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?

1 Answer
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/(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.