24 - Physics 6210/Spring 2007/Lecture 24 Lecture 24 Relevant sections in text 3.5 3.6 Angular momentum eigenvalues and eigenvectors(cont Next we

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Physics 6210/Spring 2007/Lecture 24 Lecture 24 Relevant sections in text: § 3.5, 3.6 Angular momentum eigenvalues and eigenvectors (cont.) Next we show that the eigenvalues of J 2 are non-negative and bound the magnitude of the eigenvalues of J z . One way to see this arises by studying the relation J 2 - J 2 z = 1 2 ( J + J - + J - J + ) = 1 2 ( J - J - + J + J + ) . Now, for any operator A and vector | ψ i we have that (exercise) h ψ | A A | ψ i ≥ 0 , so that for any vector | ψ i (in the domain of the squared angular momentum operators) (exercise) h ψ | J 2 - J 2 z | ψ i ≥ 0 . Assuming the eigenvectors | a, b i are not of the “generalized” type, i.e., are normalizable, we have 0 ≤ h a, b | J 2 - J 2 z | a, b i = a - b 2 , and hence a 0 , - a b a. The ladder operators increase/decrease the b value of the eigenvector with out changing a . Thus by repeated application of these operators we can violate the inequality above unless there is a maximum and minimum value for b such that application of J + and J - , respectively, will result in the zero vector. Moreover, if we start with an eigenvector with a minimum (maximum) value for b , then by successively applying J + ( J - ) we must hit the maximum (minimum) value. As shown in your text, these requirements lead to the following results. The eigenvalues a can only be of the form a = j ( j + 1)¯ h 2 , where j 0 can be a non-negative integer or a half integer only: j = 0 , 1 / 2 , 1 , 3 / 2 , . . . . For an eigenvector with a given value of j , the eigenvalues b are given by b = m ¯ h, 1
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Physics 6210/Spring 2007/Lecture 24 where m = - j, - j + 1 , . . . , j - 1 , j. Note that if j is an integer then so is m , and if j is a half-integer, then so is m . Note also that for a fixed value of j there are 2 j + 1 possible values for m . The usual notational
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This note was uploaded on 02/18/2012 for the course PHYSICS 6210 taught by Professor M during the Spring '07 term at AIU Online.

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24 - Physics 6210/Spring 2007/Lecture 24 Lecture 24 Relevant sections in text 3.5 3.6 Angular momentum eigenvalues and eigenvectors(cont Next we

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