- We Are Going To Consider Using As A Basis The Development Seen In Class The Angular Momentum Of A Composite System Fo 1 (84.02 KiB) Viewed 103 times
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PLEASE HELP ME ANSWERING ALL THE PARAGRAPHS, IS AN IMPORTANT PROBLEM, AND PLEASE WRITE THE STEP BY STEP WITH ALL THE ALGEBRA AND I PROMISE UP VOTE.
We are going to consider, using as a basis the development seen in class, the angular momentum of a composite system. For this exercise, let us take the particular case where the angular momentum of one of the parts is j1 = 3/2 and that of the other is ja = 1, where we have used to make clear that it can be either orbital momentum or spin. Let us recall that for the first, the states with component and square of defined angular momentum have been denoted as 3), 13), 1 - 3), 13 - ) while we have written those of the second as (11), (10) and 1 - 1). a) What is the only state of the composite system that can have a total angular momentum component z equal to 5/2? b) Apply it to the state you found in a) using relation (2) that the various ascent and descent operators fulfill when acting on each factor, and normalize the resulting state. c) Show that the space of states that have a z component of the total angular momentum equal to 3/2 is two dimensional (that is, it is generated by only two elements of the basis that is obtained as the tensor product of the two originals). d) Show that the result of b) is in the space of c) and find the only, except for one phase, normalized state perpendicular to that of b). e) Apply Il to the result states of b) and d), and normalize. f) Show that the space of complete states that have a component z of the total angular momentum equal to 1/2 is three dimensional. g) Show that the states resulting from e) are in the space of f) and find the only normalized state perpendicular to both. h) Apply the it operator once, then twice, and finally three times to the result states of e) and g), normalizing the results each time. i) Determine the value of the square of the total angular momentum for all the states you found, realizing that you only need to apply Et2 = 12 +212 +ni (1) on three of them, well chosen to simplify the accounts. Choose one of the ones that don't simplify the calculations to explicitly check the value it has for this observable. j) Label all your states as jmt), that is, by its value for the component z and the square of the total angular momentum. Look up the table of Clebsch-Gordan coefficients for for 3/2 x 1 and check that your results are correct. expressed in it. k) Verify that the coefficients you needed to normalize the original states in b) and e) are compatible with the coefficients that would result from using relation (2) on the states labeled as in j). Note that you would have arrived at the same thing using the states with the maximally negative : component of the total angular momentum. Also note that 493=62402. After this, it should be clear to you how to construct any of the Clebsch-Gordan coefficient tables.
Lt = 1, + iL + L = L - iLg
LY.m = [(1 +1) - m(m 1)Y,m1 VI( =