Heisenberg’s Uncertainty Principle

♦ Study about Heisenberg’s Uncertainty Principle & Phenomenon explained by this principle ♦

Canonically Conjugate Quantities :
Those quantities, the dimensions of whose product are equivalent to those of action (i.e. J.s.) are defined as canonically conjugate quantities. The canonically conjugate quantities are energy and time, momentum and position, angular momentum and angular displacement. The minimum uncertainty in the assessment of conically conjugate quantities is of the order of h i.e. 1.054 × 10–34 J.s.

Heisenberg’s Uncertainty Principle :
The simultaneous measurement, of any two canonically conjugate quantities i.e. position and momentum, energy and time, angular momentum and angular displacement, accurately is not possible. The product of uncertainties, in the simultaneous measurement of position and momentum, energy and time, angular momentum and angular displacement accurately is not possible. The product of uncertainties, in the simultaneous measurement of position and momentum, energy and time, angular momentum and angular displacement, is of the order of ħ (or h).

Heisenberg’s Uncertainty Principle

(1)  ΔP Δx ≥ ħ where Δp = uncertainty in the measurement of momentum of particle.

(2)   ΔE Δt ≥ ħ where Δr = uncertainty in the measurement of position of particle.

(3)   ΔJ Δθ ≥ ħ where ΔE = uncertainty in the measurement of energy of particle.

(4)   Δt = uncertainty in the measurement of time of particle.

(5)   ΔJ = uncertainty in the measurement of angular momentum of particle.

(6)   Δθ = uncertainty in the measurement of angular displacement of particle.

Note :
(A)  According to this principle, the simultaneous accurate measurement of any two canonically conjugate quantities is not possible. If one of quantities is measured accurately then the uncertainty (error) in the measurement of another quantity will be very large (i.e. tending to ¥)

For example :     Δx = 0 then ΔP = ∞

ΔE = 0, then Δt = ∞

(B)   This principle is much more effective for microscopic particles because for these particles the error due to uncertainty principle more than their measurement errors whereas for macroscopic particles this error is much smaller.

(C)   The errors in the measurement of canonically conjugate quantities are due to the dual nature of particles and not due to instrumental error.

(D)  According to this principle, electron cannot exist inside the nucleus because the energy of b-particles emitted by the nuclei us 97 MeV whereas according to Heisenberg’s uncertainty principle this energy must be 4 MeV.

(E)   For a wave packet ΔK Δx ≈ 1 where ΔK = uncertainty in the measurement of wave vector.

(F)   For photonΔn Δt » 1 where Δn = uncertainty in the measurement of frequency.

(G)  (i) Δx ΔPx ≥ ħ          (ii) Δy ΔPy ≥ ħ             (iii) Δz ΔPx ≥ ħ

(iv) Δx ΔPy = 0              (v) Δx ΔPy = 0             (vi) Δy ΔPx = 0

(vii) Δy ΔPz = 0              (viii) Δz ΔPx = 0           (ix) Δz ΔPy = 0

(x) Δx ΔE = 0              (xi) ΔP Δt = 0              (xii) ΔP ΔJ = 0

(xiii) ΔP Δq = 0              (xiv) Δx Δq = 0           (xv) Δx Δt = 0

  Phenomena explained by this principle :
 Finite width of spectral lines, non-existence of electrons inside the nucleus, size of hydrogen atom, Gaussian wave group, theory of probability in quantum mechanics.

Special Point :
(1)   If the wavelength of spectral line of width Δλ is λ , then the time for the atom to remain in the excited state is
Heisenberg’s Uncertainty Principle

(2)  The uncertainties of finding a proton within a nucleus of radius r and in the measurement of its momentum are respectively given by Δx = 2r and

Heisenberg’s Uncertainty Principle