91影视

Using uncertainty principle$\mathbf{鈭�}\mathit{E}\mathbf{鈭�}\mathit{t}\mathbf{鈮�}\frac{\mathbf{h}}{\mathbf{2}}$

Where,$\mathbf{鈭�}\mathit{E}$is the uncertainty in energy,$\mathbf{鈭�}\mathit{t}$is the uncertainty in time,his Planck鈥檚

constant.

Formula used,

$\mathit{c}\mathbf{=}\mathit{位}\mathit{f}$

where, is the speed of light, fis the frequency,$\mathit{位}$is the wavelength.

(a)

Time

$$\begin{gathered} 鈭�t=1ns \\ =1脳{10}^{-9}s \end{gathered}$$

Wavelength

$$\begin{gathered} 位=1060\mathrm{nm} \\ =1060脳{10}^{-9}\mathrm{m} \end{gathered}$$

Therefore,

$$\begin{gathered} 鈭�E鈭�t鈮�\frac{h}{2} \\ 鈭�\left(hf\right)鈭�t鈮�\frac{\left({\displaystyle \frac{h}{2\mathrm{蟺}}}\right)}{2} \\ h鈭�f鈭�t鈮�\frac{h}{4\mathrm{蟺}} \\ 鈭�f鈭�t鈮�\frac{1}{4\mathrm{蟺}} \\ 鈭�f鈮�\frac{1}{4\mathrm{蟺}鈭�\mathrm{t}} \end{gathered}$$

Plugin the values

$$\begin{gathered} 鈭�f鈮�\frac{1}{4\mathrm{蟺}\left(1脳{10}^{-9}\right)} \\ 鈮�0.079脳{10}^9 \\ 鈮�7.9脳{10}^7{s}^{-1} \end{gathered}$$

$鈭�f鈮�7.9脳{10}^7{s}^{-1}$is the range of frequencies in a 1ns pulse of 1060nm infrared laser light.

(b)

Time

$$\begin{gathered} 鈭�t=1ns \\ 鈭�t=1脳{10}^{-9}s \end{gathered}$$

Frequency

$$\begin{gathered} \mathrm{f}=100\mathrm{MHz} \\ \mathrm{f}=100脳{10}^6\mathrm{Hz} \end{gathered}$$

Therefore,

$$\begin{gathered} 鈭�E鈭�t鈮�\frac{h}{2} \\ 鈭�\left(hf\right)鈭�t鈮�\frac{\left({\displaystyle \frac{h}{2\mathrm{蟺}}}\right)}{2} \\ h鈭�f鈭�t\frac{h}{4\mathrm{蟺}} \\ 鈭�f鈭�t鈮�\frac{1}{4\mathrm{蟺}} \\ 鈭�f鈮�\frac{1}{4\mathrm{蟺}鈭�\mathrm{t}} \end{gathered}$$

Plugin the values

$$\begin{gathered} 鈭�f鈮�\frac{1}{4\mathrm{蟺}\left(1脳{10}^{-9}\right)} \\ 鈮�0.079脳{10}^9 \\ 鈮�7.9脳{10}^7{s}^{-1} \\ 鈭�f鈮�7.9脳{10}^7{s}^{-1} \end{gathered}$$

$鈭�\mathrm{f}鈮�7.9脳{10}^7{\mathrm{s}}^{-1}$ is the range in frequencies a 1ns pulse of 1060nm infrared laser light.

(c)

For infrared laser light.

Substitute the given value in the equation

$$\begin{gathered} f=\frac{c}{位} \\ f=\frac{3脳{10}^8}{1.06脳{10}^{-6}} \\ f=2.83脳{10}^{14}{s}^{-1} \end{gathered}$$

The relative size of uncertainty in frequency to the approximate frequency can be find as:

$$\begin{gathered} \frac{螖f}{f}=\frac{7.96脳{10}^1{s}^{-1}}{2.83脳{10}^{14}{s}^{-1}} \\ \frac{鈭�f}{f}=2.81脳{10}^{-7} \end{gathered}$$

For radio waves

$$\begin{gathered} \frac{鈭�f}{f}=\frac{7.96脳{10}^{\text{'}}{s}^{-1}}{1脳{10}^8{s}^{-1}} \\ \frac{鈭�f}{f}=0.796 \end{gathered}$$

For radio waves, the uncertainty in frequency relative to its approximate value is $2.81脳{10}^{-7}$while for infrared laser light the uncertainty in frequency relative to its approximate value is 0.796.

Therefore, uncertainty in frequency relative to its approximate value is larger for radio waves.