Problem 11.69

11.69    Measurement is made of the sum of the x and y components of the spin of an electron. What are the possible results of this experiment? After this measurement, the z component of spin is measured. What are the respective probabilities of obtaining the values ?


Partial Solution

Here we have no information about the initial state of the system, but it is not necessary. We are being asked to find what will be obtained after measurement of  = Sx + Sy, or to what state can the system fall after measurement of Sx + Sy. The "possible results of this experiment" are thus the eigenfunctions (jn) and the corresponding eigenvalues (an) of Â. Since spin is 1/2,






we get for Â:



To find the eigenvalues, we need to solve the equation,

Âjn = anjn


or,

det[Â - an1] = 0


where 1 is the identity matrix. The eigenvalues are thus,



with eigenvectors j1, j2.


To find probabilities of finding after measurement of z:

For spin - 1/2 particles, there are two eigenvectors for Sz:  |+1/2ñ and |-1/2ñ. We can use the superposition principle to find the probabilities of getting .

Say the system is in eigenstate j1 of Â. The probability of finding in this state is:


The probablity of finding is thus:


We can find the probabilities of finding the Sz eigenvalues for the eigenstate j2 in the same manner.

References

Liboff, R. L. Introductory Quantum Mechanics, Fourth Edition. San Francisco, CA: Addison Wesley, p. 543, 2003.

Peleg, Y.; Pnini, Reuven; Zaarur, Elyahu Schaum's Outline and Theory and Problems of Quantum Mechanics. New York, NY: McGraw-HIll, pp. 122, 127-128, 1998.