A Simple Question on wave-particle duality

Question

This was an mcq question in which only 1 option is correct . The question stated -

Two Photons having -

1. equal wavelength have equal linear momenta

2. equal energy have equal linear momenta

3. equal frequency have equal linear momenta

4. equal linear momenta have equal wavelengths

Now i was able to rule to 2 and 3 option easily.

But I am not able to decide btw 1 and 4. Aren't they the same options ,if one is true other has to be true.

The book gave answer as option D but no reason is mentioned.

Can anybody provide me reason that why option (1) is not true

Answer 1

Option 1 is that equal wavelength implies equal momenta. Option 4 is that equal momenta implies equal wavelength.

They are not the same statement. The implication goes in opposite directions. (Similar to how "human" implies "mammal", but "mammal" does not imply "human".)

From the overall momentum vector, you can infer the magnitude of the momentum, and from that, the wavelength. But from the wavelength you can infer the magnitude of the momentum, but not the direction.