Draw the sketch of localised (a)wave packets and (b)extended wave with fixed wavelength correspond to heisenberg uncertainty principle.

Wave packet is defined as the disturbance which is obtained from the superposition of many different waves usually having different amplitudes and slightly different wavelengths or frequencies. Wave packets are used in quantum mechanics that represents the wave-function of a localised particle.

If a simple one-dimensional de Broglie wave having fixed amplitude and wavelength is considered and is extended to infinity along the x-direction then this kind of wave corresponds to a particle whose momentum magnitude is absolutely known. But, unluckily such kind of a wave is not localised in space because its amplitude is the same everywhere and therefore it does not pass any information about the position of the particle. Rather, if a wave of finite extent is produced having some implied localization of the particle then we should construct a wave packet by superposing or adding the waves and by arranging their superposition to reduce sharply outside the expected range of particle positions ∆x. Therefore while arguing this procedure, it is suitable to use the variable called as the angular wave number k rather than the wavelength λ and is mathematically expressed as,
$k=\frac{2\pi }{\lambda }$
Following figure shows the construction of a finite wave packet by the superposition of waves of properly selected amplitudes and wavelengths of the superposing waves.


The figure shows that, for the corresponding wave packet, the greater the spread of angular wave numbers $\Delta k$ the width will be more contracted. Thus, it can be concluded that a spread in angular wave number corresponds to the spread in particle momentum $p$. This means that the wave packet corresponding to a particle whose position is known within the range $\Delta x$ should composed of de Broglie waves related to the particle momenta having the range as,
$$\begin{gathered} \Delta p=\frac{h}{2\pi }\Delta k \\ \approx \frac{h}{2\pi }\frac{1}{\Delta x} \end{gathered}$$
This equation shows the way to the Heisenberg uncertainty principle which is mathematically expressed as,
$\Delta x\Delta p_x\ge \frac{h}{2\pi }$
Where, $\Delta p_x$ represents the uncertainty in the x-component of the momentum of a particle which is accepted to be localised within the extension of $\Delta x$.