Sakurai, modern quantum mechanics, addisonwesley 1993. Chapter 5 harmonic oscillator and coherent states 5. Quantum physics ii, lecture notes 6 mit opencourseware. He obtained the quantum states by solving the di erential equation satis ed by the evolution operator. Quantum harmonic oscillator from ladder operators to coherent states. Its time evolution can be easily given in closed form. Because of its close relation to the timeevolution of a system, it is of fundamental importance in most formulations of quantum theory. The time dependence of this state becomes then j ti 1 p 2 e i. Kicked frequency in the probability representation of quantum mechanics. More generally, the time evolution of a harmonic oscillator with a timedependent frequency. Threedimensional harmonic oscillator and time evolution.
November 8, 2018 the time evolution of thermal states of. It is a solvable system and allows the explorationofquantum dynamics in detailaswell asthestudy ofquantum states with classical properties. Coherent states of the harmonic oscillator in these notes i will assume knowledge about the operator method for the harmonic oscillator corresponding to sect. Chapter 8 the simple harmonic oscillator a winter rose. Time evolution of quantummechanical harmonic oscillator with timedependent frequency. A new method for analyzing the time evolution of quantum. Request pdf threedimensional harmonic oscillator and time evolution in quantum mechanics the problem of defining a time or phase operator for a threedimensional harmonic oscillator has. We generalize the classical time evolution problem of a harmonicoscillator chain with periodic and fixedend boundary conditions to the quantum mechanical case. We show that nonspreading wave packets exist in this system in addition to. The harmonic oscillator described by the schroumldinger equation is a central textbook example in quantum mechanics its time evolution can be easily given in closed form more generally the time evolution of a harmonic oscillator with a timedependent frequency can also be given in quadratures this allows the efficient solution of the. It is shown explicitly that the hightemperature limits of our quantum mechanical treatment give the known classical results. The varying mass and frequency of the system are reduced to constant mass and frequency, and the corresponding eigenvalues and eigenvectors are derived. The quantum harmonic oscillator with timedependent mass and frequency is analyzed by using the canonical transformation method. Box 2, h1502 budapest, hungary received 19 april 1994.
The rain and the cold have worn at the petals but the beauty is eternal regardless. He also investigated the time evolution of a charged oscillator with a time dependent mass and frequency in a timedependent eld. Time evolution of harmonic oscillator thermal momentum. Amazing but true, there it is, a yellow winter rose. The harmonic oscillator is a model which has several important applications in both classical and quantum mechanics. The harmonic oscillator is important in physics since any oscillatory motion is harmonic by approximation as long as the amplitude is small.
Pdf energy evolution in timedependent harmonic oscillator. The system so defined is the quantum harmonic oscillator. Any classical driving force acting on a quantum oscillator. Ehrenfest theorem 4 symmetry in quantum mechanics 5 heisenberg representation 6 example. Continuous variable quantum optical simulation for time. Quantum harmonic oscillator with timedependent mass and. At a couple of places i refefer to this book, and i also use the same notation, notably xand pare operators, while the correspondig eigenkets. It serves as a prototype in the mathematical treatment of such diverse phenomena as elasticity, acoustics, ac circuits, molecular and crystal vibrations, electromagnetic fields and optical properties of matter. In order to solve the quantum system we attempt to factorize the hamiltonian. The equation that governs time evolution is called the schrodinger equation. Time evolution of states in quantum mechanics1 uio. We can write the quantum hamiltonian in a similar way. To make a dissipative quantum harmonic oscillator, one can just make the value of the mass or the spring constant or both slightly complex, with a small imaginary part.
An harmonic oscillator is a particle subject to a restoring force that is proportional to the displacement of the particle. It is useful to exhibit the solution as an aid in constructing approximations for more complicated systems. Fermis golden rule e k e l transition probability per second. Time evolution operator in interaction picture harmonic oscillator with time dependent perturbation 2. A quantum harmonic oscillator is the simplest system for which an exact expression for the wigner function n w x p, is obtained, where n is the state number 41. Simple harmonic oscillator february 23, 2015 one of the most important problems in quantum mechanics is the simple harmonic oscillator, in part. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Volume 192, issues 56, 12 september 1994, pages 3115. In the hamiltonian description of classical mechanics, the system. The equation for the quantum harmonic oscillator is a second order differential equation that can be solved using a power series. We illustrate this method by exactly solving the system of driven harmonic oscillator. Time evolution of quantum harmonic oscillator youtube. Exact time evolution of a quantum harmonicoscillator. Time evolution of quantummechanical harmonic oscillator.
Then, if the signs of the imaginary parts are chosen correctly, the time. Time evolution of states in quantum mechanics1 the time evolution from time t 0 to tof a quantum mechanical state is described by a linear operator ut. The hamiltonian of a qho is, in which h is the plank constant. The time evolution of a quantum harmonic oscillator with a series of sudden jumps of the mass or the frequency is determined in the form of a recursion relation.
The quantum harmonic oscillator university physics volume 3. Janszky research laboratory for crystal physics, hungarian academy of sciences, p. Chapter 15 time evolution in quantum mechanics 201 15. The quantum harmonic oscillator is the quantummechanical analog of the classical harmonic oscillator. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the weinorman lie algebraic approach. The harmonic oscillator is an ubiquitous and rich example of a quantum system. Evolution of the quantum mechanical system is governed by the associated hamiltonian. Pdf time evolution for harmonic oscillators with position. We generalize the classical time evolution problem of a harmonic oscillator chain with periodic and fixedend boundary conditions to the quantum mechanical case. The quantum harmonic oscillator stephen webb the importance of the harmonic oscillator the quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems that can really be solved in closed form, and is a very generally useful solution, both in approximations and in exact solutions of various. The time evolution from time t0 to t of a quantum mechanical state is.
To obtain this result we shall study the lecture notes in relativistic quantum mechanics from l. It models the behavior of many physical systems, such as molecular vibrations or wave packets in quantum optics. Elsevier 12 september 1994 physics letters a 192 1994 3115 timeevolution of a harmonic oscillator. The timedependent wave function the evolution of the ground state of the harmonic oscillator in the presence of a timedependent driving force has an exact solution. The hamiltonian is named after william rowan hamilton, who created a revolutionary reformulation of newtonian mechanics, now called hamiltonian mechanics, which is also important in quantum physics. The harmonic oscillator is a system where the classical description suggests clearly the. In an open system, the qho will interact with the environment and under the rotating wave approximation the hamiltonian is given by, where the second and third terms. Heisenbergpicture approach to the exact quantum motion of. Time evolution of harmonic oscillator thermal momentum superposition states ole steuernagel school of physics, astronomy and mathematics, university of hertfordshire, hat eld, al10 9ab, uk dated. The harmonic oscillator hamiltonian is not explicitly dependent on time so we can use eq. It is seen that the richness of quantum physics will greatly a ect the future generation technologies in many aspects.
More elegant solution of the quantum harmonic oscillator diracs method all properties of the quantum harmonic oscillator can be. The time dependent wave function the evolution of the ground state of the harmonic oscillator in the presence of a time dependent driving force has an exact solution. Pdf time evolution of harmonic oscillators with time. When considering the harmonic oscillator, the righthand side of the moyal equation is equal to zero, and the moyal equation transforms into the liouville. Lecture notes weng cho chew1 september 23, 20 1the author is with u of illinois, urbanachampaign.