Modern Physics

 Heisenberg Uncertainty principle Statement According to uncertainty principle it is impossible to measure the momentum (i.e velocity) and position of the particle simultaneously with accuracy.  The product of uncertainties in position and momentum is equal to or greater than Planck's constant or h/2π or h/4π. ∆x.∆p ≥ h .....(1) A more exact derivation ∆x.∆p≥ h/2π Where ∆x is the uncertainty in position and ∆p is the uncertainty in momentum. From eqn(1) it is evident that if we try to measure the position of particle with utmost accuracy i.e ∆x➝ 0, the corresponding uncertainty in momentum becomes very large ∆p➝∞ ∆p≥h/∆x ∆p≥h/0 ∆p≥∞ and vice versa. Its extension to Energy and time. The time energy uncertainty principle states that it is impossible to determine the energy and time of particle simultaneously with a desire accuracy. Let ∆E be the uncertainty in energy and ∆t be the uncertainty in time then ∆E.∆t  ≥ h A more exact derivation ∆E.∆t ≥ h/2π ∆E.∆t ≥ h/4π Derivatio

 The concept of Wave packets and group velocities; De-Broglie Wave Velocity; The De-Broglie waves associated with the particle are expected to move with the same velocity with which the particle is moving. However, the velocity of the De-Broglie waves is different from the velocity of the particle. If vₚ is the velocity of the De-Broglie waves, then vₚ=𝛎𝝀 ....(1) Where 𝛎 is the frequency of waves and 𝝀 is the wavelength. But 𝝀=h/mv ....(2) Also 𝛎=E/h ....(3) Where E is the energy of the quantum of waves and h is Planck's constant . Since, E=mc² Therefore, from eqn.(3),  𝛎=mc²/h ....(4) Using equation (2) and (4) in eqn.(1), we get vₚ=mc²/h×h/mv=c²/v or vₚ=(c/v)c ....(5) Since c >> v, it is clear from eqn.(5) that the De-Broglie wave velocity ( also called as the phase velocity) is greater than the velocity of light. vₚ > c According to theory of special relativity , speed of particle vₚ < c for any particle that has mass. The phase velocity of matter waves a

 De-Broglie Waves Particle vs Wave Difference between particle and wave. History The concept that matter behaves like a wave was proposed by De-Broglie . De-Broglie, in his 1924 PhD thesis , proposed that just as light has both wave-like and particle-like properties, electrons (or matter) also have wave-like and particle-like properties. Hypothesis  "Every moving particle has a wave associated with it". The wavelength of moving particle is given by 𝛌=h/p Where h is Planck's constant and p is the momentum of moving particle. Here 𝛌 represents wave nature , while momentum accounts for the particle nature . Thus the dual nature (i.e particle and wave nature) of matter was proposed by De-Broglie and it was experimentally verified by Davisson and Germer . Matter Wave The wave associated with moving particle is known as matter wave . Matter waves are reffered to as De-Broglie waves . De-Broglie Wavelength The wavelength associated with matter wave is known as De-Brogli

 Pair Production. The formation of an electron and positron (positively charge electron) from a photon , usually in the vacinity of an atomic nucleus is called pair production . Pair production. In this process electromagnetic energy is changed into matter .                𝜸⁰⟶ e⁺ + e⁻ The inverse of pair production ( pair annihilation ) occurs when a positron is near an electron and the two come together under the influence of their opposite electric charges.                                               e⁺+ e⁻ ⟶ 𝜸 + 𝜸 No nucleus or other particle is needed for pair annihilation to take place. In pair production any law of conservation is not violated . The sum of the charges of the electron (q=-e) and of the positron (q=+e) is zero, as is the charge of photon; the total energy, including rest energy, of the electron and positron equals the photon energy and linear momentum is also conserved. Thus during pair production law of conservation of energy, linear momentum and elect

Compton Effect Compton Effect; The scattering of photon by an electron is called Compton Effect . During this effect energy and momentum are conserved . Thus the energy of scattered photon is less than that of incident photon. Compton Effect derivation. Compton Effect derivation. Conclusion; The quantity ∆𝛌 is called Compton Shift and it depends only on the angle of scattering ϴ. 1) When ϴ=0, the Compton Shift ∆𝛌 is zero. 2) When ϴ=90°, the Compton Shift is equal to h/mₒc i.e ∆𝛌=h/mₒc. 3) When ϴ=180°, the Compton Shift ∆𝛌 is maximum i.e ∆𝛌=2h/mₒc. Questions; Fill in the blanks; 1) The Scattering of photon by an electron is called ------------. 2) During Compton Effect energy and momentum are ---------. 3) Thus the energy of scattered photon is ----- than that of incident photon. 4) The energy of incident photon/ photon before collision is -----. 5) The momentum of incident photon/before collision is -----. 6) The energy of electron before collision is ----. 7) The momentum of e

 Einstein Explanation of Photoelectric Effect: According to Einstein , electromagnetic radiation of frequency 𝛎 consist of photons, each of energy h𝛎. When a photon of energy h𝛎 is incident on the surface of material, some of its energy is spent in making the electron free and the rest appears as kinetic energy of electron. The electrons at the surface of the material are most loosely bound and require minimum energy for their liberation. This energy is called the work function 𝛟 of material. The maximum kinetic energy of photoelectrons, ejected from the surface is given by;  K.E ₘ ₐ ₓ=  h𝛎 - 𝛟  The electrons which are tightly bound are ejected with less kinetic energy. Thus the kinetic energy of electron depends on the fact whether it is on the surface of material or it is deeper inside the material. If  𝛎₀ is the frequency of incident radiation such that the photon energy h𝛎₀ is just sufficient to make the electron free from the material, then the ejected electron has zero

 Photoelectric effect; The emission of electrons by a substance under the action of light is called Photoelectric effect. Photoelectric Effect. Experimental Setup: The phenomenon of photoelectric effect can be studied with the help of an apparatus shown in Figure below. Within an evacuated glass jacket two electrodes R and S are enclosed and the light radiation is allowed to enter the jacket through a quartz window. The radiation falls on electrode R, called cathode. The electrode S can be kept at desired (positive or negative) potential with respect to the cathode. A sensitive ammeter is put in the circuit to record current resulting from photoelectrons. The potential difference between the cathode and anode can be measured by voltmeter . Experimental Setup. Experimental observations The experimental observation of photoelectric effect may be summarised as follows; 1) Effect of Intensity of light on Photoelectric current; For a constant potential difference between the cathode and

Planck's radiation law; Conclusion of Rayleigh-Jeans law; According to Rayleigh-Jeans law as the frequency of radiation tends to infinity the spectral energy density must also tend to infinity. But in reality as the frequency increase spectral energy density decreases. Thus Rayleigh-Jeans law agrees with experimental results at low frequencies but strongly disagrees at high frequencies. This inconsistency between observations and the predictions of classical physics is commonly known as ultraviolet catastrophe. Ultraviolet catastrophe.   Planck's law Planck's hypothesis; This failure of classical theory (Rayleigh-Jeans law) led Max Planck to develop a law known as Planck's law for the study of Blackbody radiation. He put forward the following assumptions (Planck's hypothesis) to derive the Blackbody radiation formula. 1) Blackbody radiation contain infinite number of harmonic oscillators. Each oscillator have its own natural frequency. 2) Blackbody does not emit or