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1、1.The Mathematical Description of Spin2.Wave Function with Spin3.The Pauli Equation Chapter 12 Spin (自旋自旋)The doublet splitting of sodium atom. The transition of valence electron from the first excited state to the ground state (2p2s) lead to two adjacent spectral lines of 589.0 nm and 589.6 nm.The
2、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), (2)Experimental demonstration of spin(1)The magnitude of the magnetic moment of the ele
3、ctron 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 2l1 possibilities. For the doublet of sodium atom, 2ls1 =2, so Spin angular momentum is only Spin is an angular momentum, and its description is analogo
4、us to the orbital angular momentum. Its three components are Their commutation relations are1. The Mathematical Description of SpinWe introduce Pauli matrices , and define them in the following Their commutation relations are Since the unit matrix unchanged when the representation changes, soTo obta
5、in the matrices of and in the representation of , we use Multiplication from the left and from the right by , and addition, we getIn its eigenrepresentayion (能量表象), the eigenvalues of z is 1, and they can be written by the form of diagonal matrix, namelyAccording toWe getUsing the same method, we ca
6、n obtain the relations of other componentsThe Pauli matrices are anticommuting (反对易反对易).setUsing the their anticommuting relations of Pauli matrices, we getSo Since the Pauli matrix must be Hermitian, namelySo we obtain a21=a*12So ( is real)Using the relation ofUsing the same method, we also get the
7、 matrix The total spin isFinally we get three components of Pauli matrix in representation,Example 1: In the presentation z , solve the eigenfunction and eigenvalue of x.Solution ：In the presentation z , Set the eigenfuction of x can be written byIts corresponding eigenvalue is , therefore we getnam
8、ely We obtainWhen =1, a = b. According to the condition of normalization, we get the eigenfunction When = -1, a = -b. According to the condition of normalization, we get the eigenfunction Problem: In the presentation z , solve the eigenfunction and eigenvalue of yExample 2：set operator A commutes wi
9、th Pauli matrix , prove the following expressionSolution:Each component of operator A commutes with that of Pauli matrix , namelySo Therefore we obtainAccording to the same method, we get Example 3 spin angular momentum projects (投影) of an electron in the direction of (sincos, sinsin, cos) is Proble
10、m: solve the eigenfunction and eigenvalue of SnSolution: In the presentation of Sz, the matrices of Sx, Sy and Sz can be written by So Since the values which the electron spin projects in arbitrary direction have only two values, namely , the eigenvalue of Sn is Set the corresponding eigenfunction i
11、s According to the normalized condition, we get So we get WhenWhenSince the spin component Sz can take only two values, namely Sz 2. Wave Function with Spin (自旋波函数自旋波函数)After considering spin, the wave function of a particle can be written by So the spin wave function has only two components, i.e. T
12、he complete spin wave function is described by 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.whereis the probability finding an electron with spin down at r and t.In the absence of spin, the Hamiltonian for the mo
13、tion of an electron in an electromagnetic field can be writtenSpin interacts with the magnetic field, and the magnetic moment is 3. The Pauli EquationwhereThe potential in magnetic field is The Schrodinger equation of a particle with spin is is called the spinor wave function.So where4. The simple Z
14、eeman effectIn 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
15、 angular moment, we have g=1. However g =2 for the spin. Set So we get Where H0 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 Where L is the Larmor frequency, So we obtainA
16、ccording to the following equations So we getforforIn 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. Example: Only considering spinor motion, set an electron is in magnetic fie
17、ld B(0, B, 0). When t= 0, the electron is in the state of “spin up”, namely, its spinor wave function is Problem: (1) solve the spinor function in case of t 0 (2) How long does the time go through when the state of the electron turns from “spin up” to “spin down”? (3) At the time t, the probability
18、of finding an electron in the state of “spin up” or “spin down”Solution: (1) In the case of only considering spinor motion, Schrodinger equation satisfiesWhere Hs is the Hamiltonian of the spinor motion, and it can be written by Where B is Bohr magneton, B is the magnitude of magnetic field. At the
19、time t, set the spinor function of an electron is After the above differential equation are solved, we getAccording to the initial condition, We obtain(2) From the spinor wave function (t), we can easily get“Spin up”“Spin down”“Spin up”Therefore, when the period The direction of spin of the electron
20、 changes, namely from “spin up” to “spin down” or “spin down” to “spin up”. The spinor state of the electron oscillates between 1/2 and -1/2, the oscillation period is （3）Therefore the probability of finding an electron in the state of “spin up” is The probability of finding an electron in the state
21、 of “spin down” is 1. Set operators A and B commute with respectively, prove the following expression.Exercise 2. Only considering spinor motion, set an electron is in magnetic field B(0, 0, B). When t= 0, the electron is in the state of “spin down”, namely, its spinor function is solve the spinor f
22、unction in case of t 0*3. Only considering spinor motion, when t0, the direction of magnetic field is along z, namely B0 =(0, 0, B0). When t 0, another magnetic field B1(t) is applied to the system, its direction is perpendicular to that of B0, namelyProblem: (1) solve the spinor function in case of t 0 (2) How long does the time go through when the state of the electron turns from “spin up” to “spin down”?
