Question

What do the Broglie equations relate?

Answer 1

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

You can find the equations at What is the De Broglie equation?

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