\right) \\ d位=1.14脳{10}^{-14}\mathrm{m} \end{gathered}$$

Hence, the range of wavelengths for the 656 nm spectral line is $1.14脳{10}^{-14}\mathrm{m}$.

(c)

The absolute value of the change in wavelength

$d位=\frac{{位}^2}{hc}dE$

For finite change, $鈭�位=\frac{{位}^2}{hc}鈭�E$

Take the value of $\frac{hc}{{位}^2}d位\mathrm{for}鈭�E,\mathrm{and}\frac{h}{2\mathrm{蟺}}\mathrm{for}\sout{h}$in the equation $鈭�E鈭�t鈮�\frac{\sout{h}}{2}$

$$\begin{gathered} \left(\frac{hc鈭�位}{{位}^2}\right)鈭�t鈮�\frac{\left({\displaystyle \frac{h}{2\mathrm{蟺}}}\right)}{2} \\ \frac{c鈭�位鈭�t}{{位}^2}鈮�\frac{1}{4\mathrm{蟺}} \\ 鈭�位鈭�t鈮�\frac{{位}^2}{4\mathrm{蟺肠}} \end{gathered}$$

Hence, the relationship between the $鈭�位$and $鈭�t$is $鈭�位鈭�t鈮�\frac{{位}^2}{4\mathrm{蟺肠}}$.