## LESSON 8. Mathematical Derivation for Uncertainty Priciple

(1) Determining of the position of a particle by microscope

For the measured of position of a particle (electron) in the range of microscope the resolving power of the microscope can be measure the smallest distance between the two points,

Δ$x={\lambda\over{2\sin \theta }}$……..(1)

Δx - Uncertainty in determining the position of particle

$\lambda$ - Wavelength of light used

$\theta$ - Semi vertical angle of the cone of light

To observed the particle (position of electron), it is necessary a photon strikes the electron (particle) and scattered inside the microscope. When a photon of initial momentum ρ = ${h \over \lambda }$ after scattering (as a momentum is conserved in the collision) a particle enters in the field of view of microscope. A particle may be anywhere within angle 2$\theta$

Therefore, the momentum(x-component) in microscopic region is p sin $\theta$ and -p sin $\theta$

Hence,

Δp = p sin $\theta$ - (-p sin $\theta$ )

Δp = 2p sin $\theta$

But , p =   ${h \over \lambda }$

Δp  = ${{2h\sin\theta}\over\lambda}$……..(2)

From (1) and (2)

Δx Δp = h

Product of uncertainties in position and momentum is of order of Plank’s Constant.

(2) Diffraction by single slit

Suppose a narrow beam of electrons passes through a single narrow slit (Δy) and produces a diffraction pattern on the screen.

Hence diffraction equation

$d\sin \theta=n\lambda$
For 1st order

$d\sin\theta=\lambda$
d - Width of the slit = Δy

Δy sin  $\theta$ =$\lambda$

Regarding diffraction pattern of on screen, all electrons have passed through the slit. But we can’t say that exact position of a particle within slit Hence the uncertainty can be determine the position of the electron equal to the width  Δy.

Δy =${\lambda \over{\sin \theta }}$

Here, initially electrons are moving along -axis and they have no component of momentum along -axis. After diffraction at the slit electrons are deviated from their initial path to form diffraction pattern and have a component . The  y - component of momentum may lie between p sin $\theta$  and -p sin$\theta$.

The uncertainties in y – component of momentum = Δp = 2p sin$\theta$

But, p  =  ${h \over \lambda }$

Δp = 2p sin$\theta$ = ${{2h\sin\theta}\over\lambda }$

Δy Δp = ${\lambda\over{\sin \theta }}$  X ${{2h\sin\theta }\over\lambda }$

Δy Δp = 2h

Or

Δy Δp h

It shows the product of uncertainties in position and momentum is of order of Plank’s constant.