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A beam of spin‐1/2 particles is prepared in the state:     20 2...

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Electrical Engineering

A beam of spin‐1/2 particles is prepared in the state:    20 2 20 4  i a) What are the possible results of a measurement ofthe spin component Sz and with what probabilities would they occur?b) What are the possible results of a measurement of the spincomponent SX and with what probabilities would they occur? c)Suppose that the SZ measurement yields the result SZ = ‐ħ/2.Subsequent to that result a second measurement is performed tomeasure the spin component SX. What are the possible results ofthat measurement and with what probabilities would they occur? d)Draw a schematic diagram depicting the successive measurements asdescribed in part c. Use the Stern‐Gerlach diagrams as drawn in thebook showing the possible numbers of events when measured.

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4.3 Ratings (573 Votes)
a If we measure the component Sz of theintrinsic angular momentum of a silver atom in its ground state wecan find only one or the other of two values corresponding to thedeflections HN1 and HN2 We have to reject the classical pictureand have to conclude that Sz is a quantized physical quantitywhose discrete spectrum includes only two eigenvalues 2 and 2Theoretical descriptionWe associate an observable Sz with Sz According to theexperiment it has to satisfy the eigenvalue equationsSz 2 Sz 2 where and are normalized andorthogonal and thus form a basis of the twodimensional Hilbertspace of onespin 12 particle 1PROBABILITIESIn this case we get Z up or downwith equal probabilities Measuring X has erased our originalmeasurement of Z Similarly if we started with a definite state ofX and measure Z we erase the original value of X By a morecomplicated arrangement we can measure the component of spin inthe direction Y If we do we find that measuring Y erases thevalue of either X or Z measuring X erases Y or Z and measuring Zerases X or Y Suppose that a spin is up in the Zdirection If we measure the spin component along an axis at angle to the Z axis we find the spin up along that axis withprobability pup cos 2 2 This is not special to the Zdirection If the spin is initially up along a given axismeasuring along an axis at angle has outcomes up or down withprobabilities cos 2 2 and sin 2 2 We need a mathematicalframework to encompass these resultsSuppose we multiply the state bya pure phase expi ei ei This doesnt change theprobabilities for a measurement along the Z axis    See Answer
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