Electron wavepackets: unveiling group velocity and dispersion
Continuing my post on wavepackets here, I further explore the nuanced dynamics of electron wavepackets in the realm of quantum mechanics. In this analysis, I specifically focus on free electrons, scrutinizing their behavior through the lens of the Schrödinger equation for a potential V(x) = 0, which is expressed as:
-\frac{\hbar^2}{2m} \frac{\mathrm d^2\psi(x)}{\mathrm d x^2} = E\psi(x)
The solution to this equation unveils a critical aspect of quantum mechanics: the phase velocity (v_p) and group velocity (v_g) of an electron wavepacket are distinct, leading to profound implications for the motion of electrons. Specifically, the relationship between energy (E) and the wave vector (k) is quadratic, as shown by:
E = \frac{\hbar^2 k^2}{2m}
From this, I deduce the group velocity of a free electron to be:
v_g = \frac{\hbar k}{m}
This result aligns with the classical definition of kinetic energy, suggesting that electrons indeed move at the group velocity. This observation is pivotal in understanding the dispersion of electron wavepackets, a phenomenon that emerges due to the non-linear relationship between \omega and k.
To elucidate this further, I constructed a Gaussian-modulated wavepacket, illustrating how group velocity dispersion manifests in a quantum mechanical context. The spreading of the wavepacket over time visually demonstrates the dispersion effect, which is inherently tied to the quadratic nature of the \omega(k) relationship.
For more insights into this topic, you can find the details here.