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Physics / Classical Mechanics 0 / Chapter 16 / Problem 16.16

Classical Mechanics | 0th Edition | ISBN: 9781891389221 | Authors: John R Taylor

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Let f (r) be any spherically symmetric function; that is,

Chapter 16, Problem 16.16

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QUESTION:

Let f (r) be any spherically symmetric function; that is, when expressed in spherical polar coordinates, (r, 8, 0), it has the form f (r) = f (r), independent of 8 and 0. (a) Starting from the definition (16.38) of V2, prove that 1 8, V2f = -r 0r2(rf). (b) Prove the same result using the formula inside the back cover for V2 in spherical polar coordinates. (Obviously, this second proof is much simpler, but the hard work is hidden in the derivation of the formula for V2.)

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QUESTION:

Let f (r) be any spherically symmetric function; that is, when expressed in spherical polar coordinates, (r, 8, 0), it has the form f (r) = f (r), independent of 8 and 0. (a) Starting from the definition (16.38) of V2, prove that 1 8, V2f = -r 0r2(rf). (b) Prove the same result using the formula inside the back cover for V2 in spherical polar coordinates. (Obviously, this second proof is much simpler, but the hard work is hidden in the derivation of the formula for V2.)

ANSWER:

Step 1 of 4

Part (a)

Since is the spherical system then,

Here, and are the cartesian coordinates.

Now,

Solve further as,

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