Fisika Modern 08 schrodinger eqinhydrogenatom

r and  appear only on the left side and  appears only on the right side. The left side of the equation cannot change as  changes. The right side cannot change with either r or  . Each side needs to be equal to a constant for the equation to be true. Set the constant to be − m ℓ 2 Solution of the Schrödinger Equation for H azimuthal equation 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

satisfies the azimuthal equation for any value of m ℓ . The solution must be single valued to be a valid solution for any  : Solution of the Schrödinger Equation for H m ℓ must be an integer (positive or negative) for this to be true. Specifically: So: 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

[object Object],Solution of the Schrödinger Equation for H Now, the left side depends only on r , and the right side depends only on  . We can use the same trick again! Rearrange it and divide by sin 2 (  ) : 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | − m ℓ 2

[object Object],Solution of the Schrödinger Equation for H Radial equation Angular equation We’ve separated the Schrödinger equation into three ordinary second-order differential equations, each containing only one variable. 07/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

Fisika Modern 08 schrodinger eqinhydrogenatom

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Fisika Modern 08 schrodinger eqinhydrogenatom