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Central potential and 3 dimensional Schrodinger equation

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The Central Potential

The 3-D time independent Schrodinger equation for a particle of mass m in a spherically symmetric potential is

-- Equation----

a) By writing grad^ in terms of r, theta and phi show that this can be written as

-- Equations---

(b) Writing shi = R Y show that Y must satisfy L^2 Y = c Y where c is a constant, and where

-- Equation----

(c) Writing R = f(r)/r, show that f(r) satisfies the equation

-- Equation----

(d) This is of exactly the same form as a 1-D time independent Schrodinger equation with V replaced by V + c/2mr^2. So can we conclude that the spectrum in the original 3D problem is the same as for this 1D problem?

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(a)
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