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CRC Press eBooks

Harmonic Oscillators

Tao Xiang

Published Mar 31, 2022• 0 citations
Publisher

Abstract

The quantum harmonic oscillator is a quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary 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. Coherent state is physically interesting because it is a minimum uncertainty state at which the Heisenberg uncertainty relation reaches its minimum. If a charged particle of charge moves in an electromagnetic field, there is a Lorentz force acting on the particle. The electric and magnetic are invariant under the gauge transformation. In the presence of an external vector potential, the Hamiltonian is changed, so is the probability current operator.

DOI
10.1201/9781003174882-4

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•Jul 20, 1993•0 citations
The quantum harmonic oscillator is a quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary 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. Coherent state is physically interesting because it is a minimum uncertainty state at which the Heisenberg uncertainty relation reaches its minimum. If a charged particle of charge moves in an electromagnetic field, there is a Lorentz force acting on the particle. The electric and magnetic are invariant under the gauge transformation. In the presence of an external vector potential, the Hamiltonian is changed, so is the probability current operator.
Here a special case of perturbation in quantum harmonic oscillator is studied. Here we assume the perturbed potential to be a Harmonic Oscillator that has been shifted in the position space.We construct the new creation and annihilation operators for the new Hamiltonian to find out its energy eigenstates. What is interesting about the solution of this system is that we find out that the ground state of this system is a coherent state of the ground state of the original harmonic oscillator. We know that the Hamiltonian of electromagnetic field is similar to that of the harmonic oscillator where the electric field and magnetic field are expressed in terms of time evolution of annihilation and creation operators. According to the quantum electromagnetic field theory, photon states are expressed as the coherent states of the ground states of the unperturbed harmonic oscillator.In the previous research works,it has been established how coherent states provide a better picture of Electric field compared to the energy eigenstates. This paper mainly deals with why the coherent states provide a more complete picture of electric and magnetic field theory compared to just energy eigenstates. Here we find how our perturbation act as a coherent state generator and adds a new term in the Hamiltonian. The Perturbation is thought to arise from the only available interaction i.e. observation. The perturbation also explains how observation leads to coherence of photon states. We will find out how the new perturbation affects the time evolution of electromagnetic field. This also explains how the classical electromagnetic field theory takes into account the factor of observation, due to which the classical laws are in perfect agreement with the quantum laws.
Following the development of the quantum theory of radiation and with the advent of the laser, the states of the field that most nearly describe a classical electromagnetic field were widely studied. In order to realize such ‘classical’ states, we will consider the field generated by a classical monochromatic current, and find that the quantum state thus generated has many interesting properties and deserves to be called a coherent state . An important consequence of the quantization of the radiation field is the associated uncertainty relation for the conjugate field variables. It therefore appears reasonable to propose that the wave function which corresponds most closely to the classical field must have minimum uncertainty for all times subject to the appropriate simple harmonic potential. In this chapter we show that a displaced simple harmonic oscillator ground state wave function satisfies this property and the wave packet oscillates sinusoidally in the oscillator potential without changing shape as shown in Fig. 2.1. This coherent wave packet always has minimum uncertainty, and resembles the classical field as nearly as quantum mechanics permits. The corresponding state vector is the coherent state |α〉, which is the eigenstate of the positive frequency part of the electric field operator, or, equivalently, the eigenstate of the destruction operator of the field. Classically an electromagnetic field consists of waves with welldefined amplitude and phase. Such is not the case when we treat the field quantum mechanically.
The gauge-invariant formulation of quantum mechanics is applied to the charged harmonic oscillator in an electromagnetic field in the electric dipole approximation to obtain the probability that the oscillator is in a particular state at time t. This probability is compared with the corresponding 'probability' calculated from the conventional approach to the interaction of radiation with matter using the interaction A.p. The probabilities do not agree with each other. From the principle of gauge invariance it is concluded that the probability calculated from the gauge-invariant formulation is correct in this case, and that the conventional approach in general is incorrect.
In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables and I > 0 which played a prominent role in the “old” Bohr-Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (φ,I)-model for the HO: the transformation is only locally symplectic and singular for (q,p) = (0,0). Globally the phase space {(q,p)} has the topological structure of the plane ℝ2, whereas the phase space {(φ,I)} corresponds globally to the punctured plane ℝ2 -(0,0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = Icos φ and h2 = -Isin φ as the basic coordinates on {(φ,I)}, where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(φ,I)} compared to the usual one for {(q,p)}: In the quantum mechanics for the (φ,I)-model of the HO the three hj correspond to the self-adjoint generators Kj, j = 0,1,2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy of the (φ,I)-Hamiltonian . For an m-fold covering the lowest possible value for k is k = 1/m, which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is “erased” when passing to the simpler (Q,P)-model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mulliken-type experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.
The time dependent harmonic oscillator is solved explicitly for quantum mechanics by the operator method with an auxiliary condition as the classical solution. Two classical invariant quantities which determine whether or not the system is bound are derived by the classical equation of motion. We obtain the invariant operator from one classical invariant quantity. Its eigenfunction is related to the solution of Schrodinger equation of the system and its eigenvalue is related to another classical quantity. The wave function is evaluated exactly by the eigenfunction of the invariant operator but it is not the eigenfunction of the Hamiltonian of the system. The uncertainty which calculates with the wave function is not a minimum one. We will confirm that the function which holds minimum uncertainty is a eigenfunction of the Hamiltonian.