ADDITION OF ANGULAR MOMENTUM QM 1 I Total angular momentum Recall that quantum mechanical particles can possess both orbital, L, and spin, S, angular momentum The state of such a particle may be written in a basis of the eigenvectors of L, L z, S, and Ŝ z : ψ χ = l m s, m l, m ; s, m, l s l s A Consider an electron in the angular momentum state l m ; s, =,0; 1, 1 1 Determine four quantities related to angular momentum that are well-defined for this electron, and the value of each Explain What must be true about these four quantities if they are all well-defined? 3 For the state above, do the x- or y-components of the orbital angular momentum have well-defined values? Explain Is your answer the same for the x- and y-components of the spin angular momentum? The total angular momentum, J, for a particle is defined as the vector sum of all forms of angular momentum For a single quantum mechanical particle, this takes the form J = L + S B Using J = L + S, write down separate equations for each of the three components of (eg, ) in terms of the components of L and S z 1 Is the z-component of the total angular momentum z for the electron in part A welldefined? If so, give the value; if not, list all possible values Explain McDermott, Heron, Shaffer, and PEG, U Wash Preliminary First Edition, 015
QM Addition of angular momentum The questions on this page refer to the electron with the state l m ; s, =,0; 1, 1 Is the x-component of the total angular momentum x for this electron well-defined? If so, give the value; if not, list all possible values Explain 3 Is the y-component of the total angular momentum y for this electron well-defined? If so, give the value; if not, list all possible values Explain 4 Check that your answers are consistent with the fact that only a single component of angular momentum can well-defined at the same time Consider the quantum numbers j and m j associated with and J z, respectively Note that obeys the same formalism as other forms of angular momentum, such as L and S C Determine the value of m j for this electron Is this quantity well-defined? Explain D For the electron in this state, will j be an integer or a half-integer? Explain E Predict whether the magnitude of the total angular momentum vector, J, for this electron will be well-defined Explain Check your results with a tutorial instructor McDermott, Heron, Shaffer, and PEG, U Wash Preliminary First Edition, 015
Addition of angular momentum QM 3 II Addition of angular momentum In this section, you will make a classical analogy as a guide to determining the allowed values of the quantum number j associated with the magnitude of the total angular momentum A Consider two classical vectors, A and B of lengths 5 and, respectively There is no knowledge of the individual components or directions of the two vectors Let C = A+ B 1 What is the largest possible value for the magnitude of the vector C? Draw an alignment of the vectors A and B that could result in this value What is the smallest possible value for the magnitude of the vector C? alignment of the vectors A and B that could result in this value Draw an 3 What are all the possible values for the magnitude of the vector C? Illustrate the possible alignments of the vectors A and B that could result in these values Explain B Consider again the electron in the state l m ; s, =,0; 1, 1 1 Determine the length of the orbital angular momentum vector, L value to one decimal in terms of ħ (Hint: It is not just lħ) Approximate this Determine the length of the spin angular momentum vector, S to one decimal in terms of ħ Approximate this value 3 Suppose were a classical vector Determine both the largest and the smallest possible values for its magnitude Also, find the largest and possible values for Approximate these values to one decimal in terms of ħ and enter them in the table at right Quantity Largest value Smallest value McDermott, Heron, Shaffer, and PEG, U Wash Preliminary First Edition, 015
QM 4 Addition of angular momentum C For each of the given values for the quantum number j, fill in the table below for the value of Approximate each value to one decimal in terms of ħ Quantum number j 0 1/ 1 3/ 5/ 3 7/ Value of j(j+1) Approximation Is it possible to measure values for that do not appear in the table (ie, values that lie in between the values shown)? Explain Consider again the electron in the state l m ; s, =,0; 1, 1 D Recall your answer to question D from section I Can j be a half-integer for this electron? Can it be an integer? Cross out any values in the table above that are not allowed for this electron E Cross out any values in the table above that would not be classically allowed for this electron Check that your answer is consistent with the table in question B3 F For this electron, list all the possible values of Explain and its corresponding quantum number, j G Consider the discussion between two students below Student 1: Originally, we knew that l = and s = 1/ Since J = L + S, we just add l and s together to get j, which would be equal to 5/ for this particle Student : You can t do that because J, L, and S are vectors Since L and S could point in any direction, the magnitude of J could be any number between the magnitude of L S and the magnitude of L + S, which for this particle would be 15 < J < 33 Both students are incorrect Identify the flaws in each student s reasoning Explain Check your results with a tutorial instructor McDermott, Heron, Shaffer, and PEG, U Wash Preliminary First Edition, 015
Addition of angular momentum QM 5 III The total angular momentum basis Since angular momentum is quantized, the quantum number j associated with the total angular momentum vector J L = + S can take values from l s to l + s in integer steps A Use this formula to determine the possible values of the quantum number j for an electron in the state l, m ; s, =,0; 1 1, l m s Is your answer consistent with your answer from the previous section? B Is there a single, well-defined value for this quantum number for this electron? Explain The state of a particle with both spin and orbital angular momentum may be written not only in the l, m ; s, m basis, but also in the following different basis: ψ = l s; j, m l s, j C Which four quantities related to angular momentum can be well-defined for a particle written in this basis? Explain why this is possible D Write down a general expression (ie, with undetermined coefficients) in the l, s; j, m j basis for the total angular momentum state of the electron defined at the top of the page Explain (Hint: How many terms should there be in your expression?) McDermott, Heron, Shaffer, and PEG, U Wash Preliminary First Edition, 015
QM 6 Addition of angular momentum E Should the probabilities associated with each term in your expression be equal? F The undetermined coefficients in your expression are known as Clebsch-Gordon coefficients; they can be found in a table on page 188 of Introduction to Quantum Mechanics nd Edition by David Griffiths Use this table to find the coefficients in your expression above Resolve any inconsistencies with your other answers The formalism introduced in this tutorial for the addition of angular momentum is valid for any sum of two (or more) angular momentum states Expressions such as L 1 + L, S 1 + S, and S 1 + S + S 3 + all obey this formalism McDermott, Heron, Shaffer, and PEG, U Wash Preliminary First Edition, 015