Consider the following three particles: Rank them according to the wavelengths of their matter waves, least to greatest.
A)1, 2, 3
B)3, 2, 1
C)2, 3, 1
D)1, 3, 2
E)1, then 2 and 3 tied

Question

Consider the following three particles:  Rank them according to the wavelengths of their matter waves, least to greatest.
A)1, 2, 3 
B)3, 2, 1 
C)2, 3, 1 
D)1, 3, 2 
E)1, then 2 and 3 tied

Quizplus · Accepted Answer

The wavelength of a matter wave is given by the de Broglie wavelength formula: 
λ = h/p
where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. 
Higher momentum results in smaller wavelengths.

Particle 1: 
No information given about the momentum or mass of the particle. 
Cannot compare its wavelength to the other particles.

Particle 2: 
Mass: 6.64 x 10^-27 kg 
Speed: 4 x 10^4 m/s 
Momentum: p = mv = (6.64 x 10^-27 kg)(4 x 10^4 m/s) = 2.656 x 10^-22 kg·m/s 
λ = h/p = (6.626 x 10^-34 J·s)/(2.656 x 10^-22 kg·m/s) ≈ 2.5 x 10^-12 m

Particle 3: 
Mass: 9.11 x 10^-31 kg 
Speed: 2 x 10^7 m/s 
Momentum: p = mv = (9.11 x 10^-31 kg)(2 x 10^7 m/s) = 1.822 x 10^-23 kg·m/s 
λ = h/p = (6.626 x 10^-34 J·s)/(1.822 x 10^-23 kg·m/s) ≈ 3.6 x 10^-11 m

Since particle 3 has higher momentum than particle 2, its wavelength is smaller. Therefore, we have Particle 2 > Particle 3. 
And we cannot compare the wavelength of Particle 1 with either particles 2 or 3.

Therefore, the correct ranking is 3, 2, 1 or choice B.

Consider the Following Three Particles: Rank Them According to the Wavelengths