de Broglie Wavelength — Revision Notes

⚡ 30-Second Revision

De Broglie Wavelength: — $\lambda = h/p$
Momentum (non-relativistic): — $p = mv$
In terms of Kinetic Energy: — $\lambda = h/\sqrt{2mE_k}$
For Charged Particle (charge $q$, mass $m$) accelerated by $V$: — $\lambda = h/\sqrt{2mqV}$
For Electron accelerated by $V$: — $\lambda = 12.27/\sqrt{V}\ \text{Å}$ (V in Volts, $\lambda$ in Angstroms)
For Thermal Neutron (mass $m$, temperature $T$): — $\lambda = h/\sqrt{3mkT}$ (k = Boltzmann's constant)
Planck's Constant: — $h = 6.626 \times 10^{-34}\ \text{J\cdot s}$
Wave-Particle Duality: — All matter exhibits both wave and particle properties.

2-Minute Revision

The de Broglie wavelength is a key concept in quantum mechanics, asserting that every moving particle has an associated wave. This 'matter wave' has a wavelength ($\lambda$) inversely proportional to the particle's momentum ($p$), given by $\lambda = h/p$, where $h$ is Planck's constant.

For non-relativistic particles, $p = mv$. This means heavier or faster particles have shorter wavelengths, making their wave nature less observable. For particles with kinetic energy $E_k$, the formula becomes $\lambda = h/\sqrt{2mE_k}$.

A common application involves charged particles (like electrons) accelerated through a potential difference $V$, where $E_k = qV$, leading to $\lambda = h/\sqrt{2mqV}$. For electrons, a handy simplified formula is $\lambda = 12.

27/\sqrt{V}\ \text{Å}$. Thermal neutrons, having kinetic energy$E_k = \frac{3}{2}kT$, have a wavelength$\lambda = h/\sqrt{3mkT}$. The wave nature of matter is experimentally confirmed by electron diffraction, but it's not observable for macroscopic objects due to their extremely small wavelengths.

Remember, matter waves are distinct from electromagnetic waves.

5-Minute Revision

The de Broglie wavelength is a cornerstone of quantum mechanics, extending the concept of wave-particle duality to all matter. Louis de Broglie proposed that any moving particle, whether an electron or a proton, has an associated wave, known as a matter wave.

The wavelength of this matter wave is given by the fundamental relation $\lambda = h/p$, where $\lambda$ is the de Broglie wavelength, $h$ is Planck's constant ($6.626 \times 10^{-34}\ \text{J\cdot s}$), and $p$ is the particle's momentum.

For non-relativistic particles, momentum $p = mv$.

This formula has several crucial variations for different scenarios:

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In terms of Kinetic Energy ($E_k$): — Since $p = \sqrt{2mE_k}$, the wavelength can be expressed as $\lambda = h/\sqrt{2mE_k}$. This is useful when the particle's kinetic energy is known.

* *Example:* If an electron has $E_k = 1.6 \times 10^{-19}\ \text{J}$, its $\lambda = (6.626 \times 10^{-34}) / \sqrt{2 \times (9.1 \times 10^{-31}) \times (1.6 \times 10^{-19})} \approx 1.0 \times 10^{-9}\ \text{m}$.

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For Charged Particles Accelerated by Potential Difference ($V$): — When a particle with charge $q$ and mass $m$ is accelerated from rest through a potential difference $V$, its kinetic energy gained is $E_k = qV$. Substituting this, we get $\lambda = h/\sqrt{2mqV}$.

* *Example:* For an electron ($m_e$, $e$) accelerated by $100\ \text{V}$, $\lambda = 12.27/\sqrt{100}\ \text{Å} = 1.227\ \text{Å}$.

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For Thermal Neutrons: — For neutrons in thermal equilibrium at temperature $T$, their average kinetic energy is $E_k = \frac{3}{2}kT$ (where $k$ is Boltzmann's constant). The de Broglie wavelength is then $\lambda = h/\sqrt{3mkT}$.

Key takeaways for NEET: Understand that $\lambda$ is inversely proportional to momentum and inversely proportional to the square root of kinetic energy or accelerating potential. Macroscopic objects do not exhibit observable wave properties because their large mass results in an extremely small de Broglie wavelength.

Matter waves are distinct from electromagnetic waves; they are not oscillations of electric and magnetic fields. The Davisson-Germer experiment provided experimental proof of electron diffraction, confirming de Broglie's hypothesis.

Prelims Revision Notes

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De Broglie Hypothesis: — Every moving particle has an associated wave, called a matter wave.
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De Broglie Wavelength Formula: — $\lambda = h/p$, where $h$ is Planck's constant ($6.626 \times 10^{-34}\ \text{J\cdot s}$) and $p$ is the momentum of the particle.
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Momentum: — For non-relativistic particles, $p = mv$ (mass $\times$ velocity).
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Wavelength in terms of Kinetic Energy ($E_k$): — $\lambda = h/\sqrt{2mE_k}$. Remember $E_k = p^2/(2m)$.
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**Wavelength for Charged Particles Accelerated by Potential ($V$):**

* Kinetic energy gained: $E_k = qV$ (charge $\times$ potential difference). * Formula: $\lambda = h/\sqrt{2mqV}$. * For Electron: $\lambda_e = 12.27/\sqrt{V}\ \text{Å}$ (where $V$ is in Volts, $\lambda$ in Angstroms, $1\ \text{Å} = 10^{-10}\ \text{m}$). This is a crucial shortcut.

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Wavelength for Thermal Neutrons:

* Average kinetic energy: $E_k = \frac{3}{2}kT$ (where $k$ is Boltzmann's constant, $T$ is absolute temperature). * Formula: $\lambda = h/\sqrt{3mkT}$.

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Proportionalities:

* $\lambda \propto 1/p$ * $\lambda \propto 1/v$ * $\lambda \propto 1/\sqrt{E_k}$ * $\lambda \propto 1/\sqrt{V}$ (for charged particles) * $\lambda \propto 1/\sqrt{T}$ (for thermal particles)

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Wave-Particle Duality: — Matter exhibits both particle and wave properties. De Broglie wavelength quantifies the wave aspect.
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Experimental Evidence: — Davisson-Germer experiment (electron diffraction) confirmed the wave nature of electrons.
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Macroscopic Objects: — Their de Broglie wavelength is negligibly small due to large mass and momentum, so wave properties are not observed.
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Distinction: — Matter waves are NOT electromagnetic waves. They are probability waves associated with particles.

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