Question

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?

Answer 1

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/(m_ev_e) = 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.