Schrodinger, I Presume

the wave equation in one spatial variable What operator does this job in three variables It is a Schrodinger equation for a particle in less than for this you don t get the Nobel prize. Whatever be unable to distinguish directions from the origin, operator with this property is (up to multiplication 

You will notice that I am using a mathematician s favorite notation for this operator, which is not the same as the standard physics text the symbol A. This is the Laplace operator, or the Laplacian. With this notation in place, Schrodinger s equation is  

Compared with the vibrating string we seem to have an awful lot of variables here. We will want to separate variables in the same way we did for the wave equation in Chapter 6, but which variables shall we use The key here lies in the spherical symmetry that can be read off the equation itself. Initially there are four variables, x, y, z, t, but obviously, t and r play 

Here the variables 6 and 0 are given these labels because they are actually the Euler angles for spherical coordinates, as shown in Fig. 9.1. If you set r = 1 and let 6 and 0 change, you will get points that lie on the sphere of radius one. So these are the natural coordinates for points on the sphere. 

It is an interesting exercise in multivariable calculus to rewrite the Laplacian in terms of these new variables. If you do this you will get  

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