(III) Show that the uncertainty principle holds for a "wave...

(III) Show that the uncertainty principle holds for a "wave packet" that is formed by two waves of similar wavelength $\lambda_{1}$ and $\lambda_{2} .$ To do so, follow the argument leading up to $D=\left[2 A \cos 2 \pi\left(\frac{f_{1}-f_{2}}{2}\right) t\right] \sin 2 \pi\left(\frac{f_{1}+f_{2}}{2}\right) t,$ (8) $$ \begin{array}{l}{\text { but use as the two waves } \psi_{1}=A \sin k_{1} x \text { and }} \\ {\psi_{2}=A \sin k_{2} x . \text { Then show that the width of each "wave }}\end{array} $$ $$ \begin{array}{l}{\text { packet" is } \Delta x=2 \pi /\left(k_{1}-k_{2}\right)=2 \pi / \Delta k \text { (from } t=0.05 \mathrm{s}} \\ {\text { to } t=0.15 \mathrm{s} ) . \text { Finally, show that } \Delta x \Delta p=h \text { for this simple }} \\ {\text { situation. }}\end{array} $$

(III) Show that the uncertainty principle holds for a "wave
packet" that is formed by two waves of similar wavelength
$\lambda_{1}$ and $\lambda_{2} .$ To do so, follow the argument leading up to
$D=\left[2 A \cos 2 \pi\left(\frac{f_{1}-f_{2}}{2}\right) t\right] \sin 2 \pi\left(\frac{f_{1}+f_{2}}{2}\right) t,$ (8) $$
\begin{array}{l}{\text { but use as the two waves } \psi_{1}=A \sin k_{1} x \text { and }} \\ {\psi_{2}=A \sin k_{2} x . \text { Then show that the width of each "wave }}\end{array}
$$ $$
\begin{array}{l}{\text { packet" is } \Delta x=2 \pi /\left(k_{1}-k_{2}\right)=2 \pi / \Delta k \text { (from } t=0.05 \mathrm{s}} \\ {\text { to } t=0.15 \mathrm{s} ) . \text { Finally, show that } \Delta x \Delta p=h \text { for this simple }} \\ {\text { situation. }}\end{array}
$$

Key Concepts

Uncertainty Principle

The uncertainty principle in quantum mechanics establishes that the product of the uncertainties in position (?x) and momentum (?p) must be above a certain bound (of the order of Planck's constant, h). This fundamental concept implies that a particle cannot have both a precisely defined position and momentum simultaneously, which is reflected in the relation ?x ?p = h for the specific case of the wave packet under discussion.

Wave Packet

A wave packet is a localized disturbance that results from the superposition of multiple waves with slightly different wavelengths or wave numbers. It represents a particle in quantum mechanics by providing a spatially confined probability distribution, where the width of the packet (?x) is inversely related to the range of wave numbers (?k) contributing to the packet.

Superposition Principle

The superposition principle allows for the combination of different wave functions to form a new, resultant wave. In the context of wave packets, the interference between two waves of similar wavelengths leads to an envelope structure that modulates the oscillatory behavior, enabling an analysis of the spatial extent and momentum distribution of the packet.

Wave Number

The wave number (k) is a measure of spatial frequency and is directly related to a particle's momentum via the de Broglie relation (p = ?k). A small difference in wave numbers (?k) between the constituent waves of a packet results in a relatively large spatial width (?x), emphasizing the inverse relationship between position and momentum uncertainties.

Beat Phenomenon

When two waves of slightly different frequencies combine, they create a beat pattern characterized by a rapidly oscillating carrier wave under a slowly varying amplitude envelope. This modulation is key to determining the spatial extent of a wave packet, where the envelope’s width is directly related to the uncertainty in position (?x), while the difference in wave numbers (?k) contributes to the uncertainty in momentum.

Transcript

Please give Ace some feedback