Linearity and Superposition.

 Linearity and Superposition:

Wave functions add, not probabilities.

The important property of Schrodinger's equation is that it is linear in wavefunction Ψ. By this we meant that the equation is in terms of Ψ and its derivatives, and no term is independent of Ψ or involves higher powers of Ψ or its derivatives. As a result, the linear combination of solutions of Schrodinger's equation is itself a solution. If Ψ₁ and Ψ₂ are solutions of Schrodinger's equation then

Ψ=a₁Ψ₁+a₂Ψ₂

is itself a solution. Where a₁ and a₂ are constants. Thus wavefunction Ψ₁ and Ψ₂ obeys superposition principle.

Let us apply superposition principle to the diffraction of electron beam. The fig(a) represents two slits (1&2) from which a beam of electrons passes to a viewing screen. If the slit 1 is open as shown in fig (b), then the corresponding probability density is

P₁=|Ψ₁|² = Ψ₁*Ψ₁

If the slit 2 is open as shown in fig (c), then the corresponding probability density is

P₂ = |Ψ₂|² = Ψ₂*Ψ₂

We might suppose that if both slits are open as in fig(d) then the corresponding probability density is

P=P₁+P₂

But it is not the case, in quantum mechanics wavefunction add, not probabilities.Instead the result with both slits open is shown in fig(e).

Fig(e) is due to superposition Ψ of wavefunction Ψ₁ and Ψ₂  of the electron beams passing through the slits 1 and 2.

Ψ=Ψ₁+Ψ₂

Diffraction of electron.

Thus, the probability density is given by

P= |Ψ|²

 = (Ψ₁*+Ψ₂*)(Ψ₁+Ψ₂)

 = Ψ₁*Ψ₁+Ψ₂*Ψ₂+Ψ₁*Ψ₂+ Ψ₂*Ψ₁

 = P₁+P₂+Ψ₁*Ψ₂+ Ψ₂*Ψ₁

The two terms on the right side of the equation explains the difference between Fig(d) and Fig(e).