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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
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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 Hypothesis
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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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What is De Broglie Hypothesis?
De broglie's hypothesis says that matter consists of both the particle nature as well as wave nature. de broglie wavelength λ is given as λ = h p , where p represents the particle momentum and can be written as: λ = h m v where, h is the planck's constant, m is the mass of the particle, and v is the velocity of the particle. from the above relation, it can be said that the wavelength of the matter is inversely proportional to the magnitude of the particle's linear momentum. this relation is applicable to both microscopic and macroscopic particles the de broglie equation is one of the equations that is commonly used to define the wave properties of matter. electromagnetic radiation exhibits the dual nature of a particle (having a momentum) and wave (expressed in frequency, and wavelength)..
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The mathematical formulation of de Broglie wavelength is. λ= h p λ = h p. We can replace the momentum by p = mv to obtain. λ = h mv λ = h m v. Unit. 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.
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 = ℎ/𝑝 . where: 𝜆 is the wavelength, ℎ is Planck's constant, approximately 6.626×10⁻³⁴ Joule seconds, 𝑝 is the momentum of the electron.
The de Broglie equation states that matter can act as waves much like light and radiation, which also behave as waves and particles. The equation further explains that a beam of electrons can also be diffracted just like a beam of light. In essence, the de Broglie equation helps us understand the idea of matter having a wavelength.
De Broglie proposed that a moving material particle of total energy E and momentum p has a wave associated with it (analogous to a photon). He suggested a relation between properties of the wave, like frequency and wavelength, with that of a particle, like energy and momentum. v E h v = E h and h p h mv λ = h p = h mv …. (1)
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.
De Broglie gave the following equation which can be used to calculate de Broglie wavelength, \lambda λ, of any massed particle whose momentum is known: \lambda = \frac {h} {p}, λ = ph, where h h is the Plank's constant and p p is the momentum of the particle whose wavelength we need to find. With some modifications the following equation can ...
De Broglie's relations are usually expressed in terms of the wave vector →k, k = 2π / λ, and the wave frequency ω = 2πf, as we usually do for waves: E = ℏω →p = ℏ→k. Wave theory tells us that a wave carries its energy with the group velocity. For matter waves, this group velocity is the velocity u of the particle.
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: We call label the new wavelength and substitute the new potential difference, 25 V : Now we will manipulate this ...
Determine the de Broglie wavelength of a person of mass 70 kg moving at 2 ms-1 and comment on your answer. Step 1: Write the known values. Mass, m = 70 kg. Velocity, v = 2 m s −1. Planck's constant, h = 6.63 × 10 −34 Js. Step 2: Write the equation and substitute the values. Step 4: Write the answer to the correct number of significant ...
In 1924, Louis de Broglie proposed a new speculative hypothesis that electrons and other particles of matter can behave like waves. Today, this idea is known as de Broglie's hypothesis of matter waves. In 1926, De Broglie's hypothesis, together with Bohr's early quantum theory, led to the development of a new theory of wave quantum ...
De Broglie's hypothesis is an independent postulate about the structure of nature. In this respect, its status is the same as that of Newton's laws or the laws of thermodynamics. Nonetheless, we can construct a line of thought that is probably similar to de Broglie's, recognizing that these are heuristic arguments and not logical deductions.
While this equation was specifically for waves, de Broglie, using his hypothesis that particles can act like waves, combined the equations: E = m c 2 = h ν. Where E is energy, m is mass, c is the ...
The de Broglie wavelength of the photon can be computed using the formula: λ = h p. = 6.62607 × 10 − 34 Js 1.50 × 10 − 27 kgm / s. = 4.42 × 10 − 7 m. = 442 × 10 − 9 m. = 442 nm. The de Broglie wavelength of the photon will be 442 nm, and this wavelength lies in the blue-violet part of the visible light spectrum. Q.2.
The de- Broglie wave length of an electron moving with a speed of 6.6 × 10 5 m/s is approximately. The kinetic energy of electron in (electron volt) moving with the velocity of 4 × 10 6 m/s will be. Find the de-Broglie Wavelength for an electron moving at the speed of 5.0 × 10 6 m/s. (mass of electron is 9.1 × 10-31)
The de Broglie‐Bohr model of the hydrogen atom presented here treats the electron as a particle on a ring with wave‐like properties. λ = h mev λ = h m e v. de Broglie's hypothesis that matter has wave-like properties. nλ = 2πr n λ = 2 π r. The consequence of de Broglieʹs hypothesis; an integral number of wavelengths must fit within ...
The de Broglie wavelength is the wavelength, λ λ, associated with a object and is related to its momentum and mass. Introduction. In 1923, Louis de Broglie, a French physicist, proposed a hypothesis to explain the theory of the atomic structure. By using a series of substitution de Broglie hypothesizes particles to hold properties of waves.
An electron initially at rest is accelerated through a potential difference of. 54 volts compute 1) the velocity of the electron 2) deBroglie wavelength. ocity of the electron waveExampleWhat is the De-Broglie waveleng. ron whose energy is 1 eV.HomeworkWhat is the De-Broglie wavelength of an elect.
De Broglie's Hypothesis says that Matter consists of both the particle nature as well as wave nature. De Broglie wavelength λ is given as λ = h p, where p represents the particle momentum and can be written as: λ = h m v Where, h is the Planck's constant, m is the mass of the particle, and v is the velocity of the particle.; From the above relation, it can be said that the wavelength of the ...
According to the de-Broglie hypothesis, a moving material particle behaves like a wave at times and like a particle at other times. Every moving material particle is connected with a wave. The de-Broglie wave or matter-wave is an unseen wave associated with a moving particle that propagates in the form of wave packets with the group velocity.
The de Broglie equation gives a relation between the momentum of a moving particle and its wavelength. So from the de Broglie equation, we can say that all matter also has wave's nature.de Broglie, hypothesized that light is not the only matter which shows a wave-particle duality.
Louis Victor Pierre Raymond, 7th Duc de Broglie (/ d ə ˈ b r oʊ ɡ l i /, [1] also US: / d ə b r oʊ ˈ ɡ l iː, d ə ˈ b r ɔɪ /; [2] [3] French: [də bʁɔj] [4] [5] or [də bʁœj] ⓘ; 15 August 1892 - 19 March 1987) [6] was a French physicist and aristocrat who made groundbreaking contributions to quantum theory.In his 1924 PhD thesis, he postulated the wave nature of electrons ...