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Physics 3313 - Lecture 16

Physics 3313 - Lecture 16. Wednesday April 1, 2009 Dr. Andrew Brandt. Hydrogen Atom Wave Function Angular Momentum Orbital and Magnetic Quantum Numbers Angular Momentum Operator TEST moved to 4/27. Hydrogen Atom Wave Function.

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Physics 3313 - Lecture 16

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  1. Physics 3313 - Lecture 16 Wednesday April 1, 2009 Dr. Andrew Brandt • Hydrogen Atom Wave Function • Angular Momentum • Orbital and Magnetic Quantum Numbers • Angular Momentum Operator • TEST moved to 4/27 3313 Andrew Brandt

  2. Hydrogen Atom Wave Function • http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html#c1 • different orbital angular momentum states identified by a letter in orbital notation 3313 Andrew Brandt

  3. Angular Momentum • Radial equation (above) should only be concerned with radial motion (towards and away from nucleus), but energy could have an orbital term • If then this term would cancel out • since L=r x p =mvr, so 3313 Andrew Brandt

  4. Orbital (l) and Magnetic (ml) Quantum Numbers • l is related to orbital angular momentum; angular momentum is quantized and conserved, but since h is so small, often don’t notice quantization • Electron orbiting nucleus is a small current loop and has a magnetic field, so an electron with angular momentum interacts with an external magnetic field • The magnetic quantum number ml specifies the direction of L (which is a vector—right hand rule) and gives the component of L in the direction of the magnetic field Lz • Five ml values for l=2 correspond to five different orientations of angular momentum vector. 3313 Andrew Brandt

  5. Angular Momentum • L cannot be aligned parallel with an external magnetic field (B) because Lz is always smaller than L (except when l=0) • In the absence of an external field the choice of the z axis is arbitrary (measure projection as in any direction) • Why only Lzquantized? What about Lx and Ly? • Suppose L were in z direction, then electron would be confined to x-y plane; this implies z position is known and pz is infinitely uncertain, which is not true if part of a hydrogen atom • Therefore average values of Lx =Ly =0 and it is only necessary to specify L and Lz 3313 Andrew Brandt

  6. Precession of Angular Momentum • The direction of L is thus continually changing as it precesses around the z axis (note average values of Lx =Ly =0 ) 3313 Andrew Brandt

  7. Angular Momentum Operator • Consider angular momentum definition: so • We can define the angular momentum operator in cartesian and spherical coordinates: • with gives similarly 3313 Andrew Brandt

  8. QM Modifications to Bohr Model • In Bohr model electron has circular orbit around nucleus with and =90o and changes with time • QM mods: 1) No definite r, , and , but only probabilities due to wave nature of electron 2) |2| independent of time and varies from place to place, so can’t think of electron as orbiting probability constant independent of azimuthal angle (spherical symmetry) 3313 Andrew Brandt

  9. Radial Wave Function • Radial part of wave function varies with r and differs for different n,lcombinations • R is maximum at r=0 (inside nucleus)for s states but approaches 0 at r=0 for l>0 3313 Andrew Brandt

  10. Probabilities • Probability of finding electron in hydrogen atom in a spherical shell between r and r+dr is given by • with • since angular wave functions are normalized 3313 Andrew Brandt

  11. Probability Distributions 3313 Andrew Brandt

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