7 angular momentum angular momentum is conserved in

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Unformatted text preview: n. u༇ For the two problems we have solved in 3D, we have used 3 quantum numbers. o  not surprising that we need one quantum number per coordinate. u༇ We will need 3 quantum numbers in 3D to describe the energy eigenstates. 6 Central Force Problems: Spherical Symmetry u༇  Spherical symmetry of the poten2al V(r) implies all the components of angular will be conserved. u༇  But there is an uncertainty principle between any two components of the angular momentum. u༇  It turns out the most we can know at the same 2me is the eigenvalues of L2 and of one component of L. o  Its not surprising that the two angular degrees of freedom in spherical coordinates to give us 2 quantum numbers in our solu2on. o  we choose Lz because z axis is the special axis of the usual spherical coordinate system. 7 Angular Momentum Angular momentum is conserved in the universe due to rota2onal symmetry of space. u༇  Angular momentum is important in most solu2ons to QM problems. u༇  There is an uncertainty principle between any two components of L u༇  We can know L2 and any one component. u༇  o  we choose Lz u༇  These two quantum numbers can cover the angular dependence in any spherically symmetric problem. o  2 out of 3 dimensions determined o  Spherical harmonics (ahead) Angular momentum is quan2zed u༇  Same as the SU(2) group u༇  8 Separa2on of Variables in Spherical Coordinates u༇ We can write p2 in terms of radial deriva2ves and L2. u༇ 3D Schrödinger equa2on becomes u༇ We can separate r from angular variables; and assume the wave func2on factorizes o  note that eq. does not contain Lz (reasonable) 9 The Spherical Harmonics u༇ Eigenfunc2ons of L2 and Lz u༇ Ortho- normal u༇ Any angular func2on can be wriOen as a sum u༇ The spherical harmonics are a powerful mathema2cal tool. 10 The Spherical Harmonics u༇ Orthonormal u༇ higher l implies higher order in cosθ u༇ φ dependence simple since Lz operator is u༇ The constant gives the correct normaliza2on. u༇  The phase is conven2onal 11 Quan2za2on of Angular Momentum u༇  Since we can’t know that the other two components are 0 due to uncertainty, the z component is always less than the total magnitude of angular momentum. u༇  Spherical Harmonics are the eigenfunc2ons of angular momentum. u༇  Measurements of angular mom...
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