Harmonic oscillator operator formulation

An important advantage of formulating harmonic oscillators problems in terms of raising and lowering operators is that these operators evolve very simply in time. Using the Heisenberg equations of motion (2.66), the expression (2.155) and the commutation relations for a and leads to... 

As an example for the use of this formulation let us calculate the (in-principle time-dependent) variance, (Ax(Z)2), defined by Eq. (2.149) for a Harmonic oscillator in its ground state. Using the expression for position operator in the Heisenberg representation from Eq. (2.166) and the fact that (0 Ax(Z)2 0) = <0 x(Z) 0 for an oscillator centered at the origin, this can be written in the from... 

We now need a systematic way to evaluate matrix elements like — Rgf v y. This is provided by the second quantization formulation [5] of the one-dimensional harmonic oscillator problem, which parallels in some ways the ladder operator treatment of angular momentum. The harmonic oscillator Hamiltonian is... 

The Schrodinger equation for the harmonic oscillator happens to be a well-studied differential equation that mathematicians had solved long before the quantum mechanical problem was formulated. There are an infinite number of functions of x that turn out to be valid eigenfunctions of the Hamiltonian operator. A subscript n is used, therefore, to distinguish one solution from another. Each eigenfunction has its own energy eigenvalue. The differential equation at hand is... 

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