KNOW PHYSICS

Planck Blackbody Radiation Formula The spectral energy density of blackbody radiation as a function of frequency is given by Planck's radiation law: \[ u(\nu)d\nu = \frac{8\pi h\nu^3}{c^3} \frac{1}{e^{h\nu/kT}-1} \, d\nu \] where \(h = 6.626 \times 10^{-34} \, \text{J·s}\) is Planck's constant \(u(\nu)\) is the energy density of radiation per unit frequency interval \(\nu\) is the frequency \(T\) is the temperature \(k\) is the Boltzmann constant In these notes we will derive the above formula in a step by step manner. Waves in a Box Consider an electromagnetic wave travelling with the speed of light in some arbitrary direction represented by the position coordinate \(x\). If the wavelength of the wave is \(\lambda\), the amplitude of the wave along the \(x\)-direction can be written as \[ A(x) = A_0 \sin \left(\frac{2\pi x}{\lambda}\right) \] This expression can also be written in terms of the wave number \(k\): \[ A(x) = A_0 \sin(kx) \] where the ...

Coherent State for the Harmonic Oscillator and its properties Discovered by R. J. Glauber in 1963. Glauber received the Nobel Prize in 2005 for the relevance of coherent states in quantum optics. The state describing a laser beam can be briefly characterized as: An indefinite number of photons. A precisely defined phase. Laser dynamics → coherent state. Normal light → unpolarized / incoherent. Uncertainty relation: \[ \Delta N \, \Delta \Phi \ge \frac{1}{2} \] Here \( \Delta N \) : fluctuation in occupation number \( \Delta \Phi \) : fluctuation in phase For a laser: \( \Delta \Phi \) → very small \( \Delta N \) → large For normal light: \( \Delta N \) → fixed / small \( \Delta \Phi \) → large (not in the same phase) Definition of Coherent States A coherent state \( |\alpha\rangle \) (also known as a Glauber state ) is defined as an eigenstate of the annihilation operator \( \hat{a} \) with eigenvalue \( \alp...

Quantum Mechanics #3 Black Body A perfectly black body is one which absorbs totally all the radiation of any wavelength which fall on it. As it neither reflects nor transmits any radiation, it appears black; whatever be the colour of incident radiation. The main characteristic of such a body is that when heated to a suitable high temperature, it emits full or total radiation. As it is a perfect absorber, it is also a perfect radiator, its emission being the greatest possible for every wavelength at any given temperature. BLACK BODY Black Body in Practice  In practice, a perfectly black body is not available. Lamp-black and Platinum black are the nearest approach to a black body. However, a body showing close approximation to a perfectly black body can be constructed. Black Body absorber                   Black body emitter A closed chamber, say a hollow sphere (known as hollow spherical cavity) whose in...

INTRODUCTION  The failure of classical mechanics led Planck(in 1900) to the discovery that radiation is emitted in quanta whose energy is E=hν. In 1901, Planck was able to derive an empirical formula to explain the experimentally observed distribution of energy in the spectrum of a black body, on the basis of his revolutionary hypothesis known as quantum theory of heat radiation. This was the  Origin of quantum Mechanics(click here) Max Karl Ernst Ludwig Planck According to this theory, the energy distribution is given by This realtion agrees (and hence completely fit) with the experimental curves obtained. This formula of distribution of energy with wavelengths, on the basis of his quantum concept, was deduced using following assumptions, which may be called as Planck's quantum postulates. PLANCK'S QUANTUM POSTULATES  1) A black body contains the atomic oscillators capable of vibrating with all possible frequencies. An oscillator of frequency ν can...

INTRODUCTION What is Quantum mechanics?| Quantum Mechanics is the theory which attempts to explain the behaviour  of matter and interaction with energy on the scale of atoms and atomic particles i.e., particles of the size of the order of   1/10¹⁰ metres. Phenomena such as motion of mechanical objects involving distances larger than about  1/10⁶ metres can be explained satisfactorily by laws of classical theoretical physics which is based on the following basic laws :  1) Newton's Laws of motion,  2) Newton's inverse square law of gravitational attraction between two bodies,  3) Coulomb's inverse square law of attraction or repulsion between two electrically charged bodies,  4) The law of force on a moving charge in a magnetic field, i.e., the Lorentz force. However, certain phenomena such as spectral distribution of energy in blackbody radiation, Photoelectric effect etc; and phenomena involving distances of the order o...