The harmonic oscillator is fundamental to a wide range of physics, including the electromagnetic ﬁeld, spectroscopy, solid state physics, coherent state theory and SUSY-QM. The broad application of the harmonic oscillator stems from the raising and lowering ladder operators which are used to factor the system Hamiltonian.

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In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator.

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Chapter 15 Harmonic oscillators and photons 354 15.1 Harmonic oscillator and raising and lowering operators 354 15.2 Hamilton’s equations and generalized position and momentum 360 15.3 Quantization of electromagnetic fields 361 15.4 Nature of the quantum mechanical states of an electromagnetic mode 366 15.5 Field operators 367

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Abstract: Two vector operators aimed at shifting angular momentum quantum number l in spherical harmonics ｜lm〉,primarily proposed by X.L.Ka in 1999,are in fact first rank irreducible tensor operators.For a given magnetic quantum number m,specific state ｜lm〉 in spherical harmonics with the lowest angular momentum quantum number l is obtained and how to use this state to generate whole ...

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The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator.Because an arbitrary potential can 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.

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Quantum Harmonic Oscillator Ladder Operators David G. Simpson January 1, 2007 Here a is the demotion (annililation, lowering) operator; and aé is the promotion (creation, raising) operator for the quantum-mechanical simple harmonic oscillator. One Operator a jniD p n jn 1i aé jniD p n C1 jnC1i Two Operators aa jniD p n.n 1/ jn 2i aaé jniD ...