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...