(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 Eq. $16-8,$ but use as the two waves $\psi_{1}=A \sin k_{1} x$ and $\psi_{2}=A \sin k_{2} x .$ Then show that the width of each "wave packet" is $\Delta x=2 \pi /\left(k_{1}-k_{2}\right)=2 \pi / \Delta k$ (from $t=0.05 \mathrm{~s}$ to $t=0.15 \mathrm{~s}$ in Fig. $16-17)$. Finally, show that $\Delta x \Delta p=h$ for this simple situation.

(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 Eq. $16-8,$ but use as the two waves $\psi_{1}=A \sin k_{1} x$ and $\psi_{2}=A \sin k_{2} x .$ Then show that the width of each "wave packet" is $\Delta x=2 \pi /\left(k_{1}-k_{2}\right)=2 \pi / \Delta k$ (from $t=0.05 \mathrm{~s}$
to $t=0.15 \mathrm{~s}$ in Fig. $16-17)$. Finally, show that $\Delta x \Delta p=h$ for this simple situation.

Key Concepts

Wavenumber-Momentum Relationship

In quantum mechanics, there is a direct relationship between a particle’s momentum and the wavenumber of its corresponding wave function, given by p = ?k. This relationship allows the conversion of a spread in wavenumbers (?k) into a momentum uncertainty (?p). It plays a critical role in demonstrating the uncertainty principle, as it forms the bridge connecting the mathematical description of wave packets in terms of k to the physical observable of momentum.

Uncertainty Principle

The uncertainty principle is a fundamental concept in quantum mechanics that sets a lower limit on the product of uncertainties in conjugate variables, such as position and momentum. It implies that as one tries to measure the position of a particle more precisely, its momentum becomes correspondingly less well-determined, and vice versa. This principle is typically expressed as ?x?p ? h (or h/2? when using ?), revealing the intrinsic limitations imposed by the quantum nature of particles.

Wave Packet

A wave packet is a localized wave function formed by the superposition of multiple waves with different wavelengths or wavenumbers. In quantum mechanics, it is used to represent particles with an inherent dispersion in position and momentum. The spatial width of a wave packet is related to the spread of the associated wavenumbers, demonstrating that a narrow spatial localization requires a wide range of momentum components, in accordance with the uncertainty principle.

Superposition Principle

The superposition principle states that when two or more wave functions are added together, the result is also a valid wave function. This principle is essential in constructing wave packets, as it allows the combination of waves with slightly different wavelengths or momenta to form an overall localized probability distribution. This construction is the basis for understanding phenomena such as interference and the relationship between spatial and momentum uncertainties.

Interference Effects

Interference effects arise when waves of similar but different wavelengths are superposed, resulting in an amplitude modulation or beat pattern. This modulation forms an envelope that defines the overall spatial extent or width of the wave packet. The periodicity of the interference pattern is directly related to the difference in wavenumbers, illuminating the connection between ?x and ?k, and consequently linking spatial and momentum uncertainties.

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