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Today we know that every particle exhibits both matter and wave nature. This is called wave-particle duality . The concept that matter behaves like wave is called the de Broglie hypothesis , named after Louis de Broglie, who proposed it in 1924.

De Broglie Equation

Explanation of bohr's quantization rule.

De Broglie gave the following equation which can be used to calculate de Broglie wavelength, \(\lambda\), of any massed particle whose momentum is known:

\[\lambda = \frac{h}{p},\]

where \(h\) is the Plank's constant and \(p\) is the momentum of the particle whose wavelength we need to find.

With some modifications the following equation can also be written for velocity \((v)\) or kinetic energy \((K)\) of the particle (of mass \(m\)):

\[\lambda = \frac{h}{mv} = \frac{h}{\sqrt{2mK}}.\]

Notice that for heavy particles, the de Broglie wavelength is very small, in fact negligible. Hence, we can conclude that though heavy particles do exhibit wave nature, it can be neglected as it's insignificant in all practical terms of use.

Calculate the de Broglie wavelength of a golf ball whose mass is 40 grams and whose velocity is 6 m/s. We have \[\lambda = \frac{h}{mv} = \frac{6.63 \times 10^{-34}}{40 \times 10^{-3} \times 6} \text{ m}=2.76 \times 10^{-33} \text{ m}.\ _\square\]

One of the main limitations of Bohr's atomic theory was that no justification was given for the principle of quantization of angular momentum. It does not explain the assumption that why an electron can rotate only in those orbits in which the angular momentum of the electron, \(mvr,\) is a whole number multiple of \( \frac{h}{2\pi} \).

De Broglie successfully provided the explanation to Bohr's assumption by his hypothesis.

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de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that profoundly explains particle behavior at the quantum level. According to de Broglie hypothesis, particles like electrons, atoms, and molecules exhibit wave-like and particle-like properties.

This concept was introduced by French physicist Louis de Broglie in his doctoral thesis in 1924, revolutionizing our understanding of the nature of matter.

de Broglie Equation

A fundamental equation core to de Broglie hypothesis establishes the relationship between a particle’s wavelength and momentum . This equation is the cornerstone of quantum mechanics and sheds light on the wave-particle duality of matter. It revolutionizes our understanding of the behavior of particles at the quantum level. Here are some of the critical components of the de Broglie wavelength equation:

1. Planck’s Constant (h)

Central to this equation is Planck’s constant , denoted as “h.” Planck’s constant is a fundamental constant of nature, representing the smallest discrete unit of energy in quantum physics. Its value is approximately 6.626 x 10 -34 Jˑs. Planck’s constant relates the momentum of a particle to its corresponding wavelength, bridging the gap between classical and quantum physics.

2. Particle Momentum (p)

The second critical component of the equation is the particle’s momentum, denoted as “p”. Momentum is a fundamental property of particles in classical physics, defined as the product of an object’s mass (m) and its velocity (v). In quantum mechanics, however, momentum takes on a slightly different form. It is the product of the particle’s mass and its velocity, adjusted by the de Broglie wavelength.

The mathematical formulation of de Broglie wavelength is

We can replace the momentum by p = mv to obtain

The SI unit of wavelength is meter or m. Another commonly used unit is nanometer or nm.

This equation tells us that the wavelength of a particle is inversely proportional to its mass and velocity. In other words, as the mass of a particle increases or its velocity decreases, its de Broglie wavelength becomes shorter, and it behaves more like a classical particle. Conversely, as the mass decreases or velocity increases, the wavelength becomes longer, and the particle exhibits wave-like behavior. To grasp the significance of this equation, let us consider the example of an electron . 

de Broglie Wavelength of Electron

Electrons are incredibly tiny and possess a minimal mass. As a result, when they are accelerated, such as when they move around the nucleus of an atom , their velocities can become significant fractions of the speed of light, typically ~1%.

Consider an electron moving at 2 x 10 6 m/s. The rest mass of an electron is 9.1 x 10 -31 kg. Therefore,

These short wavelengths are in the range of the sizes of atoms and molecules, which explains why electrons can exhibit wave-like interference patterns when interacting with matter, a phenomenon famously observed in the double-slit experiment.

Thermal de Broglie Wavelength

The thermal de Broglie wavelength is a concept that emerges when considering particles in a thermally agitated environment, typically at finite temperatures. In classical physics, particles in a gas undergo collision like billiard balls. However, particles exhibit wave-like behavior at the quantum level, including wave interference phenomenon. The thermal de Broglie wavelength considers the kinetic energy associated with particles due to their thermal motion.

At finite temperatures, particles within a system possess a range of energies described by the Maxwell-Boltzmann distribution. Some particles have relatively high energies, while others have low energies. The thermal de Broglie wavelength accounts for this distribution of kinetic energies. It helps to understand the statistical behavior of particles within a thermal ensemble.

Mathematical Expression

The thermal de Broglie wavelength (λ th ) is determined by incorporating both the mass (m) of the particle and its thermal kinetic energy (kT) into the de Broglie wavelength equation:

Here, k is the Boltzmann constant, and T is the temperature in Kelvin.

• de Broglie Wave Equation – Chem.libretexts.org  
• de Broglie Wavelength – Spark.iop.org
• de Broglie Matter Waves – Openstax.org
• Wave Nature of Electron – Hyperphysics.phy-astr.gsu.edu

Article was last reviewed on Friday, October 6, 2023

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De Broglie Hypothesis

Does All Matter Exhibit Wave-like Properties?

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The De Broglie hypothesis proposes that all matter exhibits wave-like properties and relates the observed wavelength of matter to its momentum. After Albert Einstein's photon theory became accepted, the question became whether this was true only for light or whether material objects also exhibited wave-like behavior. Here is how the De Broglie hypothesis was developed.

De Broglie's Thesis

In his 1923 (or 1924, depending on the source) doctoral dissertation, the French physicist Louis de Broglie made a bold assertion. Considering Einstein's relationship of wavelength lambda to momentum p , de Broglie proposed that this relationship would determine the wavelength of any matter, in the relationship:

lambda = h / p

recall that h is Planck's constant

This wavelength is called the de Broglie wavelength . The reason he chose the momentum equation over the energy equation is that it was unclear, with matter, whether E should be total energy, kinetic energy, or total relativistic energy. For photons, they are all the same, but not so for matter.

Assuming the momentum relationship, however, allowed the derivation of a similar de Broglie relationship for frequency f using the kinetic energy E k :

f = E k / h

Alternate Formulations

De Broglie's relationships are sometimes expressed in terms of Dirac's constant, h-bar = h / (2 pi ), and the angular frequency w and wavenumber k :

p = h-bar * kE k

= h-bar * w

Experimental Confirmation

In 1927, physicists Clinton Davisson and Lester Germer, of Bell Labs, performed an experiment where they fired electrons at a crystalline nickel target. The resulting diffraction pattern matched the predictions of the de Broglie wavelength. De Broglie received the 1929 Nobel Prize for his theory (the first time it was ever awarded for a Ph.D. thesis) and Davisson/Germer jointly won it in 1937 for the experimental discovery of electron diffraction (and thus the proving of de Broglie's hypothesis).

Further experiments have held de Broglie's hypothesis to be true, including the quantum variants of the double slit experiment . Diffraction experiments in 1999 confirmed the de Broglie wavelength for the behavior of molecules as large as buckyballs, which are complex molecules made up of 60 or more carbon atoms.

Significance of the de Broglie Hypothesis

The de Broglie hypothesis showed that wave-particle duality was not merely an aberrant behavior of light, but rather was a fundamental principle exhibited by both radiation and matter. As such, it becomes possible to use wave equations to describe material behavior, so long as one properly applies the de Broglie wavelength. This would prove crucial to the development of quantum mechanics. It is now an integral part of the theory of atomic structure and particle physics.

Macroscopic Objects and Wavelength

Though de Broglie's hypothesis predicts wavelengths for ​matter of any size, there are realistic limits on when it's useful. A baseball thrown at a pitcher has a de Broglie wavelength that is smaller than the diameter of a proton by about 20 orders of magnitude. The wave aspects of a macroscopic object are so tiny as to be unobservable in any useful sense, although interesting to muse about.

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De Broglie Hypothesis

• Practice Questions

The De Broglie Hypothesis is a fundamental concept in proposed by the French physicist Louis de Broglie in 1924. This groundbreaking idea introduced the wave-particle duality of matter, suggesting that not only light (previously understood to exhibit both wave-like and particle-like properties) but all forms of matter have wave-like characteristics.

De Broglie Equation Derivation

Louis de Broglie hypothesized that if light can display dual characteristics (both wave-like and particle-like properties), then particles, such as electrons, might also exhibit similar dual characteristics. His derivation was based on the parallels between the equations for energy and momentum in both light and material particles.

Step 1: Relating Energy and Momentum for Light

For photons (light particles), the energy (𝐸 E ) and momentum (𝑝 p ) are related by the equations:

Here, ℎ h is Planck’s constant, 𝑓 f is the frequency of the photon, and 𝑐 c is the speed of light. By substituting the energy equation into the momentum equation, we get:

Since the wavelength ( λ ) of a photon is related to its frequency by 𝑐 = 𝜆𝑓, we can rewrite 𝑓 as:

Substituting back, the momentum of a photon can be expressed as:

Step 2: Applying the Concept to Material Particles

De Broglie proposed that if light (which was known to have wave-like properties) has a wavelength given 𝜆 = ℎ/𝑝​, then particles, such as electrons, should also have a wavelength describable by a similar relationship, even though they have mass. Thus, he extended the equation to all matter, proposing that:

where p is now the momentum of the particle, which for a non-relativistic particle is given by:

Here, m is the mass of the particle and v is its velocity.

Step 3: De Broglie Wavelength of Particles

Combining the expressions, the de Broglie wavelength for any particle is thus given by:

This equation implies that every moving particle has a wave associated with it, and the wavelength of that wave is inversely proportional to the particle’s momentum. This groundbreaking idea led to the development of wave mechanics and has been fundamental in many areas of quantum physics, such as the theory behind quantum fields and elementary particles.

De Broglie Wavelength for an Electron

To calculate the De Broglie wavelength of an electron, we use the formula derived by Louis de Broglie which relates a particle’s wavelength to its momentum. The formula is:

• 𝜆 is the wavelength,
• ℎ is Planck’s constant, approximately 6.626×10⁻³⁴ Joule seconds,
• 𝑝 is the momentum of the electron.

Calculating Momentum

The momentum 𝑝 p of an electron can be calculated using the formula: 𝑝=𝑚𝑣 p = m v where:

• 𝑚 is the mass of the electron, approximately 9.109×10⁻³¹ kg,
• 𝑣 is the velocity of the electron.

Significance of the De Broglie Equation

The De Broglie equation , 𝜆 = ℎ/𝑝 ​, is a cornerstone in quantum mechanics, providing a profound understanding of the wave-particle duality of matter. Its implications extend far beyond theoretical physics, impacting various scientific fields and technologies.

Fundamental to Quantum Mechanics

The equation integrates wave-like behavior into the description of elementary particles, bridging a gap between classical and quantum physics. This wave-particle duality is essential for the development of quantum mechanics, influencing the theoretical framework that describes how subatomic particles behave.

Basis for Modern Physics Theories

De Broglie’s insights laid the groundwork for Schrödinger to formulate his wave equation, which uses the concept of wavefunctions to describe the statistical behavior of systems. The wave-particle duality concept is integral to quantum field theory, which extends quantum mechanics to more complex systems including fields and forces.

Experimental Validation and Applications

The equation has been empirically validated through experiments such as electron diffraction and neutron diffraction, which demonstrate that particles exhibit wave-like behavior under certain conditions. These experiments are pivotal for technologies such as electron microscopes, which rely on electron waves to achieve high-resolution imaging beyond the capability of traditional optical microscopes.

Technological Impact

Understanding the wave properties of particles enables the exploitation of phenomena such as quantum tunneling, utilized in devices like tunnel diodes and the scanning tunneling microscope. These applications are crucial in electronics and materials science, where quantum effects are significant.

Educational and Conceptual Influence

The De Broglie equation has also profoundly impacted educational approaches in physics, providing a fundamental concept that challenges and expands our understanding of the natural world. It encourages a more nuanced view of matter, essential for students and researchers delving into quantum physics.

Relation between De Broglie Equation and Bohr’s Hypothesis of Atom

De broglie’s equation.

Louis de Broglie introduced his theory of electron waves in 1924, which proposed that particles could exhibit properties of waves. His famous equation relates the wavelength of a particle to its momentum: 𝜆 = ℎ/𝑝 where 𝜆is the wavelength, ℎ is Planck’s constant, and 𝑝 p is the momentum of the particle.

Bohr’s Hypothesis of the Atom

Niels Bohr proposed his model of the atom in 1913. His key hypothesis was that electrons orbit the nucleus in distinct orbits without radiating energy, contrary to what classical electromagnetism would predict. To explain the stability of these orbits, Bohr introduced the concept of quantization:

• Electrons can only occupy certain allowed orbits.
• The angular momentum of electrons in these orbits is quantized, specifically, it is an integer multiple of the reduced Planck constant
• (ℏ): 𝐿 = 𝑛×ℎ/2𝜋 = 𝑛ℏ
• where 𝐿 L is the angular momentum, n is a positive integer (quantum number), and h is Planck’s constant.

Integrating De Broglie’s Equation with Bohr’s Model

De Broglie’s theory was revolutionary because it provided a theoretical justification for Bohr’s quantization condition by interpreting the electron not just as a particle, but as a wave that must form a standing wave pattern around the nucleus. For the electron wave to be stable and not interfere destructively with itself, the circumference of the electron’s orbit must be an integer multiple of its wavelength:

where 𝑟 is the radius of the electron’s orbit, and n is an integer. This condition ensures that the wave ‘fits’ perfectly into its orbital path around the nucleus.

Substituting De Broglie’s Equation

By substituting De Broglie’s expression for the wavelength into the condition for a stable orbit, we get:

Using the expression for momentum 𝑝=𝑚𝑣 p = mv and the definition of angular momentum 𝐿=𝑚𝑣𝑟 L = mvr , we can relate this to Bohr’s quantization of angular momentum:

Thus, De Broglie’s hypothesis not only supported Bohr’s model but also suggested a deeper wave nature of the electron. It bridged the gap between the quantized orbits of Bohr’s atom model and the wave-like behavior of particles, paving the way for modern quantum mechanics, which would further refine and expand these ideas in the Schrodinger equation and beyond.

Examples of De Broglie Hypothesis

Electron Diffraction

One of the first confirmations of De Broglie’s hypothesis was the observation of electron diffraction patterns. When electrons are passed through thin metal foils or across a crystal, they produce diffraction patterns similar to those produced by light waves, confirming that electrons behave as waves under certain conditions.

Scanning Tunneling Microscope (STM)

The scanning tunneling microscope, which can image surfaces at the atomic level, operates based on the quantum tunneling of electrons between the microscope’s tip and the surface. The wave nature of electrons, as predicted by De Broglie, is fundamental to the operation of this instrument.

Bohr Model of the Atom

De Broglie’s ideas extended the Bohr model by providing a theoretical basis for the quantization of electron orbits in atoms. His hypothesis suggested that electrons form standing wave patterns around the nucleus, which only occur at certain discrete (quantized) orbits.

Matter Waves

The concept of matter waves is essential in fields like quantum mechanics and has led to further developments in wave mechanics. This includes the use of neutrons, atoms, and molecules in wave-like applications, similar to how light and electrons are used.

Neutron Interferometry

Neutron beams, used in neutron interferometry, exhibit wave-like interference effects. These experiments have provided precise measurements of neutron properties and fundamental quantum phenomena, supporting De Broglie’s hypothesis at larger scales.

Atomic Force Microscopy (AFM)

AFM, like STM, uses the principles of quantum mechanics and the wave-like properties of atoms on a surface to achieve high-resolution imaging. The forces between the tip’s atoms and the sample’s atoms are influenced by their wave functions.

How was the De Broglie Equation derived?

Louis de Broglie proposed that particles of matter, like electrons, could exhibit wave-like properties similar to light. Combining Einstein’s equation relating energy and mass (𝐸 = 𝑚𝑐²) with Planck’s equation relating energy and frequency (𝐸 = ℎ𝑓), and considering the wave equation (𝑐 = 𝑓𝜆), De Broglie derived his hypothesis that matter behaves as waves.

Why is the De Broglie Equation important?

The De Broglie Equation is crucial for understanding quantum mechanics as it introduces the concept of wave-particle duality. This concept states that every particle or quantum entity can exhibit both particle-like and wave-like behavior. It forms the basis for the development of quantum theory, particularly in the formulation of wave mechanics.

Can the De Broglie Equation be applied to all objects?

While theoretically applicable to all matter, in practice, the wave-like properties described by the De Broglie Equation are significant only for very small objects, like subatomic particles. For larger objects, the wavelengths calculated by the equation become so small that they are not detectable with current technology.

What is wave-particle duality?

Wave-particle duality is a fundamental concept of quantum mechanics that suggests that every particle or quantum entity may be partly described in terms not only of particles, but also of waves. It means that elementary particles such as electrons and photons exhibit both particle-like and wave-like properties, depending on the experimental setup.

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• De Broglie Equation

Introduction

The wave nature of light was the only aspect that was considered until Neil Bohr’s model. Later, however, Max Planck in his explanation of quantum theory hypothesized that light is made of very minute pockets of energy which are in turn made of photons or quanta. It was then considered that light has a particle nature and every packet of light always emits a certain fixed amount of energy. 

By this, the energy of photons can be expressed as:

E = hf = h * c/λ

Here, h is Plank’s constant

F refers to the frequency of the waves

Λ implies the wavelength of the pockets

Therefore, this basically insinuates that light has both the properties of particle duality as well as wave. 

Louis de Broglie was a student of Bohr, who then formulated his own hypothesis of wave-particle duality, drawn from this understanding of light. Later on, when this hypothesis was proven true, it became a very important concept in particle physics. 

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What is the De Broglie Equation?

Quantum mechanics assumes matter to be both like a wave as well as a particle at the sub-atomic level. The De Broglie equation states that every particle that moves can sometimes act as a wave, and sometimes as a particle. The wave which is associated with the particles that are moving are known as the matter-wave, and also as the De Broglie wave. The wavelength is known as the de Broglie wavelength. 

For an electron, de Broglie wavelength equation is:       

λ = \[\frac{h}{mv}\]

Here, λ points to the wave of the electron in question

M is the mass of the electron

V is the velocity of the electron

Mv is the momentum that is formed as a result

It was found out that this equation works and applies to every form of matter in the universe, i.e, Everything in this universe, from living beings to inanimate objects, all have wave particle duality. 

Significance of De Broglie Equation

De Broglie says that all the objects that are in motion have a particle nature. However, if we look at a moving ball or a moving car, they don’t seem to have particle nature. To make this clear, De Broglie derived the wavelengths of electrons and a cricket ball. Now, let’s understand how he did this.  

De Broglie Wavelength 

1. De Broglie Wavelength for a Cricket Ball

Let’s say,Mass of the ball  = 150 g (150 x 10⁻³ kg),

Velocity = 35 m/s, 

and  h = 6.626 x 10⁻³⁴ Js

Now, putting these values in the equation 

λ = (6.626 * 10 to power of -34)/ (150 * 10 to power of -3 *35) 

This yields

λBALL = 1.2621 x 10 to the power of -34 m,

Which is 1.2621 x 10 to the power of -24 Å.

We know that Å is a very small unit, and therefore the value is in the power of 10−24−24^{-24}, which is a very small value. From here, we see that the moving cricket ball is a particle.

Now, the question arises if this ball has a wave nature or not. Your answer will be a big no because the value of λBALL is immeasurable. This proves that de Broglie’s theory of wave-particle duality is valid for the moving objects ‘up to’ the size (not equal to the size) of the electrons.

De Broglie Wavelength for an Electron

We know that me  = 9.1 x 10 to power of -31 kg

and ve = 218 x 10 to power of -6 m/s

Now, putting these values in the equation  λ = h/mv, which yields λ = 3.2 Å. 

This value is measurable. Therefore, we can say that electrons have wave-particle duality. Thus all the big objects have a wave nature and microscopic objects like electrons have wave-particle nature.

E  = hν  = \[\frac{hc}{\lambda }\]

The Conclusion of De Broglie Hypothesis

From de Broglie equation for a material particle, i.e.,  

λ = \[\frac{h}{p}\]or \[\frac{h}{mv}\], we conclude the following:

i. If v = 0, then λ = ∞, and

If v = ∞, then λ = 0

It means that waves are associated with the moving material particles only. This implies these waves are independent of their charge. 

FAQs on De Broglie Equation

1.The De Broglie hypothesis was confirmed through which means?

De Broglie had not proved the validity of his hypothesis on his own, it was merely a hypothetical assumption before it was tested out and consequently, it was found that all substances in the universe have wave-particle duality. A number of experiments were conducted with Fresnel diffraction as well as a specular reflection of neutral atoms. These experiments proved the validity of De Broglie’s statements and made his hypothesis come true. These experiments were conducted by some of his students. 

2.What exactly does the De Broglie equation apply to?

In very broad terms, this applies to pretty much everything in the tangible universe. This means that people, non-living things, trees and animals, all of these come under the purview of the hypothesis. Any particle of any substance that has matter and has linear momentum also is a wave. The wavelength will be inversely related to the magnitude of the linear momentum of the particle. Therefore, everything in the universe that has matter, is applicable to fit under the De Broglie equation. 

3.Is it possible that a single photon also has a wavelength?

When De Broglie had proposed his hypothesis, he derived from the work of Planck that light is made up of small pockets that have a certain energy, known as photons. For his own hypothesis, he said that all things in the universe that have to matter have wave-particle duality, and therefore, wavelength. This extends to light as well, since it was proved that light is made up of matter (photons). Hence, it is true that even a single photon has a wavelength. 

4.Are there any practical applications of the De Broglie equation?

It would be wrong to say that people use this equation in their everyday lives, because they do not, not in the literal sense at least. However, practical applications do not only refer to whether they can tangibly be used by everyone. The truth of the De Broglie equation lies in the fact that we, as human beings, also are made of matter and thus we also have wave-particle duality. All the things we work with have wave-particle duality. 

5.Does the De Broglie equation apply to an electron?

Yes, this equation is applicable for every single moving body in the universe, down to the smallest subatomic levels. Just how light particles like photons have their own wavelengths, it is also true for an electron. The equation treats electrons as both waves as well as particles, only then will it have wave-particle duality. For every electron of every atom of every element, this stands true and using the equation mentioned, the wavelength of an electron can also be calculated.  

6.Derive the relation between De Broglie wavelength and temperature.

We know that the average KE of a particle is:

                       K = 3/2 k b T

Where k b is Boltzmann’s constant, and

T   = temperature in Kelvin

The kinetic energy of a particle is  ½ mv²

The momentum of a particle, p = mv = √2mK

= √2m(3/2)KbT = √2mKbT 

de Broglie wavelength, λ = h/p = h√2mkbT 

7.If an electron behaves like a wave, what should determine its wavelength and frequency?

Momentum and energy determine the wavelength and frequency of an electron.

8. Find λ associated with an H 2 of mass 3 a.m.u moving with a velocity of 4 km/s.

Here,  v = 4 x 10³ m/s 

Mass of hydrogen = 3 a.m.u = 3 x 1.67 x 10⁻²⁷kg = 5 x 10⁻²⁷kg    

On putting these values in the equation λ = h/mv we get

λ = (6.626 x 10⁻³⁴)/(4 x 10³ x 5 x 10⁻²⁷) = 3 x 10⁻¹¹ m.

9. If the KE of an electron increases by 21%, find the percentage change in its De Broglie wavelength.

We know that  λ = h/√2mK

So,  λ i = h/√(2m x 100) , and λ f = h/√(2m x 121)

% change in λ is:

Change in wavelength/Original x 100 = (λ fi - λ f )/λ i = ((h/√2m)(1/10 - 1/21))/(h/√2m)(1/10) 

On solving, we get

% change in λ = 5.238 %

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De-Broglie Wavelength Formula: Equations, Solved Examples

De-Broglie Wavelength Formula – Einstein proposed that any electromagnetic radiation, including light which was, till then, considered an electromagnetic wave, in fact, showed particle-like nature. He coined the word “photon” for the quanta or particle of light. Soon, scientists began to wonder if other particles could also have a dual wave-particle nature.

In 1924, French scientist Louis de Broglie derived an equation, known as the De Broglie Wavelength Formula, that described the wave nature of any particle. Thus, establishing the wave-particle duality for the matter. Microscopic particle-like electrons also proved to possess this dual nature property. Let us learn about the equation proposed by de-Broglie in detail in this article.

De Broglie’s Hypothesis and Equation

Louis-de-Broglie explained the concept of de-Broglie waves in the year 1923. In his thesis, he suggested that any moving particle, whether microscopic or macroscopic, will be related to a wave character. This was later experimented with and proved by Davisson and Germer within the year 1927. The waves associated with matter were called ‘Matter Waves’. These waves explain the character of the wave associated with the particle. We know that electromagnetic radiation exhibit the dual nature of a particle (having a momentum) and wave (expressed in frequency, and wavelength).

He further proposed a relation between the speed and momentum with the wavelength if the particle had to behave as a wave. Since, at non-relativistic speeds, the momentum of a particle will be adequate to its mass \(\text {m}\), multiplied by its velocity \(\text {v}\). Thus, according to de-Broglie, the wavelength \(\left( \lambda \right)\) of any moving object is given by:

\(\lambda = \frac{{\rm{h}}}{{\rm{p}}}\)

Where \(\text {h}\) is Planck’s constant and \(\text {p}\) is the momentum of the particle.

Derivation of De Broglie’s Wavelength

From Einstein’s relation of mass-energy equivalence, we know that,

\({\text{E}} = {\text{m}}{{\text{c}}^2} \cdots (1)\)

\(\text {E}=\) energy of the particle

\(\text {m}=\) mass of the particle

\(\text {c}=\) speed of light

According to Planck’s theory, every quantum of a wave has a discrete amount of energy associated with it, and he gave the equation:

\({\text{E}} = {\text{hf}} \cdots (2)\)

\(\text {E}:\) energy of the particle

\(\text {h}=6.62607 \times 10^{-34} \mathrm{Js}:\) Planck’s constant

\(\text {f}=\) frequency

De-Broglie’s hypothesis suggested that particles and waves behave as similar entities. Thus, he equated the energy relation for both particle and wave; equating equations \((1)\) and \((2)\), we get:

\(\text {mc}^{2}=\text {hf}\)

Since the particles generally do not travel at the speed of light, De Broglie substituted the speed of light \(\text {c}\), with the velocity of a real particle \(\text {v}\), and obtained:

\({\text{m}}{{\text{v}}^2} = {\text{hf}} \cdots (3)\)

If \(\lambda \) be the wavelength of the wave, then the frequency will be: \({\rm{f}} = \frac{{\rm{v}}}{\lambda }\)

 Substituting this in equation \((3)\), we get:

\({\rm{m}}{{\rm{v}}^2} = \frac{{{\rm{hv}}}}{\lambda }\)

\(\lambda = \frac{{\rm{h}}}{{{\rm{mv}}}}\)

or, \(\lambda = \frac{{\rm{h}}}{{\rm{p}}}\,\,\, \cdots (4)\)

Where \(\text {p}\) is the momentum of the particle.

De Broglie Wavelength and Kinetic Energy

The kinetic energy of an object of mass \(\text {m}\) moving with velocity \(\text {v}\) is given as:

\({\text{K}} = \frac{1}{2}{\text{m}}{{\text{v}}^2}\)

or, \({\text{K}} = \frac{1}{2}{\text{mv}} \cdot {\text{v}}\)

\({\text{m}} \cdot {\text{K}} = \frac{1}{2}{({\text{mv}})^2}\)

Since, \(\text {p}=\text {mv}\), Thus:

\({\text{m}}.{\text{K}} = \frac{1}{2}{({\text{p}})^2}\)

From equation \((4),\,{\rm{p}} = \frac{{\rm{h}}}{\lambda }\)

\( \Rightarrow {\rm{m}} \cdot {\rm{K}} = \frac{1}{2}{\left( {\frac{{\rm{h}}}{\lambda }} \right)^2}\)

\({\lambda ^2} = \frac{{{{\rm{h}}^2}}}{{2\,{\rm{mK}}}}\)

\(\lambda = \frac{{\rm{h}}}{{\sqrt {2\,{\rm{mK}}} }}\)

De-Broglie Wavelength and Potential

When a charged particle, having a charge \(\text {q}\) is accelerated through an external potential difference \(\text {V}\), the energy of the particle can be given as:

\({\text{E}} = {\text{qV}} \cdots ({\text{i}})\)

According to Planck’s equation,

\(\text {E}=\text {hf}\)

Since, \({\rm{f}} = \frac{{\rm{v}}}{\lambda }\)

Therefore, \({\rm{E}} = {\rm{h}}\frac{{\rm{v}}}{\lambda }\,\,\, \ldots ({\rm{ii}})\)

Equating the equations \(\left({\text{i}} \right)\) and \(\left({\text{ii}} \right)\),

\({\rm{qV}} = {\rm{h}}\frac{{\rm{v}}}{\lambda }\)

or, \(\lambda = \frac{{{\rm{hv}}}}{{{\rm{qV}}}}\)

Thermal De Broglie Wavelength

There exists a relation between the De-Broglie equation and the temperature of the given gas molecules, and the thermal de Broglie wavelength gives it \(\left( {{\lambda _{{\rm{Th}}}}} \right).\) The Thermal de Broglie equation represents the average value of the de Broglie wavelength of the gas particles at the specified temperature in an ideal gas.

The expression gives the thermal de Broglie wavelength at temperature \(\text {T}\):

\({\lambda _{{\rm{Th}}}} = \lambda  = \frac{{\rm{h}}}{{\sqrt {2\,{\rm{m}}{{\rm{k}}_{\rm{B}}}{\rm{T}}} }}\)

\(h=\) Planck constant

\(m=\) mass of a gas particle

\({{\text{k}}_{\text{B}}} = \) Boltzmann constant

De Broglie Wavelength of Day-to-Day Objects

According to the hypothesis, all particles have a wave associated with them. That is true for us humans and the objects around us. To get an idea of the de-Broglie wavelength associated with macroscopic particles:

Let us find the wavelength of a wave associated with a car of mass \(1000 \mathrm{~kg}\) moving with the velocity of \(10 \mathrm{~m} / \mathrm{s}.\)

The wavelength associated with the car will be: \(\lambda = \frac{{\rm{h}}}{{{\rm{mv}}}}\)

\(\lambda=\frac{6.62607 \times 10^{-34} \mathrm{Js}}{1000 \mathrm{~kg} \times 10 \mathrm{~m} / \mathrm{s}}=6.6 \times 10^{-30} \mathrm{~m}=6.6 \times 10^{-21} \mathrm{~nm}\)

Thus, the value of wavelength associated with this car is insignificant.

Similarly, for other macroscopic objects with large mass values, the wavelength associated with them is so small that it can not be detected.

De-Broglie Wavelength of an Electron

As we have seen above, the matter waves associated with real objects is so small that it is of no good use to us. But for sub-atomic particles with negligible masses, the value of de-Broglie wavelength is substantial. To calculate the de-Broglie wavelength associated with a microscopic particle,

Let us take an electron of mass \({\rm{m}} = 9.1 \times {10^{ – 31}}\;{\rm{kg}},\) moving with the speed of light, i.e., \(\text {c}=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\), then the de-Broglie wavelength associated with it can be given as:

\(\lambda = \frac{{\rm{h}}}{{{\rm{mc}}}}\)

\(\lambda=\frac{6.62607 \times 10^{-34} \mathrm{Js}}{9.1 \times 10^{-31} \mathrm{~kg} \times 3 \times 10^{8} \mathrm{~m} / \mathrm{s}}=0.7318 \times 10^{-11} \mathrm{~m}=0.073 \mathrm{~A}^{\circ}\)

This is a substantial value. Thus, the de-Broglie wavelength associated has a significant value, and it can be detected. 

Relation Between De-Broglie Wavelength and Potential (for an electron)

The expression for the de-Broglie wavelength of an electron,

If the electron having a charge e is moving under an external potential \(\text {V}\), then,

The kinetic energy of the electron, \({\text{K}} = {\text{eV}}\)

Substituting this expression in the above equation,

\(\lambda = \frac{{\rm{h}}}{{\sqrt {2\,{\rm{meV}}} }}\)

Put, \(h=6.62607 \times 10^{-34} \mathrm{Js}\)

\(\text {e}=1.6 \times 10^{-19} \mathrm{C}\)

\(\text {m}=9.1 \times 10^{-31} \mathrm{~kg}\)

\(\lambda = \frac{{12.27}}{{\sqrt {\rm{V}} }}{\rm{A}}^\circ \)

De Broglie Wavelength- Solved Problems

Q.1. A certain photon has a momentum of \(1.50 \times 10^{-27} \mathrm{~kg} \mathrm{~m} / \mathrm{s}\). What will be the photon’s de Broglie wavelength?

Ans: \(\text {p}=1.50 \times 10^{-27} \mathrm{~kg} \mathrm{~m} / \mathrm{s}\) Plank’s constant, \(\text {h}=6.62607 \times 10^{-34} \mathrm{Js}\) The de Broglie wavelength of the photon can be computed using the formula: \(\lambda = \frac{{\rm{h}}}{{\rm{p}}}\) \(=\frac{6.62607 \times 10^{-34} \mathrm{Js}}{1.50 \times 10^{-27} \mathrm{kgm} / \mathrm{s}}\) \(=4.42 \times 10^{-7} \mathrm{~m}\) \(=442 \times 10^{-9} \mathrm{~m}\) \(=442 \mathrm{~nm}\) The de Broglie wavelength of the photon will be \(442 \mathrm{~nm}\), and this wavelength lies in the blue-violet part of the visible light spectrum.

Q.2. What is the de Broglie wavelength of an electron which is accelerated through a potential difference of \(10\, \mathrm{kV}\) ?

Ans:  If the electron having a charge \(\text {e}\) is moving under an external potential \(\text {V}\), then the expression for the de-Broglie wavelength of an electron is: \(\lambda = \frac{{12.27}}{{\sqrt {\rm{V}} }}{\rm{A}}^\circ \) We are given, \(\text {V}=10 \mathrm{kV}=10 \times 10^{3} \mathrm{~V}=10^{4} \mathrm{~V}\) \(\lambda = \frac{{12.27}}{{\sqrt {{{10}^4}} }}{\rm{A}}^\circ \) \(\lambda = \frac{{12.27}}{{100}}{\rm{A}}^\circ = 0.1227{\mkern 1mu} {\rm{A}}^\circ \)

According to de-Broglie, the wavelength \(\left( \lambda \right)\) of any moving object is given by: \(\lambda = \frac{{\rm{h}}}{{\rm{p}}},\) Where \(\text {h}\) is Planck’s constant and \(\text {p}\) is the mass of the particle.

The relation between de-Broglie wavelength and the kinetic energy of an object of mass \(\text {m}\) moving with velocity \(\text {v}\) is given as: \(\lambda = \frac{{\rm{h}}}{{\sqrt {2\,{\rm{mK}}} }}\)

When a charged particle having a charge \(\text {q}\) is accelerated through an external potential difference \(\text {V}\), de-Broglie wavelength, \(\lambda = \frac{{{\rm{hv}}}}{{{\rm{qV}}}}\)

The expression for the de-Broglie wavelength of an electron, \(\lambda = \frac{{\rm{h}}}{{\sqrt {2\,{\rm{mK}}} }}\) or \(\lambda = \frac{{\rm{h}}}{{\sqrt {2\,{\rm{meV}}} }}\)

Frequently Asked Questions on De Broglie Wavelength Formula

The most commonly asked questions on De Broglie Wavelength Formula are answered here:

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de Broglie hypothesis for matter waves in quantum theory

Hypothesis is a predicted theory that satisfies other phenomena but isn’t verified experimentally. Scientist de Broglie gave a such type of theory for the matter particles which plays important roles in Quantum mechanics. In this article, we are going to discuss de Broglie hypothesis for matter waves in quantum mechanics , its explanation and formula and some numerical problems based on de Broglie’s hypothesis formula.

Statement of de Broglie hypothesis

Equation of de broglie hypothesis.

If m is the mass of the matter particle which is moving with a speed v , then de Broglie hypothesis gives the equation for the wavelength of the matter wave as \small {\color{Blue} \lambda =\frac{h}{mv}}

E xplanation of de Broglie hypothesis

de Broglie hypothesis helps people to know the wave and particle duality nature of matter particles. Scientist de Broglie predicts that all the matter particles like electron, proton, atom, molecules, etc. which have very very small masses behave like waves. Therefore, matter particles have another name matter waves.

Some questions and numerical problems on de Broglie’s hypothesis

1. find de broglie wavelength of an electron moving in the first orbit of hydrogen atom..

Velocity of the electron in the first orbit of a hydrogen atom is v = 2.18×10 6 m/s

or, the de Broglie wavelength of the electron is 3.33×10 -10 meter or 3.33 Angstrom.

2. Which experiment confirms de Broglie hypothesis experimentally?

3. what is the expression for de broglie wavelength of a photon.

or, \small {\color{Blue} \lambda =\frac{hc}{E}}

Where c is the speed of light and E is the energy of photon and E=pc

2 thoughts on “de Broglie hypothesis for matter waves in quantum theory”

Comments are closed.

De Broglie's Hypothesis ( AQA A Level Physics )

Revision note.

De Broglie's Hypothesis of Wave-Particle Duality

What was debroglie's hypothesis.

• Louis DeBroglie hypothesised that all particles can behave both like waves and like particles, following Einstein's work with photons
• By equating two equations from Einstein, he derived an equation for the momentum of a photon:
• Where  h  is Planck's constant,  c   is the speed of light, m   is mass, λ  is wavelength and  f   is frequency
• mc   is the momentum, p , of a photon - DeBroglie extended this idea to particles with mass to obtain the relation you should recall from Particles & Radiation:

Finding the Wavelength of Accelerated Particles

• Finding their momentum directly is difficult, but recall from The Discovery of the Electron that the work done on an electron by an electric field ( eV ) is equal to its kinetic energy - this can be used to find the electron's speed:
• This can be substituted into the momentum term in DeBroglie's hypothesis to then find wavelength:
• The wavelength of the electron depends on the work done on it by the electric field, eV
• From this equation, as  eV   increases, λ   decreases
• When the electron is accelerated to a higher speed , its DeBroglie wavelength decreases

Worked example

An electron is accelerated through an electric field and is found to have a DeBroglie wavelength of  λ . The potential difference across the electric field then increases by a factor of 25. Write the new wavelength of the electron in terms of λ .

Step 1: Write out the equation for an accelerated particle's wavelength from your data and formulae sheet:

• The wavelength of an accelerated particle is:

Step 2: Label the new wavelength and substitute the new potential difference:

• Now we will manipulate this expression until we can pull out the original expression for  λ  :
• Therefore the new wavelength is:
• This checks out with common sense - the particle is moving faster under a stronger potential difference so, as was mentioned above, its new wavelength should be smaller

This equation requires some confidence in algebra involving square roots. Remembering that you can combine square roots when multiplying or combining will help a great deal:

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• Use of SI Units & Their Prefixes
• Limitation of Physical Measurements
• Atomic Structure & Decay Equations
• Classification of Particles
• Conservation Laws & Particle Interactions
• The Photoelectric Effect
• Energy Levels & Photon Emission
• Longitudinal & Transverse Waves
• Stationary Waves
• Interference

Author: Dan MG

Dan graduated with a First-class Masters degree in Physics at Durham University, specialising in cell membrane biophysics. After being awarded an Institute of Physics Teacher Training Scholarship, Dan taught physics in secondary schools in the North of England before moving to SME. Here, he carries on his passion for writing enjoyable physics questions and helping young people to love physics.

What is deBroglie's hypothesis? Write the expression for the wavelength associated with a moving particle.

De broglie hypothesis says that all matter has both particle and wave nature. the wave nature of a particle is quantified by de broglie wavelength defined as, λ = h m v where h is plank's constant, m is mass of the particle and v is velocity of the particle..