How to Derive Velocity in Wave-Particle Duality

The wave-particle duality is a fundamental concept in quantum mechanics that describes the dual nature of particles, where they exhibit both wave-like and particle-like properties. To derive the velocity in this duality, we can start with the de Broglie postulate and the wave equation.

Understanding de Broglie’s Postulate

According to de Broglie’s postulate, the wavelength of a particle is given by the equation:

λ = h/p

Where:
– λ is the wavelength of the particle
– h is Planck’s constant
– p is the momentum of the particle

This equation establishes the relationship between the particle’s wavelength and its momentum.

Calculating the Wave Velocity (vp)

The frequency of the wave associated with the particle is given by:

f = E/h

Where:
– f is the frequency of the wave
– E is the energy of the particle
– h is Planck’s constant

Using the wave equation, the velocity of the wave (vp) can be calculated as:

vp = fλ
vp = (E/h) * (h/p)
vp = E/p

This equation shows that the wave velocity (vp) is directly proportional to the energy (E) of the particle and inversely proportional to its momentum (p).

Relating Velocity to Relativistic Energy-Momentum

The energy of a particle can be related to its mass and velocity using the relativistic energy-momentum relation:

E^2 = (mc^2)^2 + (pc)^2

Where:
– E is the energy of the particle
– m is the rest mass of the particle
– c is the speed of light
– p is the momentum of the particle

Substituting this relation into the equation for vp, we get:

vp = c^2 * p / E
vp = c^2 * p / √[(mc^2)^2 + (pc)^2]

This equation gives the wave velocity (vp) in terms of the particle’s mass, momentum, and the speed of light.

Velocity of Massless Particles (Photons)

For a massless particle, such as a photon, the energy is given by:

E = hf

Where:
– E is the energy of the photon
– h is Planck’s constant
– f is the frequency of the photon

Substituting this into the equation for vp, we get:

vp = c^2 * p / hf
vp = c^2 / c
vp = c

This shows that for a massless particle, the wave velocity (vp) is equal to the speed of light (c).

Group Velocity (vg)

The velocity of the particle itself, often called the group velocity (vg), is given by:

vg = dω/dk

Where:
– ω is the angular frequency of the wave
– k is the wave number of the wave

It can be shown that the group velocity (vg) is always less than the speed of light, and it is equal to the particle velocity for waves with a narrow frequency range.

Theorem and Formulas

Theorem: The velocity in wave-particle duality is given by:
– vp = c^2 * p / E for a particle with mass
– vp = c for a massless particle, such as a photon

The group velocity (vg), which is the velocity of the particle, is always less than the speed of light.

Physics Formulas:
– vp = c^2 * p / E
– vg = dω/dk

Examples and Numerical Problems

Physics Examples:
– A particle with mass m and velocity v has a momentum p = mv and energy E = √[(mc^2)^2 + (pc)^2].
– A photon with frequency f has energy E = hf and momentum p = hf/c.

Physics Numerical Problems:
– A particle with mass 1 kg and velocity 1 m/s has a momentum of 1 kg m/s and an energy of 1.0000000000000002 kg m^2/s^2.
– A photon with frequency 1 Hz has an energy of 6.62607004 × 10^-34 J and a momentum of 2.1886963 × 10^-35 kg m/s.

Figures, Data Points, Values, and Measurements

The velocity of a particle with mass m and velocity v is given by vp = c^2 * p / E, where p = mv and E = √[(mc^2)^2 + (pc)^2]. The velocity of a photon with frequency f is given by vp = c.

Reference Links

1. Wave-Particle Duality – Physics – The Information Philosopher
2. Chapter 3 Wave Properties of Particles – MST.edu
3. Wave–particle duality quantified for the first time: « The experiment …