Substituting this back into the integral, we can separate the terms: The speed of the individual ripples inside the envelope.
[ \phi(k) = \left( \frac2\alpha\pi \right)^1/4 e^-\alpha (k - k_0)^2 ]
The probability density:
Group ( k^2 ) terms: ( -\alpha k^2 - i \beta k^2 = -(\alpha + i\beta) k^2 ) wave packet derivation
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
Substituting our Gaussian $A(k)$: $$ \Psi(x,0) = \left( \frac2\alpha^2\pi \right)^1/4 \frac1\sqrt2\pi \int_-\infty^\infty e^-\alpha^2 (k-k_0)^2 e^ikx , dk $$
Since $A(k)$ is sharply peaked around $k_0$, the integral is dominated by values of $k$ near $k_0$. We can approximate $\omega(k)$ by expanding it in a Taylor series around $k_0$: Substituting this back into the integral, we can
[ = \left( \frac2\alpha\pi \right)^1/4 \cdot \frac1\sqrt2 \cdot \frac1\sqrt\alpha = \left( \frac2\alpha\pi \right)^1/4 \cdot \frac1\sqrt2\alpha ]
Define: [ \omega_0 = \omega(k_0), \quad v_g = \omega'(k_0) \quad \text(group velocity) ] Let (k = k_0 + \kappa), where (\kappa) is small.
Ψ(x,t)=12π∫−∞∞A(k)ei(kx−ω(k)t)dkcap psi open paren x comma t close paren equals the fraction with numerator 1 and denominator the square root of 2 pi end-root end-fraction integral from negative infinity to infinity of cap A open paren k close paren e raised to the i open paren k x minus omega open paren k close paren t close paren power d k 3. The Gaussian Wave Packet This post walks through the mathematical derivation of
To describe a localized particle, we use a superposition of many plane waves. This mathematical construction is the wave packet. 1. The Starting Point: The Plane Wave A free particle with a definite momentum and energy is represented by the wave function:
The constant ( \left( \frac2\alpha\pi \right)^1/4 ) ensures normalization: ( \int |\phi(k)|^2 dk = 1 ). The parameter ( \alpha ) controls the width: a larger ( \alpha ) means a narrower spread in ( k )-space, and thus a broader wave packet in position space (due to the uncertainty principle).