The hamiltonian of a simple harmonic oscillator with m. By comparison, coherent states also called glauber states of the harmonic oscillator are more elaborate superpositions that maintain the welldefined shape of the ground state distribution while exhibiting the ki classical oscillatory motion that is absent in the previous examples. In this paper, dynamics and quantum mechanical coherent states of a simple harmonic oscillator are considered in the framework of the generalized uncertainty principle gup. Harmonic oscillator and coherent states ethz photonics. Coherent states of the harmonic oscillator consider the following initial state of a simple harmonic oscillator.
In this paper dynamics and quantum mechanical coherent states of a simple harmonic oscillator are considered in the framework of generalized uncertainty principle gup. The eigenstates of a nonhermitian operator are given by, where are the harmonicoscillator eigenstates. Chapter 8 the simple harmonic oscillator a winter rose. Coherent states coherent states, or as they are sometimes called glauber coherent states are the eigenstates of the annihilation operator here. The coherent states of the twodimensional isotropic.
Changing the notation, it is not hard to show that jzi eiim ze jz j2 2 ez a yj i 7 which generalizes eq. Harmonic oscillator, coherent states, and feynman path. Coherent states of the 1d harmonic oscillator the very wellknown coherent states of the 1d harmonic oscillator, labelled by z 2c, satisfy a jzi zjzi. Coherent states of harmonic oscillator and generalized uncertainty. The quantum harmonic oscillator is the quantummechanical analog of the classical harmonic oscillator. Quantum physics ii, lecture notes 6 mit opencourseware. Coherent states of the harmonic oscillator physics forums. Except for the ground state, the correspondence between the quantum energy eigenstates of the simple harmonic oscillator and the classical states in which a pendulum swings back and forth is not completely clear. Algebraic extensions of the notion of qoscillator coherent states have been discussed in 11, 12 and applications in. Coherent states of the harmonic oscillator wolfram. Harmonic oscillators with di erent timedependent parameters can be related through unitary transformations.
A coherent state is a speci c kind of quantum state of the harmonic oscillator. It is a solvable system and allows the explorationofquantum dynamics in detailaswell asthestudy ofquantum states with classical properties. 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. The harmonic oscillator is an ubiquitous and rich example of a quantum system. 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. On the other hand, there are quasiclassical states for the quantum ho coherent states which illustrates the relation between. To define the notation, let us briefly recap the dynamics of the classical oscillator.
Furthermore, it is one of the few quantummechanical systems for which an exact. They are the most classical states of the harmonic oscillator. Its wavefunction will satisfy the heisenberg uncertainty principle with equality and the expectation values of position and momentum satisfy the classical equations of motion of a harmonic oscillator. Solution of timeindependent schrodinger equation for a. The rain and the cold have worn at the petals but the beauty is eternal regardless of season. Equations of motion for the simple harmonic oscillator are derived and some of their new implications are discussed. The quantum mechanical states of harmonic oscillator do not have definite phase i. These remarkable quasiclassical states are called coherent states, and were discovered by schrodinger himself. The topic of coherent states is relevant to harmonic oscillators of every kind, including mechanical oscillators, electronic oscillators, and everything else. To discuss the harmonic oscillator with the hamiltonian. The coherent state of the simple harmonic oscillator considered by schriodinger 5 is close to a classical wave packet.
But for the oscillator, the potential somehow keeps the wave packet together, a minimum uncertainty wave packet at all times. A simple physical meaning is that measurement of a small portion of a coherent state does not change the coherent state as long as the eld is strong. In this section we want to compare the quantum oscillator with predictions we. Abstract in this thesis we construct coherent states in the l2r hilbert space, these states minimize the quantum mechanical uncertainty between xand pand obey the classical equations of motion for the harmonic oscillator. We have encountered the harmonic oscillator already in sect. Physics 570a homework 6 coherent states of the harmonic. Classical mechanics of the simple harmonic oscillator. By comparison, coherent states also called glauber states of the harmonic oscillator are more elaborate superpositions that maintain the welldefined shape of the ground state distribution while exhibiting the kind of classical oscillatory motion that is absent in the previous examples. Furthermore, it is easy to see that the displaced ground state. Dressed coherent states of the anharmonic oscillator. Coherent states of the simple harmonic oscillator galileo. The dressed by the drive coherent state of the t 0 k harmonic oscillator is simply a familiar coherent state of the undriven harmonic oscillator. Coherent states of a harmonic oscillator are wavepackets that have the shape of the ground state probability distribution but undergo the motion of a classical oscillator of arbitrary energy.
A class of the well known coherent states is those of the simple harmonic oscillator 1,2,3,4. Pdf coherent states of harmonic and reversed harmonic. Coherent states of the harmonic oscillator semantic scholar. Coherent states of harmonic oscillator physics forums. In these notes i will assume knowledge about the operator method for the harmonic oscillator corresponding to sect. Coherent states of general timedependent harmonic oscillator. There is a large number of quantum systems which can be approximated, at least in the limit of small amplitudes, by the ho equations. Why look for eigenstates of the annihilation operator.
Alright, im sure im missing something extremely simple, but in griffiths and another text im reading coherent states are mentioned as eigenfunctions of the annihilation operator. Coherent states coherent states are special states of bosonic systems including the quantum harmonic oscillator, whose excitation quanta can be regarded as bosonic particles whose dynamics are highly similar to classical oscillator trajectories. Using the caldirolakanai hamiltonian for the damped harmonic oscillator, exact coherent states are constructed. The existence of generalized coherent states for a simple harmonic oscillator can then be interpreted as the result of a formal invariance under a unitary transformation which relates. This equation tells us that coherent states may be generated starting with arbitrary coherent states upon application of an appropriate creation operator. Coherent states of the harmonic oscillator in these notes i will assume knowledge about the operator method for the harmonic oscillator corresponding to sect. When anharmonicity and an external driving force are both present, we solve the master equation numerically, starting typically from a coherent state of the corresponding harmonic oscillator. Coherent states of harmonic oscillator and generalized uncertainty principle kourosh nozari and tahereh azizi department of physics, faculty of basic science, university of mazandaran, p. Our task here is to construct and analyze these coherent states and to find how they relate to the usual energy eigenstates of the oscillator. Quasiclassical states of the simple harmonic oscillator draft version introduction. In this paper dynamics and quantum mechanical coherent states of a simple har monic oscillator are considered in the framework of. Coherent superpositions for the harmonic oscillator. Amazing but true, there it is, a yellow winter rose.
Thermodynamics of the harmonic oscillator using bogoliubov. Simple harmonic oscillator february 23, 2015 one of the most important problems in quantum mechanics is the simple harmonic oscillator, in part. These new coherent states satisfy the properties which coherent states should generally have. Lecture 7 of my quantum theory course at mcgill university, fall 2012. Coherent states of harmonic and reversed harmonic oscillator article pdf available in symmetry 86.
Using coherent states, we show that the treatment of the system is simplified, that the relationship between the classical and quantum solutions becomes transparent, and that the evolution operator of the system can be calculated easily as the free evolution operator of the harmonic oscillator followed by a displacement operator that depends on. Analyzing generalized coherent states for a free particle. A coherent state, also known as a glauber state or a squeezed quantum state, is an eigenfunction of the harmonic oscillator annihilation operator, where for simplicity. Coherent states of the harmonic oscillator youtube. Coherent states for the isotropic and anisotropic 2d. Coherent states of harmonic oscillator and generalized. The mathematics of coherent states are motivated by a simple relation, namely. They provide an important link between quantum and classical harmonic oscillators. The particular case of fcoherent states called also as nonlinear coherent states for the function f expressed in terms of laguerre.
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