C CS Phys 191 Spin operators spin measurement spin initialization Fall 2005 10 13 05 Lecture 14 1 Readings Liboff Introductory Quantum Mechanics Ch 11 2 Spin Operators Last time 0 1 1 state representing ang mom w z comp up 2 1 state representing ang mom w z comp down 2 These are the eigenvectors and eigenvalues of the spin for a spin 21 system like an electron or proton 0 and 1 are simultaneous eigenvectors of S2 and Sz 1 1 3 h 2 s s 1 0 h 2 1 0 h 2 0 2 2 4 3 h 2 s s 1 1 h 2 1 4 1 1 h m 0 h 0 m 2 2 1 1 h m 1 h 0 m 2 2 S2 0 S2 1 Sz 0 Sz 1 Results of measurements S2 34 h 2 Sz 2h 2h Since Sz is a Hamiltonian operator 0 and 1 form an orthonormal basis that spans the spin 12 space which is isomorphic to C 2 1 So the most general spin 2 state is 0 1 Question How do we represent the spin operators S2 Sx Sy Sz in the 2 d basis of the Sz eigenstates 0 and 1 Answer They are matrices Since they act on a two dimensional vectors space they must be 2 d matrices We must calculate their matrix elements 2 S s211 s212 s221 s222 Sz C CS Phys 191 Fall 2005 Lecture 14 sz11 sz12 sz21 sz22 Sx sx11 sx12 sx21 sx22 etc Sy 1 Calculate S2 matrix We must sandwich S2 between all possible combinations of basis vectors This is the usual way to construct a matrix 3 2 h 4 3 0 S2 1 0 h 2 4 3 2 1 S 0 1 h 2 4 3 1 S2 1 1 h 2 4 s211 0 S2 0 0 s212 s221 s222 So S2 3 2 4 h 1 0 0 1 3 0 h 2 4 1 0 0 0 3 1 h 2 4 Find the Sz matrix s2z11 s2z12 s2z21 s2z22 So Sz h 2 h 2 h 0 Sz 1 0 2 h 1 Sz 0 1 2 h 1 Sz 1 1 2 0 Sz 0 0 1 0 0 1 0 h 2 1 0 0 0 1 h 2 Find Sx matrix This is more difficult What is Sx11 0 Sx 0 0 is not an eigenstate of Sz so it s not trivial Use raising and lowering operators S Sx iSy Sx 12 S S Sy Sx11 0 1 2 S S 1 2i S S 0 S 0 0 since 0 is the highest Sz state But what is S 0 Since S is the lowering operator we know that S 0 1 That is S 0 A 1 for some complex number A which we have yet to determine Similarly S 1 A 0 Question What is A Answer A h A h p p s s 1 m m 1 S s m A s m 1 s s 1 m m 1 S s m A s m 1 Proof First note that A 2 s m S S s m C CS Phys 191 Fall 2005 Lecture 14 2 Since S Sx iSy where Sx Sy are Hermitian S S Hence A s m A s m 1 We are free to choose a phase for the eigenstates set it so A 0 real and nonnegative Now s m S S s m S s m S s m S s m S s m A2 s m s m A2 Also S S Sx iSy Sx iSy Sx2 Sy2 i Sx Sy S2 Sz2 h Sz Therefore s m S S s m s m S2 Sz2 h Sz s m h 2 s s 1 h m 2 h h m h 2 s s 1 m m 1 using S2 s m h 2 s s 1 s m and Sz s m h m s m Thus p A s m h s s 1 m m 1 p A s m A s m 1 h s s 1 m m 1 Now we use these values A to find the desired coefficients 0 r S 0 1 1 1 1 1 1 0 h 0 h 2 2 2 2 r 1 1 1 1 1 1 1 h 1 h 2 2 2 2 0 S 1 S 0 S 1 Sx11 1 2 0 S S 0 Sx11 Sx12 Sx21 Sx22 So Sx h 2 1 2 0 S 0 S 0 1 0 0 h 1 0 2 1 0 S S 1 2 1 1 S S 0 2 1 1 S S 1 2 0 1 1 0 1 h 0 h 0 0 2 2 1 h 1 0 h 1 2 2 1 1 h 0 0 0 2 C CS Phys 191 Fall 2005 Lecture 14 3 Find Sy matrix Use Sy 2i1 S S The proceed similarly to above for Sx Check it out yourself to see that you understand the matrix and bra ket mechanics 0 i h Answer Sy 2 i 0 In summary we define 1 0 0 1 0 i 1 0 0 1 2 3 0 1 1 0 i 0 0 1 Then S2 34 h 2 0 Sx h 2 1 Sy h 2 2 Sz 2h 3 0 1 2 3 are called the Pauli Spin Matrices They are very important for understanding the behavior of two level systems Note that we have already encountered these in our discussion of qubits where they correspond to the gates X 1 Y 2 Z 3 I 0 In the next couple of lectures we shall frequently interconvert using h Sx X 2 h Sy Y 2 h Sz Z 2 3 Measuring Spin We can measure the spin state m with a Stern Gerlach device This is simply a magnet set up to generate a particular inhomogeneous B field In the figure below the field is strong near the N pole below and weaker near the S pole above If the z axis points upwards then we have a negative field gradient in the z direction i e B z 0 S up 0 N down 1 When a particle with spin state 0 1 is shot through the apparatus from the left its spin up portion is deflected upward and its spin down portion downward The particle s spin becomes entangled with its position Placing detectors to intercept the outgoing paths therefore measures the particle s spin Why does this work We ll give a semiclassical explanation mixing the classical F m a and the quantum H E which is not really correct but gives the correct intuition See Griffith s 4 4 2 pp 162 164 for a more complete argument Now the potential energy due to the spin interacting with the field is E B so the associated force is C CS Phys 191 Fall 2005 Lecture 14 Fspin E B 4 At the …