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# Lect12 - z |Y00| Angular Momentum Atomic States the Pauli...

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Angular Momentum, Atomic States, & the Pauli Principle |Y 20 | |Y 21 | |Y 22 | z |Y 10 | |Y 11 | z |Y 00 | z

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Overview Overview z Schrödinger’s Equation for the Hydrogen Atom z Radial wave functions z Angular wave functions z Angular Momentum z Quantization of L z and L 2 z Spin and the Pauli exclusion principle z Stern-Gerlach experiment z Nuclear spin and MRI
Summary of s Summary of s - - states of H states of H - atom atom z To get the exact eigenstates and energies for the “s-states” of the Coulomb potential, one needs to solve the radial SEQ ) r ( ER ) r ( R r e r r r 1 m 2 2 2 2 2 = κ = z Summary of wave functions for the “s-states”: z The zeros in the subscripts below are a reminder that these are states with zero angular momentum. (spherically symmetric) R 30 r 0 15a 0 0 3 / 2 0 0 0 , 3 3 2 2 3 ) ( a r e a r a r r R + 0 r 0 4a 0 R 10 0 / 0 , 1 ) ( a r e r R r R 20 0 10a 0 0 0 2 / 0 0 , 2 2 1 ) ( a r e a r r R

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Wavefunction Wavefunction of the H of the H - - atom atom z Solutions to the time-independent SEQ for this spherically symmetric potential can have simple forms in spherical coordinates: x y z r θ φ ) , ( ) ( ) , , ( φ θ lm nl nlm Y r R r = Ψ with quantum numbers: n l and m principal orbital magnetic (angular momentum) The Y lm ( θ,φ ) are known as “Spherical harmonics”. They are related to the angular momentum of the electron. Last lecture we studied the properties of the radial part. Today we will examine the angular part.
Quantized Angular Momentum Quantized Angular Momentum z Ordinary linear momentum depends on how fast the phase changes as you move linearly in space: z Angular momentum L Z around the z axis depends on how fast the phase changes as you rotate around the z-axis. A state with an exact value of L z is of the form: Re( ψ ) φ r p t r = L z = m = Reminder: e im φ = cos(m φ ) + i sin(m φ ) m is called the ‘orbital’ magnetic quantum number where ( ) ikx p kx e ψ =∝ = , where ( ) ( , ) im Zl m Lm rY e φ ψθφ G = An integer number , m, of wavelengths must fit around the circle: m = 0, ± 1, ± 2, ± 3, . ... Otherwise the function doesn’t have a single value. z So the angular momentum along a given axis, e.g. L Z , can have only quantized values: 0, , 2 etc Z L = ±± == You can picture this by thinking of ‘linear momentum’ h/ λ ‘around’ the loop, with m λ =r and multiplying by radius r.

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The The “l” “l” Quantum Number Quantum Number z The quantum number m reflects the component of angular momentum about a given axis. ... , , , 2 1 0 m m L z ± ± = = where = z The quantum number l in the angular wave function Y l m ( θ,φ ) tells the total angular momentum L. L 2 = L x 2 + L y 2 + L z 2 is quantized. The possible values of L 2 are: .. , 2 , 1 , 0 l where ) 1 l ( l L 2 2 = + = = States can be eigenstates of both L 2 and L Z .
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Lect12 - z |Y00| Angular Momentum Atomic States the Pauli...

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