[ \Psi(x,t) = \left( \frac2\alpha\pi \right)^1/4 \frac1\sqrt2\pi e^-\alpha k_0^2 \int_-\infty^\infty e^-(\alpha + i\beta) k^2 + (2\alpha k_0 + i x) k , dk ]
[ \Delta x , \Delta p = \frac\hbar2 ]
The Derivation of a Quantum Wave Packet wave packet is a localized disturbance consisting of a group of waves with slightly different frequencies and wavelengths that interfere constructively in a small region of space and destructively elsewhere [21, 27]. In quantum mechanics, wave packets are essential for representing particles that are localized in space, adhering to the Heisenberg Uncertainty Principle 1. Superposition of Plane Waves A single plane wave wave packet derivation
: The width grows with time. Even in free space (no forces), a wave packet inevitably spreads because different ( k )-components have different phase velocities ( v_p = \omega/k = \hbar k/(2m) ). The initially synchronized components get out of phase.
So the integral becomes:
In quantum mechanics, we can't describe a particle as a single point or a single infinite wave. Instead, we use a —a localized "burst" of wave action. This post walks through the mathematical derivation of a wave packet via the superposition of plane waves. 1. The Building Block: Plane Waves A single free particle with a definite momentum
Here, $A(k)$ is the Fourier transform of the initial spatial wave function $\Psi(x,0)$. This integral is the formal definition of a wave packet. Even in free space (no forces), a wave
To localize the particle, we combine many plane waves with slightly different wave numbers (
To get a physically meaningful packet that is localized in both position and momentum, we often choose to be a centered at Instead, we use a —a localized "burst" of wave action
To solve this, we complete the square or use standard Gaussian integral formulas. Let $q = k - k_0$, so $k = q + k_0$ and $dk = dq$. $$ \Psi(x,0) \propto e^ik_0x \int_-\infty^\infty e^-\alpha^2 q^2 e^iqx , dq $$
We define the wave packet as an integral over a range of wavenumbers . Instead of a single , we use a weighting function , typically a Gaussian distribution: