TOPIC: HEISENBERGS PRINCPCEBY:- PROF.S.V.ANGADI, PHYSICS DEPARTMENT, J.T.COLLEGE, GADAG

POWER POINT PRESENTATION

HEISENBERGS PRINCPLE

According to classical mechanics the position of the particle

and momentum are measurable with accuracy at the

same instant of time because of the particle nature.

However in quantum mechanics because of the dual

nature of the particle in motion, it is impossible to

determine the accurate position and momentum of the

particle simultaneously. This inaccuracy of measurement is

called the UNCERTAINITY.

Statement:

The principle states that it is impossible to determine the accurate position and momentum a moving particle simultaneously

OR The product of uncertainties in the

accurate measurement of position and momentum of a moving particle can never be smaller than a number

Mathematical form of the statement:

Suppose P is the momentum and x is the position of the particle at an instant of time t in motion. Let ∆𝑥 be the change in position and ∆𝑃 be the change momentum of at an instant of time t are equal to the uncertainty of measurements ∆𝑥 and ∆𝑃. According to the statement we can have, ∆𝑥. ∆𝑃 =ℏ

Where h is called Planck’s constant. Other forms of statements are

∆x. ∆k ≥ 12

∆𝐸. ∆𝑡 ≥ ℏ2

Consider a monochromatic light source emitting photons of high energy of quantization and wave length λ. Let an electron of rest mass m0 be at rest is focused by a suitable microscope. Let the photon is allowed to incident on the electron. After the collision a flash of light emitted is observed in the microscope indicating the entrance of photon into it. Everyone feel that it is possible to determine the position and momentum of moving particle simultaneously and accurately, however the laws of optics put limitations over the accuracy of these measurements.

I) ON POSITION MEASUREMENTS:

Let the two objects are closely resided, according to Abbe’s theory of resolving power of microscope, the limit of resolution of microscope is,

∆𝑥= 𝜆2𝑠𝑖𝑛𝜃

Where ∆𝑥 is the minimum distance between two points in the field of view are seen as separate, 𝜆 is the wave length of the light photon, θ be the semi vertical angle of light cone.

Suppose change in the particle position is small comparing with the

RP of the microscope, it can be determined accurately. This could

do by considering light of smaller wave length. Then the limitation of

measurement of the position is taken as the uncertainty, hence

∆𝑥= 𝜆2𝑠𝑖𝑛𝜃′

In the process of collision of electron at rest and the photon, the electron gets recoiled due to the gained momentum and with the remaining energy the photon scattered, as shown in the figure. Assuming the particle position is determined accurately by the microscope spotting the flash of light emitted in the interaction. However it cannot be recognized that the photon entered the microscopic field along the path OB or OB because of the design of the microscope, obviously the particle is in the range equal to the area of the light cone 2o can always be found .

Let the initial momentum of the photon be ℎ𝜆 and the electron at rest is zero. After the collision

electron moves with velocity v making an angle of recoil φ and the scattering angle of photon be θ.

By the law of conservation of momentum, the x direction momentum of the particle is

hνc + o = hν′c cosθ+ mvcosϕ

Since the x direction momentum of electron is transferred from the photon, then we may have

Px = mv cosϕ = hνc − hν′c cos

II) Determination of momentum:

To determine the momentum of the particle at the same instant of time, the law of conservation of momentum would have considered.

The angle inside the microscope may vary between π2 + θ to π2 − θ . Thus the

range of x direction component of momentum can be written as hνc − hν′c cosቀπ2 + θ ቁ≤ Px ≤ hνc − hν′c cosቀπ2 − θ ቁ

Let ∆𝑃 be the uncertainty in the momentum of the electron be calculated as

∆𝑃= hνc − hν′c cosቀπ2 + θ ቁ−ቈhνc − hν′c cosቀπ2 − θ ቁ

∆𝑃= −hν′c cosቀπ2 + θ ቁ−ቈ−hν′c cosቀπ2 − θ ቁ

∆𝑃= −hν′c ሺ−sinθሻ+ hν′c sin θ

∆𝑃= 2hν′c sin θ

To minimize the uncertainty the aperture of the objective must be reduce, but this leads to the Sevier demerit to cause the RP, since the RP decreases hence the uncertainty increases. To improve the accuracy of measurement the value of v should be decreased and o should be increased at the cost of the accuracy measurement

Multiplying the equation and we get

∆x.∆P = ν′2sinθ 2hν′c sin θ

∵ ν′ = cν′ ∆x.∆P = λ′2sinθ 2hccλ′ sin θ

Equation describes the Heisenberg’s uncertainty relation.

ii) DIFFRACTION OF ELECTRON AT SINGLE SLIT:

Consider a monochromatic radiation of wave length incident on a narrow slit S. due to superposition of secondary waves the diffraction take place. The diffraction pattern obtained on the screen consisting central bright maxima, on either side of it there are alternate dark and bright bands of decreasing intensities and are called secondary maxima and minima respectively.

Let ∆𝑦 be the width of the narrow slit is illuminated by electron beam of wave length. The scattered secondary (electron) waves superpose to each other produces a diffraction pattern on the screen. According to the Brag’s diffraction theory the path difference for bright band is

2Δysinθ = nλ

Where θ is the deviation in the path of the electron from their initial direction, n is order of diffraction, λ is the wave length of electron beam. For the first minimum, n=1 hence we have

2Δysinθ = λ

The electron passage through the slit causes the diffraction, however definitely it cannot be said that through point of the slit it enters. Therefore the uncertainty in the position measurement of electron along y direction is equal to the slit width dy hence from above relation we get,

Δy = λ2sinθ

Before interaction the electron was moving along x axis, thus the component of momentum along y-direction is zero. Electrons get deviated at the slit from their initial path to form the pattern, since they acquire additional momentum along y direction.

Let P be the resultant momentum of the deviated electron. The y- component of momentum of the electron may lie anywhere in the pattern between θ and – θ .then the y-component of momentum of the electron may found between Psinθ to –Psinθ. Thus the uncertainty in the measurement of electron along y axis is

Δ𝑃𝑦= 𝑃𝑠𝑖𝑛𝜃−ሺ−𝑃𝑠𝑖𝑛𝜃ሻ Δ𝑃𝑦= 2𝑃𝑠𝑖𝑛𝜃

Multiplying equation and we get,

ΔPyΔy = λ2sinθ2Psinθ

ΔPyΔy = Pλ

Where 𝑃= hλ is the momentum of the electron according to De-Broglie’s hypothesis. If h is

Planck’s constant then we have

ΔPyΔy = hλ λ

ΔPyΔy = h

This is the uncertainty relation. It may be written conveniently as.

ΔPyΔy ≥ ℎ2𝜋

END