Wolfram Alpha:

```Basic Relationships
-------------------

The Photoelectric Effect pointed to the particle
properties of light.  DeBroglie wondered if this
idea could be extended to other particles.  He
started with Einstein's formula for energy:

E = mc2 = √(p2c2 + m02c4)

and PLANCK'S HYPOTHESIS:

E = hf

Where h = Planck's constant.

Equating these 2 equations when m0 = 0 gives,

pc = hf

= hc/λ

∴ p = h/λ or λ = h/p ... the DeBROGLIE WAVELENGTH

λ is associated with waves, p is associated with
particles.

Plane Waves
-----------

The general form of a plane wave is:

A(x,t) = A0cos(kx - ωt)

We can write this in complex form as:

A(x,t) = A0exp[i(kx - ωt)]

Where k is the WAVENUMBER defined as k = 2π/λ

So if λ = 2π  k = 1

λ = π   k = 2

etc.

So A(x,t) = A0exp[i(2x - ωt)] oscillates at 2 times
A(x,t) = A0exp[i(x - ωt)].

Now from before we had p = hc/λ

But λ = 2π/k.  Therefore,

p = hk/2π = hk

Where h = h/2π is the reduced Planck's constant.

Finally, f = ω/2π Therefore,

E = hf = hω

We can rewrite the plane wave as:

ψ(x,t) = ψ0exp[i(kx - Et/h)]

≡ ψ0exp[(i/h)(px - Et)]

Where ψ(x,t) is the SPATIAL WAVEFUNCTION.```