What is the de Broglie's equation?
1 Answer
The **de Broglie equation** describes the wave-particle duality of matter, suggesting that all particles, including those of ordinary matter (such as electrons), exhibit both particle-like and wave-like properties. This principle was introduced by French physicist **Louis de Broglie** in 1924.
The de Broglie equation connects the wavelength \( \lambda \) of a particle with its momentum \( p \). The equation is given by:
\[
\lambda = \frac{h}{p}
\]
Where:
- \( \lambda \) is the de Broglie wavelength of the particle (in meters).
- \( h \) is **Planck's constant** (\( h \approx 6.626 \times 10^{-34} \, \text{J·s} \)).
- \( p \) is the **momentum** of the particle, which is the product of the particle's mass \( m \) and velocity \( v \): \( p = mv \).
Thus, for a particle of mass \( m \) moving with velocity \( v \), the de Broglie wavelength is:
\[
\lambda = \frac{h}{mv}
\]
### Significance:
- This equation shows that the wavelength is inversely proportional to the momentum of the particle. For massive objects (with large mass and/or high velocity), the wavelength is incredibly small and not detectable by conventional means.
- For very tiny particles such as **electrons**, the wavelength can be significant and noticeable. This is crucial in quantum mechanics, as particles like electrons exhibit behaviors like interference and diffraction, typically associated with waves.
### Example:
If an electron (with mass \( m = 9.11 \times 10^{-31} \, \text{kg} \)) is traveling with a velocity of \( v = 1.0 \times 10^6 \, \text{m/s} \), its de Broglie wavelength can be calculated as:
\[
\lambda = \frac{6.626 \times 10^{-34} \, \text{J·s}}{(9.11 \times 10^{-31} \, \text{kg}) (1.0 \times 10^6 \, \text{m/s})}
\]
The result will give you a wavelength that falls within the range detectable in certain quantum experiments, such as electron diffraction.
### Conclusion:
The de Broglie equation is foundational in quantum mechanics, emphasizing that **matter can exhibit wave-like behavior** under certain conditions, which was later confirmed through experiments like electron diffraction. It bridges the gap between classical mechanics (where objects are treated as particles) and quantum mechanics (where objects also exhibit wave-like properties).
The de Broglie equation connects the wavelength \( \lambda \) of a particle with its momentum \( p \). The equation is given by:
\[
\lambda = \frac{h}{p}
\]
Where:
- \( \lambda \) is the de Broglie wavelength of the particle (in meters).
- \( h \) is **Planck's constant** (\( h \approx 6.626 \times 10^{-34} \, \text{J·s} \)).
- \( p \) is the **momentum** of the particle, which is the product of the particle's mass \( m \) and velocity \( v \): \( p = mv \).
Thus, for a particle of mass \( m \) moving with velocity \( v \), the de Broglie wavelength is:
\[
\lambda = \frac{h}{mv}
\]
### Significance:
- This equation shows that the wavelength is inversely proportional to the momentum of the particle. For massive objects (with large mass and/or high velocity), the wavelength is incredibly small and not detectable by conventional means.
- For very tiny particles such as **electrons**, the wavelength can be significant and noticeable. This is crucial in quantum mechanics, as particles like electrons exhibit behaviors like interference and diffraction, typically associated with waves.
### Example:
If an electron (with mass \( m = 9.11 \times 10^{-31} \, \text{kg} \)) is traveling with a velocity of \( v = 1.0 \times 10^6 \, \text{m/s} \), its de Broglie wavelength can be calculated as:
\[
\lambda = \frac{6.626 \times 10^{-34} \, \text{J·s}}{(9.11 \times 10^{-31} \, \text{kg}) (1.0 \times 10^6 \, \text{m/s})}
\]
The result will give you a wavelength that falls within the range detectable in certain quantum experiments, such as electron diffraction.
### Conclusion:
The de Broglie equation is foundational in quantum mechanics, emphasizing that **matter can exhibit wave-like behavior** under certain conditions, which was later confirmed through experiments like electron diffraction. It bridges the gap between classical mechanics (where objects are treated as particles) and quantum mechanics (where objects also exhibit wave-like properties).