Question

The waves associated with a moving electron and a moving proton have the same wavelength \( \lambda \). It implies that they have the same :

• A. momentum
• B. angular momentum
• C. speed
• D. energy

Hint:

Remember the de Broglie relation \( \lambda = \frac{h}{p} \). If two particles have the same wavelength, they must have the same linear momentum regardless of their masses.

Answer 1

The Correct Option is A

Step 1: De Broglie wavelength relation.
According to de Broglie's hypothesis, every moving particle is associated with a wave whose wavelength is given by \[ \lambda = \frac{h}{p} \] where \( \lambda \) is the wavelength, \( h \) is Planck's constant, and \( p \) is the linear momentum of the particle.
Step 2: Relation between wavelength and momentum.
From the formula \( \lambda = \frac{h}{p} \), we see that wavelength is inversely proportional to momentum. Therefore, if two particles have the same wavelength, their momentum must be equal.
Step 3: Applying the condition in the question.
The question states that the de Broglie wavelength of a moving electron and a moving proton is the same. Therefore, \[ \lambda_e = \lambda_p \] Using the de Broglie equation: \[ \frac{h}{p_e} = \frac{h}{p_p} \] which implies \[ p_e = p_p \] Thus, both particles have equal linear momentum.
Step 4: Analysis of options.

• (A) momentum: Correct. Equal wavelength directly implies equal linear momentum.
• (B) angular momentum: Incorrect. Angular momentum is not determined by the de Broglie wavelength relation.
• (C) speed: Incorrect. Since electron and proton have different masses, equal momentum does not mean equal speed.
• (D) energy: Incorrect. Energy depends on mass and velocity, so it will not be the same for electron and proton.

Step 5: Conclusion.
Therefore, if the de Broglie wavelengths of a moving electron and proton are the same, they must have the same linear momentum.
Final Answer: momentum.