# How fast does a proton have to be moving is order to have the same de Broglie wavelength as an electron that is moving at 3.90 × 10⁶ m/s?

Jul 3, 2014

The speed of the proton must be 2.12 ×10³ m/s.

The de Broglie equation is

 λ = h/(mv)

Let ${m}_{e}$ and ${v}_{e}$ represent the mass and velocity of the electron.

Let ${m}_{p}$ and ${v}_{p}$ represent the mass and velocity of the proton.

Then

λ = h/(m_ev_e) = h/(m_pv_p)

${m}_{p} / \left({m}_{e} {v}_{e}\right) = \frac{1}{v} _ p$

v_p = v_e × m_e/m_p

${m}_{e}$ = 9.109 × 10⁻³¹ kg

${m}_{p}$ = 1.673 × 10⁻²⁷ kg

v_p = v_e × m_e/m_p = 3.90 × 10⁶ m/s × (9.109 × 10⁻³¹"kg")/(1.673 × 10⁻²⁷"kg") =
2.12 ×10³ m/s

This makes sense. A proton has about 2000 times the mass of an electron, so it would have to travel at about ¹/₂₀₀₀ the speed of an electron to have the same momentum — (3.90 × 10⁶"m/s")/2000 ≈ 2000 m/s — and the same wavelength.

Hope this helps.