Debroglie Equations for JEE Main 2025

The De Broglie equation is a cornerstone of modern physics, connecting the wave and particle nature of matter. It plays a vital role in quantum mechanics and is essential for JEE Main 2025 preparation. Understanding this equation is critical for tackling questions related to atomic structure, quantum numbers, and wave mechanics.

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What De Broglie Equation States ?

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. 

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 = $\frac{h}{mv}$

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. 

De Broglie Hypothesis Overview : 

The de Broglie equation, formulated by Louis de Broglie in 1924, establishes a relationship between a particle's momentum and its wavelength, suggesting that all matter exhibits both particle and wave characteristics. This concept is fundamental in quantum mechanics, providing insight into the behavior of microscopic particles.

Applications of The De Broglie Equations 

• One notable application of the de Broglie equation is in electron microscopy. 

• Electron microscopes utilise the wave-like nature of electrons to achieve high-resolution imaging of small structures, surpassing the capabilities of traditional light microscopes. 

• The equation also aids in predicting the behavior of particles in quantum systems, influencing the development of technologies like semiconductors and quantum computing.

Examples of De Broglie Equations 

Examples of the de Broglie Equation

To calculate the de Broglie wavelength $(\lambda)$ of a particle, use the formula:

\[\lambda = \dfrac{h}{mv}\]

where $h$ is Planck's constant $(6.626 \times 10^{-34} \, \text{Js}$, $m$ is the particle's mass, and $v$ is its velocity.

Example

Calculate the de Broglie wavelength of an electron $(mass 9.11 \times 10^{-31} \, \text{kg})$ moving at $2.0 \times 10^6 \, \text{m/s}$.

\[\lambda = \dfrac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 2.0 \times 10^6}\]

\[\lambda \approx 3.63 \times 10^{-10} \, \text{m}\]

This wavelength is on the order of X-ray wavelengths, allowing electron microscopes to resolve structures at the atomic level.

Example 2

Determine the de Broglie wavelength of a tennis ball mass $(0.057 \, \text{kg})$ moving at $30 \, \text{m/s}$.

\[\lambda = \dfrac{6.626 \times 10^{-34}}{0.057 \times 30}\]

\[\lambda \approx 3.88 \times 10^{-34} \, \text{m}\]

This wavelength is extremely small, making wave-like properties of macroscopic objects like tennis balls undetectable.

These examples illustrate that while wave-like behavior is significant for microscopic particles, it is negligible for everyday objects due to their larger mass and resulting shorter wavelengths.

Importance of the de Broglie Equation For JEE Main 2025 

• The de Broglie equation is significant because it introduces the idea that particles such as electrons have wave-like properties. 

• This duality is crucial for understanding phenomena at the quantum level, including the behavior of electrons in atoms and the principles underlying quantum mechanics. It also explains why certain particles can exhibit interference and diffraction patterns, behaviors traditionally associated with waves.

Tips for JEE Main 2025 preparation:

1. Know the Syllabus: Focus on high-weightage topics in Physics, Chemistry, and Maths.

2. Make a Study Plan: Allocate time for concept building, practice, and revision.

3. Strengthen Basics: Use NCERT textbooks to build a strong foundation.

4. Practise Regularly: Solve past papers and take topic-wise mock tests.

5. Time Management: Learn to balance time between subjects and questions.

6. Revise Often: Maintain short notes and revise formulas frequently.

7. Use Quality Resources: Refer to trusted books and platforms like Vedantu for guidance.

8. Mock Tests: Take weekly mock tests and work on weak areas.

9. Consistency: Stick to your plan and maintain discipline.

10. Self-Care: Get enough rest, eat healthily, and stay positive.

Conclusion

The de Broglie equation, highlights the wave-particle duality of matter and is a key concept for JEE Main 2025. It explains the quantum behavior of particles, with applications in electron diffraction, atomic structure, and technologies like electron microscopes. For JEE, focus on its derivation, formula applications, and problem-solving involving kinetic and potential energy. Understanding this concept solidifies your grasp of quantum mechanics and its real-world implications.

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