De Broglie Relationship

Edited By Shivani Poonia | Updated on Jul 02, 2025 05:50 PM IST

The twentieth century was a period that witnessed the quest to understand the dual nature of radiation and matter as well as new frontiers in quantum mechanics. De Broglie formulated the wave-particle duality, which challenged traditional theories about matter being made up of particles. In other words, like light exhibits particle-like and wave-like properties, particles including electrons can also exhibit wavelike characteristics.

This Story also Contains

De-Broglie Wavelength- Quantum Leap:
Some Solved Examples
Conclusion:

De Broglie Relationship

De-Broglie Wavelength- Quantum Leap:

Louis de Broglie made the wave theory of matter in 1924 and it was established as the de Broglie equation, which states the wavelength of the particle (λ) is inversely proportional to the momentum (p) of the particle. It is given in terms of Planck’s constant through the formula λ = h/p λ=h/p In this formula h represents Planck’s constant wherein p stands for the momentum of a particular particle. This equation has a certain aspect of wave-particle duality that describes quantum mechanics; nevertheless, electrons and atom particles exhibit wave properties. The equation presented by de Broglie is called the de Broglie equation proposing a method by which matter possesses wave characteristics thus affecting the fundamental approach toward the particle nature of matter in the atomic and subatomic domain. Their use cuts across all the disciplines in sciences including Physics, Chemistry, Material Sciences, and Others.

The de Broglie’s prediction was confirmed experimentally when it was found that an electron beam undergoes diffraction, a phenomenon characteristic of waves.

It needs to be noted that according to de Broglie, every object in motion has a wave character. This wavelength is quite significant for the subatomic particles which have very small masses. The wavelengths associated with ordinary objects are however so short that their wave properties cannot be detected as they have large masses.

Bohr's model and de Broglie's relation: Number of standing waves made by an electron in n^th Bohr orbit

According to Bohr's model,

$
m v r=\frac{n h}{2 \pi}
$

According to de Broglie's Relation,
$
p=\frac{h}{\lambda}
$

Combining the two

$2 \pi r=n \lambda$

So, the number of waves made by any electron in the nth orbit is equal to the principal quantum number of the orbit, i.e. n.

De-Broglie Equation-

de-Broglie proposed that just like light, matter should exhibit both particle and wave-like properties. This means that just as the photon has momentum as well as wavelength, electrons should also have momentum as well as wavelength, and he proposed the following mathematical relationship:

There is the de Broglie equation, which was given by Louis de Broglie and it defines the relationship between wavelength and momentum of the particle It is expressed as λ = h/mv. It is mathematically symbolized as λ = h/p; here h is Planck’s constant which is a natural constant. This equation implies that all particles, right from electrons to atoms, are regarded as having characteristics of both waves and particles. More precisely, it measures the wave properties of matter; it entails that particles with high momenta possess short wavelengths. This is in line with the de Broglie operation which is a substantial bar in quantum mechanical theory, which is intended to explain matter wave behaviour to give an account of the constitution of atomic and sub-atomic dimensions. This is because it retains several effects on so many science disciplines such as physics, chemistry as well as material science utilizing it to consistencies occurrences like electron interferences, atomic absorption, and movements of molecules among others.

$\lambda=\frac{h}{m v}=\frac{h}{p}=\frac{h}{\sqrt{2 m K E}}=\frac{h}{\sqrt{2 m q V}}$

where m is the mass of the particle, v is its velocity, p is its momentum,

KE is the Kinetic Energy of the particle,

V is the voltage across which the Charged particle having charge q is accelerated.

Some Solved Examples

Example 1: The de-Broglie wavelength of a particle of mass 6.63 g moving with a velocity of 100 ms^-1 is:

1) (correct) $10^{-33} \mathrm{~m}$
2) $10^{-35} \mathrm{~m}$
3) $10^{-31} \mathrm{~m}$
4) $10^{-25} \mathrm{~m}$

Solution

As discussed in concept:

De-Broglie wavelength:

$\lambda=\frac{h}{m v}=\frac{h}{p}$

- wherein

where m is the mass of the particle

v: its velocity

p: its momentum

So,

$\begin{aligned} & \lambda=\frac{6.625 \times 10^{-34}}{6.63 \times 10^{-3} \times 100} \\ & \lambda=10^{-33} \mathrm{~m}\end{aligned}$

Hence, the answer is the option (1).

Example 2: A stream of electrons from a heated filament was passed between two charged plates kept at a potential difference V esu. If e and m are the charge and mass of an electron, respectively, then the value of h/λ (where λ is wavelength associated with electron wave) is given by

1) $\mathrm{meV}$
2) $2 \mathrm{meV}$
3) $\sqrt{m e V}$
4) (correct) $\sqrt{2 m e V}$

Solution

We know that

De-Broglie wavelength:

$\lambda=\frac{h}{m v}=\frac{h}{p}$

- where

m is the mass of an electron.

v - its velocity.

p - its momentum.

$
p=\frac{h}{\lambda} \ldots \ldots
$

The kinetic energy of an electron in $\mathrm{eV}$
$
\begin{aligned}
& K \cdot E=\frac{p^2}{2 m} \\
& \therefore e V=\frac{p^2}{2 m}
\end{aligned}
$

where

V is the potential difference between two charge plates.

e is the charge of an electron.

Or $p=\sqrt{2 m e V}$
from equations (1) and (2)
$
\frac{h}{\lambda}=\sqrt{2 m e V}
$

Hence, the answer is an option (4).

Conclusion:

To sum up, de Broglie’s wavelength has had an overwhelming impact on how we understand the movement of matter at atomic and subatomic levels. By proposing that particles like electrons have a wave nature, Louis de Broglie changed the entire game of quantum mechanics and opened its doors to new revelations about the nature of particles and their interactions with each other. The de Broglie wavelength presents us with a very powerful way of describing moving objects which surpasses classical mechanics or wave theory.

Frequently Asked Questions (FAQs)

1. Explain the de Broglie wavelength.

The de Broglie wavelength is a concept used in quantum mechanics and it posits that particles in general, and electrons in specific, display wave behaviour. It postulates that every particle in motion, be it microscopic or minuscule, has a certain wavelength equal to the inverse of the momentum.

2. How do we write De Broglie equation?

De Broglie wavelength can be calculated by using the following formula: Stir = h/p where Stir is the de Broglie wavelength, h is the Planck’s constant which is equal to 6.626 x10 ^{– 34} Js, and p is the momentum which is defined by energy E divided by mc² where m is the mass of the particle and c is the speed of light in the vacuum.
- New variables introduced as De Broglie wavelength (lambda) equals h/momentum.h is Planck’s constant and p is the momentum of the particle. The equation shows that those particles travelling with more momentum have short wavelengths.

3. What does the wavelength indicated by de Broglie mean for chemistry?

In chemistry, they applied the concept of de Broglie’s wavelength to explain the motion of electrons around the nucleus or within molecules. It plays a vital role in understanding electron distribution, atomic structure, and various concepts related to chemical bonding.

4. How does the De Broglie equation relate to Heisenberg's uncertainty principle?

In regards to the uncertainty principle, The de Broglie wavelength is related to the uncertainty principle, which postulates there are certain measurable physical properties where their values cannot both be known at once. It prognosis the wave-particle duality fundamental in quantum mechanics.

5. Can you give some examples where the de Broglie wavelength can be used in physical sciences in its physical sense?

Real-life use is seen in electron microscopy to directly observe the atomic structure of objects, and in the aspects of quantum computing where the wave-particle duality of objects is employed in encoding information and encryption techniques.

6. What is the de Broglie relationship?

The de Broglie relationship is a fundamental principle in quantum mechanics that relates the wavelength of a particle to its momentum. It states that all matter exhibits both particle and wave properties, and the wavelength of a particle is inversely proportional to its momentum.

7. Who proposed the de Broglie hypothesis?

The de Broglie hypothesis was proposed by French physicist Louis de Broglie in 1924. He suggested that particles, like electrons, could exhibit wave-like properties, just as light waves could exhibit particle-like properties.

8. How is the de Broglie wavelength calculated?

The de Broglie wavelength (λ) is calculated using the formula: λ = h/p, where h is Planck's constant and p is the momentum of the particle. For a particle with mass m and velocity v, the formula becomes λ = h/(mv).

9. Why is the de Broglie relationship important in atomic structure?

The de Broglie relationship is crucial in understanding atomic structure because it explains why electrons in atoms exist in discrete energy levels. The wave nature of electrons leads to standing wave patterns around the nucleus, which correspond to the allowed electron orbitals.

10. How does the de Broglie relationship connect to Bohr's atomic model?

The de Broglie relationship provides a theoretical foundation for Bohr's atomic model. It explains why electrons can only exist in certain energy levels by showing that only specific electron wavelengths can form stable standing waves around the nucleus.

11. Can macroscopic objects have a de Broglie wavelength?

Yes, all objects have a de Broglie wavelength. However, for macroscopic objects, the wavelength is extremely small due to their large mass and is typically negligible in everyday situations. This is why we don't observe wave-like behavior in large objects.

12. How does the de Broglie wavelength change with particle velocity?

The de Broglie wavelength is inversely proportional to a particle's momentum. As the velocity (and thus momentum) of a particle increases, its de Broglie wavelength decreases. This means faster-moving particles have shorter wavelengths.

13. Why don't we observe wave-like behavior in everyday objects?

We don't observe wave-like behavior in everyday objects because their de Broglie wavelengths are extremely small due to their large mass. The wavelength is typically much smaller than the size of the object or any measuring apparatus, making it undetectable in practical situations.

14. How does the de Broglie relationship support the wave-particle duality concept?

The de Broglie relationship directly supports wave-particle duality by showing that particles can exhibit wave-like properties. It provides a mathematical link between a particle's momentum (a particle property) and its wavelength (a wave property), unifying these two seemingly contradictory aspects of matter.

15. What experimental evidence supports the de Broglie hypothesis?

The de Broglie hypothesis was experimentally confirmed by the Davisson-Germer experiment in 1927. They observed electron diffraction patterns when electrons were scattered by a nickel crystal, demonstrating the wave-like nature of electrons as predicted by de Broglie.

16. How does the de Broglie wavelength of an electron compare to that of a proton with the same kinetic energy?

For particles with the same kinetic energy, the one with less mass will have a longer de Broglie wavelength. Since electrons are much less massive than protons, an electron will have a significantly longer de Broglie wavelength than a proton with the same kinetic energy.

17. What is the relationship between de Broglie wavelength and particle size?

Generally, as the size of a particle decreases, its de Broglie wavelength becomes more significant relative to its size. This is why quantum effects are more pronounced for subatomic particles than for larger objects.

18. How does the de Broglie relationship relate to Heisenberg's uncertainty principle?

The de Broglie relationship is closely connected to Heisenberg's uncertainty principle. As a particle's position becomes more precisely known, its momentum becomes less certain, and vice versa. This is because determining position requires shorter wavelengths, which correspond to higher momenta.

19. Can the de Broglie relationship be applied to photons?

Yes, the de Broglie relationship can be applied to photons. In fact, for photons, the relationship between wavelength and momentum (E = hf = hc/λ) was already known before de Broglie's hypothesis. De Broglie extended this concept to all particles.

20. How does the de Broglie wavelength affect electron behavior in atoms?

The de Broglie wavelength of electrons in atoms determines the allowed electron orbitals. Only wavelengths that form standing waves around the nucleus are permitted, which explains the discrete energy levels and the stability of atoms.

21. What is the significance of the de Broglie relationship in electron microscopy?

The de Broglie relationship is crucial in electron microscopy because it determines the resolution limit of the microscope. Electrons with shorter wavelengths (higher energy) can resolve smaller features, allowing for higher magnification and better image quality.

22. How does temperature affect the de Broglie wavelength of gas particles?

As temperature increases, the average kinetic energy and velocity of gas particles increase. This leads to a decrease in their de Broglie wavelength. Therefore, higher temperatures result in shorter de Broglie wavelengths for gas particles.

23. What is the relationship between de Broglie wavelength and diffraction patterns?

The de Broglie wavelength determines the diffraction pattern produced when particles interact with a crystal lattice or other regular structures. Shorter wavelengths produce finer diffraction patterns, while longer wavelengths produce broader patterns.

24. How does the de Broglie relationship contribute to our understanding of quantum tunneling?

The de Broglie relationship helps explain quantum tunneling by showing that particles have wave-like properties. This wave nature allows particles to "tunnel" through potential barriers that they classically shouldn't be able to overcome, as their wave function can extend beyond the barrier.

25. What is the significance of the de Broglie relationship in the development of quantum mechanics?

The de Broglie relationship was a crucial step in the development of quantum mechanics. It bridged the gap between particle and wave theories, leading to the formulation of wave mechanics and eventually to the Schrödinger equation, which is fundamental to modern quantum theory.

26. How does the de Broglie wavelength of an electron change as it accelerates in an electric field?

As an electron accelerates in an electric field, its velocity increases, leading to an increase in its momentum. Consequently, its de Broglie wavelength decreases according to the relationship λ = h/p, where p is the increasing momentum.

27. Can the de Broglie relationship explain the stability of electron orbitals?

Yes, the de Broglie relationship helps explain the stability of electron orbitals. It shows that only certain electron wavelengths can form stable standing waves around the nucleus, corresponding to the allowed energy levels in atoms. This prevents electrons from spiraling into the nucleus.

28. How does the de Broglie wavelength relate to the uncertainty in a particle's position?

The de Broglie wavelength is inversely related to the uncertainty in a particle's position. A shorter wavelength (higher momentum) allows for more precise position measurements, while a longer wavelength (lower momentum) results in greater position uncertainty, in line with Heisenberg's uncertainty principle.

29. What role does the de Broglie relationship play in explaining atomic spectra?

The de Broglie relationship helps explain atomic spectra by providing a basis for the quantization of electron energy levels. The discrete spectral lines observed in atomic emission spectra result from electrons transitioning between these quantized energy states, which are determined by the allowed de Broglie wavelengths.

30. How does the de Broglie relationship apply to particles in a box?

For particles confined in a box, the de Broglie relationship determines the allowed wavelengths and thus the energy levels. Only wavelengths that fit perfectly within the box (forming standing waves) are permitted, leading to quantized energy states.

31. What is the connection between the de Broglie relationship and the Compton effect?

Both the de Broglie relationship and the Compton effect demonstrate the dual nature of matter and radiation. While the de Broglie relationship shows that particles can exhibit wave-like properties, the Compton effect shows that light (waves) can exhibit particle-like properties in interactions with matter.

32. How does the de Broglie wavelength of a neutron compare to its size?

The de Broglie wavelength of a neutron can be comparable to or even larger than its physical size, depending on its velocity. This is why neutrons can be used for diffraction experiments in studying crystal structures, as their wavelengths are on the order of interatomic distances.

33. What is the significance of the de Broglie relationship in nanotechnology?

In nanotechnology, the de Broglie relationship becomes increasingly important as device sizes approach the scale of electron wavelengths. It affects electron behavior in nanostructures, influencing properties like conductivity and optical characteristics, and is crucial in designing and understanding nanoscale electronic devices.

34. How does the de Broglie relationship contribute to our understanding of the photoelectric effect?

While the photoelectric effect primarily demonstrates the particle nature of light, the de Broglie relationship complements this by showing that particles can have wave-like properties. Together, they provide a complete picture of the wave-particle duality of both matter and radiation.

35. What is the relationship between de Broglie wavelength and the Schrödinger equation?

The de Broglie relationship is a key concept underlying the Schrödinger equation. The wave function in the Schrödinger equation describes the quantum state of a particle, and its wavelike nature is directly related to the de Broglie wavelength. The equation essentially describes how these matter waves evolve in time and space.

36. How does the de Broglie wavelength of an electron in a hydrogen atom compare to the atom's size?

The de Broglie wavelength of an electron in a hydrogen atom is on the same order of magnitude as the atom's size. This is why quantum mechanical effects are so important in atoms – the wave nature of the electron significantly influences its behavior and the atom's properties.

37. What is the significance of the de Broglie relationship in explaining the wave function collapse in quantum mechanics?

The de Broglie relationship helps explain wave function collapse by relating the particle and wave aspects of matter. When a measurement is made, the wave-like nature (described by the de Broglie wavelength) collapses to a particle-like state, illustrating the probabilistic nature of quantum mechanics.

38. How does the de Broglie relationship apply to molecules in diffraction experiments?

The de Broglie relationship applies to molecules in the same way it does to individual particles. In molecular diffraction experiments, the wavelength associated with the molecule's momentum determines the diffraction pattern. This allows for the study of molecular structures and dynamics.

39. What is the connection between the de Broglie relationship and the concept of matter waves?

The de Broglie relationship is the mathematical expression of the concept of matter waves. It quantifies how matter can exhibit wave-like properties by associating a wavelength with a particle's momentum, thereby directly describing matter waves.

40. How does the de Broglie relationship influence the behavior of electrons in conductors?

In conductors, the de Broglie wavelength of electrons affects their behavior. The wave nature of electrons leads to the formation of energy bands and explains phenomena like electron diffraction in crystals. It's crucial in understanding electrical conductivity and the properties of materials.

41. What role does the de Broglie relationship play in explaining the quantization of angular momentum in atoms?

The de Broglie relationship helps explain the quantization of angular momentum in atoms by showing that only certain electron wavelengths can form stable orbits around the nucleus. These allowed wavelengths correspond to specific angular momentum values, leading to the observed quantization.

42. How does the de Broglie wavelength change for relativistic particles?

For relativistic particles moving at speeds close to the speed of light, the de Broglie wavelength calculation must be modified to account for relativistic effects. The wavelength becomes λ = h / (γmv), where γ is the Lorentz factor. As speed increases, the wavelength decreases more slowly than in the non-relativistic case.

43. What is the significance of the de Broglie relationship in explaining electron diffraction in crystals?

The de Broglie relationship is crucial in explaining electron diffraction in crystals. It shows that electrons have wave-like properties with wavelengths comparable to interatomic distances in crystals. This allows electrons to produce diffraction patterns when interacting with crystal lattices, similar to X-ray diffraction.

44. How does the de Broglie relationship contribute to our understanding of quantum confinement in nanostructures?

The de Broglie relationship is key to understanding quantum confinement in nanostructures. When the size of a structure approaches the de Broglie wavelength of the confined particles (e.g., electrons), quantum effects become significant. This leads to discrete energy levels and unique optical and electronic properties in nanomaterials.

45. What is the connection between the de Broglie relationship and the concept of wave packets?

The de Broglie relationship is fundamental to the concept of wave packets. A wave packet is a superposition of waves with different wavelengths, representing a localized particle. The de Broglie relationship allows us to associate each component wave with a specific momentum, thereby describing the particle's motion and uncertainty in position and momentum.

46. How does the de Broglie relationship apply to free electrons versus bound electrons in atoms?

For free electrons, the de Broglie wavelength is directly related to their momentum. For bound electrons in atoms, the wavelength must form a standing wave pattern around the nucleus, leading to quantized energy levels. This difference explains why free electrons have a continuous energy spectrum while bound electrons have discrete energy levels.

47. What is the significance of the de Broglie relationship in explaining the double-slit experiment with electrons?

The de Broglie relationship is crucial in explaining the double-slit experiment with electrons. It shows that electrons have wave-like properties, allowing them to interfere with themselves and produce an interference pattern, just like light waves. This experiment provides direct evidence for the wave nature of particles.

48. How does the de Broglie relationship contribute to our understanding of quantum tunneling in scanning tunneling microscopy (STM)?

In STM, the de Broglie relationship helps explain how electrons can tunnel through the vacuum gap between the tip and the sample. The wave nature of electrons, as described by their de Broglie wavelength, allows their wave functions to extend beyond classical boundaries, enabling tunneling and making STM possible.

49. What is the relationship between the de Broglie wavelength and the Fermi wavelength in metals?

The Fermi wavelength in metals is related to the de Broglie wavelength of electrons at the Fermi energy. It represents the wavelength of electrons at the highest occupied energy level in the metal at absolute zero. Understanding this relationship is crucial for explaining the electronic properties of metals.

50. How does the de Broglie relationship apply to the concept of zero-point energy?

The de Broglie relationship is connected to zero-point energy through the uncertainty principle. Even at absolute zero, particles have a non-zero momentum due to their wave nature, leading to a minimum energy state. This zero-point energy is a direct consequence of the wave-particle duality described by the de Broglie relationship.

51. What role does the de Broglie relationship play in explaining the properties of superfluids?

The de Broglie relationship is important in understanding superfluids. At very low temperatures, the de Broglie wavelengths of particles in a fluid can become large enough to overlap, leading to quantum coherence effects. This quantum behavior at the macroscopic scale results in the unique properties of superfluids, such as zero viscosity.

52. How does the de Broglie relationship contribute to our understanding of the Pauli exclusion principle?

While the de Broglie relationship doesn't directly explain the Pauli exclusion principle, it provides the foundation for understanding electron behavior in atoms. The wave nature of electrons, described by their de Broglie wavelength, leads to the concept of electron orbitals, which are then subject to the Pauli exclusion principle.

53. What is the significance of the de Broglie relationship in explaining the quantization of vibrational energy in molecules?

The de Broglie relationship helps explain the quantization of vibrational energy in molecules by showing that molecular vibrations can be described as wave-like motions. Only certain vibrational wavelengths (and thus energies) are allowed, leading to the observe