Wave Packets and Group Velocity Notes

Wave Packets and Group Velocity

Phase velocity is defined as the rate at which the phase of a wave propagates in space.
The phase velocity can be a function of k, the wave number.
\omega = 2\pi/T (where T is the period).
k = 2\pi/\lambda (where \lambda is the wavelength).
Phase velocity is the wave moving one wavelength in one cycle of time.

If wave speed is constant and doesn't depend on wavelength, then \omega is a linear function of k.
This is not the case when adding waves that are not in phase.
When light goes through matter, the relationship between \omega and k becomes more complex.
Group velocity is the slope of the line from the origin on a graph of \omega vs. k at a particular value of k.
\lambda = 2\pi/k
p = h/\lambda = h/(2\pi) * k

For a single light wave, phase and group velocities are the same.
With light waves, the speed of the wave (c) does not depend on the wavelength.
c is not a function of \lambda.
Wave number k is used more in quantum mechanics because it's directly related to the wave's momentum.

Light wave in a vacuum is non-dispersive, meaning different wavelengths do not have different speeds.
In a dispersive medium, the graph of \omega vs. k is not linear.
In such cases, the phase velocity (slope from origin) is different from the group velocity (tangent to the curve).
For a linear function like y = 3x, \Delta y/\Delta x = dy/dx; however, for non-linear functions like y = 3x^2 or y = \sqrt{x}, \Delta y/\Delta x \neq dy/dx.

A wave packet is a combination of many waves with different wave numbers.
Represented as a sum of sinusoidal functions with different strengths (amplitudes).
Example: a1 \sin(k1 x) + a2 \sin(k2 x) + a3 \sin(k3 x), where k_i are different wave numbers.

Need to consider complex numbers, particularly e^{ix}, which represents sinusoidal functions in the complex plane.
e^{i \times \text{something}} represents complex sinusoidal functions.

Relates to systems with natural frequencies driven by different frequencies.
Amplitude can be positive or negative, affecting the sine of alpha.
Adding waves with different phases can result in phase velocities greater than c.
This doesn't violate relativity because it's a superposition of waves, not a transfer of the same frequency.
The added wave is a result of electrons in phase with the main wave.
No reaction time for light.

De Broglie connected wavelength to momentum in the 1920s.
The explanation of varying speed of light in different matter is not related to Einstein.

Complex Numbers: z = a(x) + b(x)i, where a(x) and b(x) are functions of x.
The complex conjugate: z^* = a - bi.
z \cdot z^* = (a + bi)(a - bi) = a^2 + b^2, which is proportional to probability density.

Fourier Transform: Used to find amplitude of contributing waves with different wave numbers.
Inverse Fourier Transform: Used to construct the function from different strengths of waves in k-space.

Approximation: \omega = \omega0 + (d\omega/dk)(k - k0)
Expanding \omega around \omega_0 using Taylor series.
Wave packet made of waves with wave numbers close to k_0.
Simplifying expression using Taylor series approximation.
e^{i(k0 x - \omega0 t)} represents the main wave.
\Omega = d\omega/dk = the group velocity as the envelope of the main function.
The envelope function modulates the amplitude of the main wave (sine or cosine function).
Different wave packets will move at different speeds, depending on the slope at k_0.