The double slit experiment proves that quantum particles can be described as a wave. They interfere with each other

The two slits are represented as path kets ∣A⟩\ket{A} (top slit) and ∣B⟩\ket{B} (bottom slit).

A particle ∣ψ⟩\ket{\psi} passes through it. The slits are like an Operator that forces its state into

∣ψ⟩=12∣A⟩+12∣B⟩\ket{\psi} = \tfrac{1}{\sqrt2}\ket{A} + \tfrac{1}{\sqrt2}\ket{B}

To find the amplitude at a certain screen distance xx it'd be.

⟨x∣ψ⟩=12⟨x∣A⟩+12⟨x∣B⟩\braket{x|\psi} = \tfrac{1}{\sqrt2}\braket{x|A} + \tfrac{1}{\sqrt2}\braket{x|B}

i.e., writing as Wave function

ψ(x)=12ψA(x)+12ψB(x)\psi(x)=\frac{1}{\sqrt{2}}\psi_A(x)+\frac{1}{\sqrt{2}}\psi_B(x)

Because they're at different distances, they interfere differently depending on the angle ϕ\phi from the thing.

ψA(x)=a eiϕA,ψB(x)=a eiϕB\psi_A(x) = a\,e^{i\phi_A}, \qquad \psi_B(x) = a\,e^{i\phi_B}

To predict the probability of a particle at (x,y)(x,y) where LL is the distance from the slit plane to the screen, λ\lambda is the wavelength of the particle. aa is the amplitude size of the wave and dd is the slit separation

ϕ=2πλ r=k r,k=2πλ\phi = \frac{2\pi}{\lambda}\, r = k\,r, \qquad k = \frac{2\pi}{\lambda}

Where rr is the distance the wave travels. The distances rA,rBr_A,r_B are the distance from slit A,BA,B to the point

rA(x,y)=L2+(x−d2)2+y2,rB(x,y)=L2+(x+d2)2+y2r_A(x,y) = \sqrt{L^2 + \left(x - \tfrac{d}{2}\right)^2 + y^2}, \qquad r_B(x,y) = \sqrt{L^2 + \left(x + \tfrac{d}{2}\right)^2 + y^2} P(x,y)=∣ψ(x,y)∣2=a2(1+cos⁡[k (rA−rB)])P(x,y) = |\psi(x,y)|^2 = a^2\Big(1 + \cos\big[k\,(r_A - r_B)\big]\Big)