Matrices, Multiple Dimensions in Quantum Mechanics

There comes to be two main approaches to Quantum Mechanics. One approach is an equations approach which uses wavefunctions, operators and sometimes eigenstages. The other approach is a linear algebra approach that uses matrices, vectors and eigenvectors to describe quantum mechanics.

matwave

Consider an example of a quantum mechanical problem that uses linear algebra for the description of particle spin:

matrqm

This allows for a more direct view of commutators as discussed in the previous article on quantum mechanics [link]. Matrices have an advantage of storing much more information elegantly and are convenient for commutations.

qmcommutate1

Matrices in fact can be written for x_hat, p_hat and other operators. Matrices are also useful for introducing more than one dimension. We can also make use of this method to give us a three-dimensional Schroedinger equation. First we will start by forming three dimensions of momentum p vectors.

qm4

 

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