Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Quantum mechanics → Wave–particle duality ↓


De Broglie wavelength


At the beginning of the 20th century, the world of physics was going through a transformational shift. Classical physics, which explained the macroscopic world incredibly well, began to struggle to provide insight into the microscopic world of atoms and subatomic particles. Around this time, a new theory arose that redefined the fundamentals: wave-particle duality. A key figure in this advancement was Louis de Broglie, who proposed the concept that matter particles, like electrons, also had wave-like properties. This led to the formula now known as the de Broglie wavelength.

The concept of wave-particle duality

To understand de Broglie wavelength, we first need to delve deeper into wave-particle duality. Classical physics has taught us that waves and particles are different. Waves, such as sound or water waves, are disturbances that travel through space and time. They are characterized by properties such as wavelength, frequency, and speed. Particles, on the other hand, are tiny localized objects that have mass and volume, such as a baseball or an electron.

However, in the early 20th century experiments such as the famous double-slit experiment began to show that light exhibits characteristics of both waves and particles. When light passes through the two slits and falls on a screen, it creates an interference pattern characteristic of waves. But when the same experiment is used to detect individual photons (particles of light), they appear to behave like particles.

These conflicting results led physicists to a surprising reality: entities at the quantum level exhibit dual properties. This theory is known as wave-particle duality. Light and other electromagnetic radiation are not the only entities to exhibit such dual properties. Electrons and other matter particles, as de Broglie proposed, also exhibit wave-like properties.

De Broglie hypothesis

Louis de Broglie suggested that all matter is of wave nature, and formulated the de Broglie wavelength formula:

λ = h / p

where λ (lambda) is the de Broglie wavelength, h is the Planck constant (about 6.626 x 10 -34 Js), and p is the momentum of the particle.

This groundbreaking hypothesis suggested that particles such as electrons have a wavelength associated with them. This wavelength is inversely proportional to the particle's speed. The implication of this hypothesis was profound: the behavior of objects we thought of as purely particles should also be analyzed with wave-like models.

Visualization of de-Broglie wavelength

To understand this concept, imagine a small billiard ball rolling on a table. According to classical physics, the ball has a certain velocity and mass, which gives it a direct momentum. Now, if we apply de Broglie's hypothesis, this ball also has a wavelength.

Let's look at this with a simple diagram showing a particle with an associated wave:

Particle Wave

The red circle in the graphic above represents a particle, and the blue wave represents the de Broglie wave associated with it. Ideally, as the particle's speed increases, its de Broglie wavelength decreases, resulting in compressible waves.

Now, let us give an example to get a clear understanding. Consider an electron moving with a velocity (v) of 2 x 10 6 m/s. The mass (m) of the electron is about 9.11 x 10 -31 kg. First, calculate the momentum p as follows:

p = m * v

On substituting the values, we get:

p = 9.11 x 10^-31 kg * 2 x 10^6 m/s = 1.822 x 10^-24 kg m/s

Using the de-Broglie formula the wavelength can be calculated as:

λ = h / p = 6.626 x 10^-34 Js / 1.822 x 10^-24 kg m/s ≈ 3.64 x 10^-10 m

This wavelength is on the order of the size of atoms, which is why wave-like properties are important on the quantum scale. For macroscopic objects such as tennis balls, the de Broglie wavelength is usually not important.

Importance of de-Broglie wavelength

The concept of de Broglie wavelength is not just theoretical. It paved the way for the development of quantum mechanics and has many practical applications. Let's take a look at some of these:

  • Electron microscopy: Because of the small de Broglie wavelength of electrons, we are able to increase the resolution of images beyond that of visible light microscopes. Electron microscopes can achieve the resolution needed to observe atoms and atomic structures.
  • Quantum computing: De Broglie's hypothesis contributed to the understanding of quantum states and the behaviour of qubits in quantum computers.
  • Diffraction and interference: The wave nature of particles makes possible phenomena such as electron diffraction, which is important in understanding and analyzing the structure of solids.

A compelling example demonstrating de Broglie's prediction is the Davisson-Germer experiment. In this experiment, electrons were shot at a nickel crystal, and the detection of the diffraction pattern confirmed their wave nature.

Electrons Crystal Diffraction Pattern

The electron beam scattered upon striking the crystal, producing an interference pattern just as predicted for waves in de Broglie's theory.

Conceptual challenges and clarifications

One area of conceptual challenge with the de Broglie hypothesis is understanding the implications of particles having a wave nature on the macroscopic scale. For most everyday phenomena, massive objects such as humans, planets or cars do not show wave-like behavior because the de Broglie wavelength associated with them is too small.

For example, consider a baseball with a mass 0.145 kg and a speed of 40 m/s. So the speed of the baseball is:

p = 0.145 kg * 40 m/s = 5.8 kg m/s

The de Broglie wavelength becomes:

λ = h / p = 6.626 x 10^-34 Js / 5.8 kg m/s ≈ 1.14 x 10^-34 m

What is worth noting is that this is much smaller than anything we can directly measure or observe, which is why everyday objects do not exhibit wave-like properties.

This concept lies at the intersection of classical and quantum physics, where our understanding of reality is transformed by how solid particles can share the properties of delicate waves.

It challenges not only our conceptual limits but also the basic structures of theoretical physics, leading us to a deeper understanding of the universe and reminding us of the complexities hidden behind surprisingly simple questions.

Conclusion

The de Broglie wavelength remains an important pillar in the field of quantum mechanics. Through the lens of wave-particle duality, it establishes a beautiful harmony between particles and waves. It has guided discoveries that have revolutionized technology and answered profound questions about the nature of the universe.

As we delve deeper into the mysteries of quantum mechanics, de Broglie's wisdom continues to inspire contemporary physicists and researchers. His hypothesis is a testament to curiosity, innovation, and the constant quest to understand the fascinating picture of the quantum world.


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De Broglie wavelength

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