Problem 1. Calculate de Broglie wavlength of an electron moving with velocity 2×104 m/s. [Given: mass of electron = 9.1 × 10-31 kg and Planck’s constant = 6.6 × 10-34Js].
(Amity Univer., May 2005)
Problem 3. Find group velocity in terms of energy and momentum.
Problem 4. If the phase velocity of a ripple is by the relation vp = c1 + c2λ, where c1 and c2 are constant. Find the group velocity of the ripple.
Problem 5. The phase velocity of a soap bubble is cλ-1/2. Find the group velocity of soap bubble. Hint: Problem 6. Use the uncertainty principle to deduce the lowest nature of energy of a particle of mass enclosed in a box of size ‘a’. Solution: The uncertainty relation is Problem 7. An electron is localised in a region of size (i) 1 cm (ii) 10-8 cm. Calculate order of kinetic energy of the electron in each case. Hint: Problem 8. Show that if the measurement of the uncertainty in the location of the particle is equal to its de Broglie wavelength, the uncertainty in its velocity is equal to its velocity.
(Δx = λ, Δ = v).
Solution: From uncertainty principle Δx Δp = h Here Δx = λ Uncertainty is the measurement of the momentum Problem 9. Show that the uncertainty principle is not observed in daily life, while it is valid for microscopic world. Solution: Case I: Uncertainty principle for macroscopic world (daily life). Let us consider a sand particle of mass m = 10-6 kg and its size is 10-4 m. Therefore uncertainty in the measurement of the position of the sand particle This gives the uncertainty in the velocity of the sand particle which is too small. Therefore uncertainty principle is not effective in daily life. Case II. Uncertainty principle in microscopic world (subatomic world): Since the matter consists of atoms and molecules and atom comprises electrons. The mass of electron is me = 9.1 × 10-31 kg The size of electron = lA = 10-10 m Fromuncertainty principle Uncertainty in the velocity of electron