Sigma Percentile
JEE Main 2021
LEVELJEE Main

Animated Solution for Physics - Dual Nature of Matter and Radiation: Given below are two statements : Statement I Two photons having equal linear momenta have equal wavelengths. Statement II If the wavelength of photon is decreased, then the momentum and energy of a photon will also decrease. In the light of the above statements, choose the correct answer from the options given below.

Select Answer:

Visualized Solution

  • Let's evaluate Statement I: Two photons having equal linear momenta have equal wavelengths.
  • For a photon, the de-Broglie wavelength is given by:

  • Given that the linear momenta are equal:
  • Substituting :
  • Thus, Statement I is true.

  • Now, let's evaluate Statement II: If the wavelength of a photon is decreased, its momentum and energy will also decrease.
  • The momentum and energy of a photon are given by:

  • From the formulas, both momentum and energy are inversely proportional to wavelength .
  • Therefore, if decreases, both and must increase.
  • Thus, Statement II is false.

  • Statement I is true.
  • Statement II is false.
  • Correct Option: (c)

The Sigma Insight: Particle Nature of Light-The photon

Solution Diagram
Welcome, future physicists! Today, we are diving into the fascinating world of quantum mechanics, specifically focusing on the dual nature of radiation and matter. We have a brilliant conceptual question from JEE Main that tests our fundamental understanding of photons, their momentum, energy, and wavelength. Let's unravel this mystery step by step.

Analyzing Statement I

The Momentum-Wavelength Connection
Let's begin by analyzing the first statement. It claims that two photons with equal linear momenta will have equal wavelengths. To verify this, we need to recall the fundamental relationship that connects the momentum of a photon to its wavelength.
According to the de-Broglie hypothesis and Planck's quantum theory, the wavelength of a photon is given by the elegant equation:
where is Planck's constant and is the linear momentum of the photon.
Since the problem explicitly states that the momenta of the two photons are equal, we can write:
Substituting our formula into this equality, we get:
Because Planck's constant is a universal constant, it simply cancels out from both sides of the equation. This leaves us with a very straightforward conclusion:
So, the wavelengths are indeed equal. This means Statement I is absolutely true. It is a direct consequence of the fundamental wave-particle duality equations.

Analyzing Statement II

The Inverse Proportionality Trap
Now, let's move on to Statement II. It claims that decreasing the wavelength of a photon will decrease both its momentum and its energy. To see if this holds up, let's write down the formulas for the momentum and energy of a photon.
We already know the momentum is given by:
And the energy of a photon is given by the famous relation:
Look closely at these formulas. In both equations, the wavelength sits firmly in the denominator. This means that both momentum and energy are inversely proportional to the wavelength .
This inverse relationship is the key to the whole statement. If the wavelength decreases (the denominator gets smaller), the overall value of the fraction must get larger. Therefore, both the momentum and the energy must actually increase, not decrease!
Think of it like a fraction: dividing a constant number by a smaller number always yields a bigger result. Because the statement claims they will decrease, Statement II is completely false.

Final Conclusion

To wrap it all up, our rigorous mathematical check has shown that Statement I is true, while Statement II is false. Looking at our given options, this perfectly matches option (c).
This problem is a beautiful reminder of why we must always trust our fundamental equations and pay close attention to whether variables are directly or inversely proportional. Keep these core concepts strong, and you'll breeze through these theoretical questions!