Question

SG 3 *** 11- SG 2 Y- SG 1 Trap Trap Figure 1: Schematic illustration of a the stream of atoms described, each individually pre- pared in the superposition V = 1 + , being measured by sequential Stern-Gerlach machines. 2. (20 points) Spin of an electron is a quantum mechanical observable, and as such, there are corresponding operators for electron spin. For example, the operator S, relates to the observable called 2-spin' and has exactly two associated eigenfunctions and eigen- values: the eigenfunction wit with associated eigenvalue +h is associate with 'z-spin up' electrons, v, with associated eigenvalue -h is associated with 'Z-spin-down' elec- trons. Similarly, there is an operator S, that has exactly two associated eigenfunctions and eigenvalues given by t+sh (for the observable 'x-spin up'), and o t h ("x-spin down'). The Stern-Gerlach apparatus can be designed to measure either the observable "z-spin' or 'x-spin'. Suppose a stream of electrons, where each individual electron starts out in a superposition state given by V = V0 + 10. goes through a Stern-Gerlach apparatus designed to measure 'z-spin' (SG 1 in the figure).

Answer 1

dy= A = ļ yt + Y Z I 2 12 the SG1 oriented along I measures the eigenstates of Sy , the outcome of are the eigenvalues and Eigenector of Sz. The state y gets projected to Yzt with probability ₂ and gets projected to Yų with probability 2. Lunefore you see two beams. (6) Note the eigenstales of of Sx is 7 4 7 beam gets again split into two beams 4th and you with each probability 1/2

Sx then If she was an eigenstate of it wouldn't split up! 0 (c) the reason why the beam slits up. at SG3 is because 64 = t yet to + 1 t , therefore SG3 oriented along L-axis measures Sy. Lue outcomes are eigenstales Of Su. eigenvalues (d) Yes, If by t passes through SG3 We would get only one beam along 47 because it is an eigenfunction of S. Note: the outcomes are eigenvalues and the state gets projected to the eiger istales * Please let me know if you have any questions!