hello friends, in my course "introductory QM" it says at the end of harmonic oscillator chapter that this may find some applicasions in electromagnetic fields and in crystal physics. now, though i havent covered solid state physics yet, but still i can visualiza the crystal being used as a combination of many oscillators one at each lattice point and any state can be described as superposition of normal modes.is this correct? but i have no idea how oscillators will find applicasions in electromagnetic fields !! i mean, we need some points of stable equilibrium about which a system fluctuates. only then will i get an oscillator quadratic hamiltonian?
The coherent states describe the states of a polarized laser. I am not sure if you have studied the number operator yet but if you have the eigenstates of x +ip (in proper units) are called coherent states. The x+ip operator is denoted a. This operator and its conjugate make up the hamiltonian in a harmonic oscillator. So through study of the harmonic oscillator they found states that describe single mode light. As for your question on crystals I am not very familiar with crystals and QM.
Almost any perturbation of a physical system can be represented as a superposition of so-called natural modes. Each of these modes (in the linear approximation) behaves like an independent linear oscillator. In fact the Hamiltonian of the perturbation is equal to a sum of oscillator quadratic Hamiltonians: [tex] \hat{H} = A\sum_\alpha \left(\frac{\hat{p}^2}{2} + \frac{\omega_\alpha^2 q^2}{2}\right). [/tex] The energy levels of an oscillator are equidistant: [tex] E_n = \hbar\omega_\alpha(n+1/2), [/tex] so we can consider it to be a set of some "particles". If the oscillator is in it's ground state (E=E_{0}) it contains no "particles". If E=E_{1} there is one "particle" and so on. When we consider electromagnetic field in a cavity the "particles" are called photons. This is the application in electrodynamics. The "particles" of acoustic oscillations in solids are called phonons. This is the application in solid-state physics. For more detailed information you can refer to 1) R. P. Feynman, Statistical Mechanics 2) Any other book where the problems of phonons in solids or electromagnetic field quantization are discussed.
many many thanks, maxim zh. i have now understood it from your post. thanks to you too, lost conjugate.
on a second thought, what if there was no cavity? i mean, i can still think of photons, can i not? thanks.
Yes you can. In unlimited space photon is a wave packet which has the proper energy depending on it's frequency.
Yes, you can still think of photons. Also, it is worth noting that that "harmonics oscillators" are useful in just every branch of physics and engineering. The reason is simply that if you start with an arbitrary -but complicated- potential and expand it in a Taylor series the second term is of course quadratic, which as it happens is just the potential for an harmonics oscillator. Hence, there are lots and lots of examples where one can simplify a problem greatly by simply looking at small deviations from an equilibrium, this in turn allow you to use an expansion which naturally leads to harmonics oscillators.