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# Spin and the Pauli equation

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Presentación sobre el espín, la ecuación de Pauli y el Efecto Zeeman.

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### Spin and the Pauli equation

1. 1. Chapter 12 Spin ( 自旋 ) <ul><li>The Mathematical Description of Spin </li></ul><ul><li>Wave Function with Spin </li></ul><ul><li>The Pauli Equation </li></ul>
2. 2. Experimental demonstration of spin The doublet splitting of sodium atom. The transition of valence electron from the first excited state to the ground state (2p  2s) lead to two adjacent spectral lines of 589.0 nm and 589.6 nm. (2) (1) The results of the experiment done by Stern-Gerlach in 1922 denote that every electron has one intrinsic ( 内禀 ) angular momentum (spin) of , which is corresponding to a magnetic moment ( 磁矩 ) of one Bohr magneton ( 玻尔磁子 ),
3. 3. The magnitude of the magnetic moment of the electron caused by the orbital motion, The z component of the orbital angular momentum is quantized by means of For each angular momentum l  , there are 2 l ＋ 1 possibilities. For the doublet of sodium atom, 2 l s ＋ 1 =2, so Spin angular momentum is only
4. 4. 2. Wave Function with Spin ( 自旋波函数 ) Since the spin component S z can take only two values, namely S z ＝ ±½ħ After considering spin, the wave function of a particle can be written by So the spin wave function has only two components, i.e. The complete spin wave function is described by
5. 5. χ ± indicate only the state of the spin, namely, “spin up” or “spin down”. is the probability finding an electron with spin up at r and t . where is the probability finding an electron with spin down at r and t .
6. 6. 3. The Pauli Equation In the absence of spin, the Hamiltonian for the motion of an electron in an electromagnetic field can be written Spin interacts with the magnetic field, and the magnetic moment is where
7. 7. The potential in magnetic field is The Schrodinger equation of a particle with spin is Ψ is called the spinor wave function. So where
8. 8. 4. The simple Zeeman effect In a weak magnetic field B , the orbital angular moment and spin will interact with magnetic. This interaction potential can be written Where M is magnetic moment. M is generally written as Where J denotes orbital angular moment or spin, q is the charge, g is Lande factor. For the orbital angular moment, we have g=1. However g =2 for the spin.
9. 9. Set So we get Where H 0 is the Hamiltonian in the absence of magnetic field. The wave function is  1 and  2 are the spatial parts of the total wave function respectively, and they can be described by
10. 10. Where  L is the Larmor frequency, So we obtain
11. 11. According to the following equations So we get for for In the magnetic field, the energy level will split, and degeneracy is missing. For the s state, which have no orbital angular moment, the energy level splits into two level.