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?

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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?

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Answer 1 · 2014-07-03T17:50:32

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.

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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?

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