What do the Broglie equations relate?

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Answer 1 · 2014-08-02T00:13:16

The de Broglie equations relate (a) the wavelength $λ$ to the momentum $p$ and (b) the frequency $f$ to the total energy $E$ .

Equation 1 is

$λ = h/p = h/(mv)$

It enables us to calculate the wavelength of a particle if we know its mass and velocity.

EXAMPLE

Calculate the wavelength of an electron with a speed of 3.00 × 10⁷ m/s.

Solution

$λ = h/p = h/(mv) = (6.626 × 10⁻³⁴"kg·m²s⁻¹")/(9.11 × 10⁻³¹"kg" × 3.00 × 10⁷"m·s⁻¹")$ = 2.42 × 10⁻¹¹ m = 24.2 pm

Equation 2 is

$f = E/h$

Since $fλ = v$ ,

$v/λ = E/h$

or

$E = (hv)/λ$

Example

What is the energy of the electron in the above example?

Solution

$E = (hv)/λ = ( 6.626 × 10⁻³⁴"J·s" × 3.00 × 10⁷"m·s⁻¹")/(2.42 × 10⁻¹¹"m")$ = 8.20 × 10⁻¹⁶ J