Engineering Physics: Wave Packet, Phase Velocity, and Group Velocity

Academic Overview and Objectives

• Course Details:

  • Course Code: 26SPH-141
  • Course Title: Engineering Physics
  • Unit Number: 1
  • Chapter Number: 1
  • Lecture Number: 4
  • Topic: Wave Packet
  • Department: Department of Engineering Foundations (1-5), University Institute of Engineering
  • Academic Programs: Bachelor of Engineering (BE) in CSE, CSE-Specialization, and Non-CSE
  • Academic Session: 2026-27 (ODD Semester, July–December 2026)
  • Instructor: Dr. Nandni Sharma (E12297), Assistant Professor

• Core Learning Objectives:

  • Objective 1: Understand the physical concept and structure of a wave packet.
  • Objective 2: State and define phase velocity ($v_p$) and group velocity ($v_g$).
  • Objective 3: Derive the mathematical relation existing between phase velocity ($v_p$) and group velocity ($v_g$).

• Prerequisites and Prior Conceptual Foundations:

  • Wave-Particle Duality: The foundational principle that matter and radiation exhibit both wave-like and particle-like behaviors.
  • Properties of Matter Waves: De-Broglie matter waves associated with moving particles, including their wavelength relationships ($\lambda = \frac{h}{p}$) and physical characteristics.
  • Numerical Applications: Quantitative evaluation of matter wave properties and particle dynamics.

Theoretical Origin of Phase Velocity and Group Velocity

• The Relativistic Wave Speed Paradox:

  • According to Einstein's theory of special relativity, the velocity of any physical material particle $v$ must always be strictly less than the speed of light $c$ ($v < c$).
  • Under the initial de-Broglie hypothesis for a single monochromatic wave associated with a moving particle, the calculated wave velocity was found to exceed the speed of light ($v_{\text{wave}} > c$).
  • This result created a major theoretical conflict: if $v_{\text{wave}} > c$, the de-Broglie wave associated with the material particle would move significantly faster than the particle itself, leaving the physical particle far behind and causing the wave and particle to become uncoupled.

• Schrödinger's Wave Packet Resolution:

  • Erwin Schrödinger resolved this theoretical difficulty by postulating that a moving material particle is not equivalent to a single monochromatic wave, but is instead equivalent to a wave packet.
  • A wave packet consists of a localized group of individual waves, each possessing slightly different wavelengths and frequencies, superimposed upon one another.
  • The overall envelope or localized amplitude region of the wave packet propagates with its own distinct velocity, designated as the group velocity ($v_g$).
  • The individual constituent waves that interfere to construct the wave packet propagate with an average velocity, designated as the phase velocity ($v_p$).
  • Equivalence Principle: The velocity of the physical material particle $v$ is identically equal to the group velocity $v_g$ of the wave packet ($v = v_g$).

Definitions of Phase Velocity and Group Velocity

• Phase Velocity ($v_p$):

  • Conceptual Definition: The velocity with which a representative point of constant phase (such as a crest or a trough) travels through space in an individual monochromatic wave forming part of a wave packet.
  • Alternative Definition: The velocity with which individual component waves forming the wave packet propagate.
  • Mathematical Expression:     $v_p = \frac{\omega}{k}$     where $\omega$ represents the angular frequency of the wave and $k$ represents the wave number ($k = \frac{2\pi}{\lambda}$).

• Group Velocity ($v_g$):

  • Conceptual Definition: The velocity with which the overall shape/envelope of the wave packet—formed due to the superposition of two or more waves of slightly different wavelengths—is transmitted through space.
  • Alternative Definition: The velocity with which the resultant wave packet formed due to the interference and superposition of constituent waves propagates.
  • Mathematical Expression:     $v_g = \frac{d\omega}{dk}$     where $d\omega$ is the differential change in angular frequency and $dk$ is the differential change in wave number.

Mathematical Derivation of the Relation Between Phase and Group Velocity

• Step 1: Expression for Phase Velocity:

  • The phase velocity $v_p$ of a wave is given by:     $v_p = \frac{\omega}{k} \quad \text{--- Equation (3.3.1)}$

• Step 2: Expressing Angular Frequency $\omega$:

  • Rewriting Equation (3.3.1) to isolate angular frequency $\omega$:     $\omega = k \, v_p \quad \text{--- Equation (3.3.2)}$

• Step 3: Differentiation with Respect to Wave Number $k$:

  • Differentiating Equation (3.3.2) with respect to wave number $k$ using the derivative product rule:     $\frac{d\omega}{dk} = v_p + k \, \frac{dv_p}{dk} \quad \text{--- Equation (3.3.3)}$

• Step 4: Substitution of Group Velocity Definition:

  • Recalling the fundamental definition of group velocity $v_g = \frac{d\omega}{dk}$, substitute $v_g$ into Equation (3.3.3):     $v_g = v_p + k \, \frac{dv_p}{dk} \quad \text{--- Equation (3.3.4)}$

• Step 5: Applying Partial Differentiation / Chain Rule:

  • Expressing the derivative $\frac{dv_p}{dk}$ in terms of wavelength $\lambda$ using the chain rule:     $v_g = v_p + k \, \frac{dv_p}{d\lambda} \, \frac{d\lambda}{dk} \quad \text{--- Equation (3.3.5)}$

• Step 6: Relationship Between Wave Number $k$ and Wavelength $\lambda$:

  • The wave number $k$ is related to wavelength $\lambda$ by:     $k = \frac{2\pi}{\lambda}$
  • Rearranging for wavelength $\lambda$:     $\lambda = \frac{2\pi}{k}$
  • Differentiating $\lambda$ with respect to $k$:     $\frac{d\lambda}{dk} = -\frac{2\pi}{k^2}$

• Step 7: Final Substitution into the Group Velocity Expression:

  • Multiplying wave number $k$ by $\frac{d\lambda}{dk}$ yields:     $k \, \frac{d\lambda}{dk} = k \left( -\frac{2\pi}{k^2} \right) = -\frac{2\pi}{k} = -\lambda$
  • Substituting $k \, \frac{d\lambda}{dk} = -\lambda$ back into Equation (3.3.5) gives:     $v_g = v_p - \lambda \, \frac{dv_p}{d\lambda} \quad \text{--- Equation (3.3.6)}$
  • Equation (3.3.6) represents the definitive general relationship between group velocity $v_g$ and phase velocity $v_p$ in terms of wavelength $\lambda$

Questions & Discussion

• Question 1:

  • Prompt: Prove that the velocity of the material particle $v$ is the same as group velocity ($v = v_g$).
  • Response Focus: Demonstrate that a moving physical particle corresponds to a quantum wave packet whose overall propagation velocity $v_g$ is equivalent to the particle velocity $v$

• Question 2:

  • Prompt: Define phase velocity and group velocity.
  • Response Focus: Provide explicit definitions for individual wave velocity ($v_p = \frac{\omega}{k}$) and envelope velocity ($v_g = \frac{d\omega}{dk}$).

• Question 3:

  • Prompt: Derive the relation between phase and group velocity.
  • Response Focus: Execute the step-by-step differentiation of $\omega = k \, v_p$ combined with the chain rule for $\lambda$ to arrive at $v_g = v_p - \lambda \, \frac{dv_p}{d\lambda}$.

Reference Materials and Learning Resources

• Recommended Textbooks:

  • Concepts of Modern Physics (Fifth Edition) by Arthur Beiser, McGraw Hill International.
  • Fundamentals of Physics by David Halliday, Robert Resnick, and Jearl Walker, Wileyplus.

• NPTEL Online Courses:

  • NPTEL Course 115104096: https://nptel.ac.in/courses/115104096
  • NPTEL Course 115103104: https://nptel.ac.in/courses/115103104
  • NPTEL Course 115101107: https://nptel.ac.in/courses/115101107
  • NPTEL Course 106106232: https://nptel.ac.in/courses/106106232

• Video Lecture Resources:

  • Lecture Video 1: https://www.youtube.com/watch?v=_VeTwdGZY_Y
  • Lecture Video 2: https://www.youtube.com/watch?v=tlM9vq-bepA

• External Web References:

  • Design like Quantum Physics: https://stradallc.com/2016/02/12/design-like-quantum-physics/
  • Branches of Physics Reference: https://en.wikipedia.org/wiki/Branches_of_physics
  • Light & Optics Calculators: http://www.sciencecalculators.org/optics/light/
  • Difference Between Phase Velocity and Group Velocity Diagram: https://pediaa.com/wp-content/uploads/2016/04/Difference-Between-Phase-Velocity-and-Group-Velocity-image-11.png