Proof of uncertainty principle

Gamma ray microscope method (Thought experiment)

Let electron whose position (x) and momentum (p) is to be determined is initially at P
From diffraction theory, the limit of resolution of microscope

Δx = λ / 2 sin θ

Δx = Distance between two points upto which they can be seen separately.
Δx = Maximum uncertainty in position of electron
Since the wavelength of 𝛾-ray is small, so we choose it because it decreases Δx
Let at least one 𝛾-ray photon be scattered by the electron into the microscope so that the electron is visible.
In this process the frequency and wavelength of the scattered photon is changed and the electron suffers a Compton recoil by gaining the momentum.
If λ = wavelength of the scattered photon, then the momentum of the scattered photon, p = h / λ
Since the scattered photon can be scattered in any direction from PA to PB, so the x-component of momentum will be from [(h / λ) sin (-θ)] to (h / λ) sin θ] i.e., from - [(h / λ) sin θ] to (h / λ) sin θ
If λ՛ = wavelength of incident photon, then momentum of incident photon, p՛ = h / λ՛
Therefore change in momentum of photon will lie between

Thus microscope is obeying the Heisenberg’s uncertainty principle during the measurement of position and momentum of the particle simultaneously.
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