The Oscillator according to Matrix Mechanics

We proceed to illustrate the fundamental ideas of matrix mechanics by means of an example, namely, the linear harmonic oscillator. We start from the classical expression for the energy, 

The difference between classical mechanics and quantum mechanics, as was explained in the text, lies in the fact that the quantities p and q are no longer regarded as ordinary functions of the time, but stand for matrices, the element q,nm of which denotes the quantum amplitude associated with the transition from one energy-level to another, E. Its square, just like q, the square of the amplitude in classical mechanics, is a measure of the intensity of the line of the spectrum emitted in this transition. When we introduce the matrices into the classical equations of motion, we must also bring in the commutation law 

These equations are easily solved. If the matrix is to satisfy the equation of motion q H- = 0, this equation must be satisfied by each element of the matirix independently  

This choice of order will be found convenient later, as individual rows (or columns) of the matrices with higher suflSjses correspond to states of higher energy. We would, however, expressly emphasize the fact that this fixing of the order does not destroy the generality of the solution in any way. 

The momentum matrix has an analogous form from p == it follows that 

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