05 - Physics 6210/Spring 2007/Lecture 5 Lecture 5 Relevant...

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Physics 6210/Spring 2007/Lecture 5 Lecture 5 Relevant sections in text: § 1.2–1.4 An alternate third postulate Here is an equivalent statement of the third postulate.* Alternate Third Postulate Let A be a Hermitian operator with an ON basis of eigenvectors | i i : A | i i = a i | i i . The only possible outcome of a measurement of the observable represented by A is one of its eigenvalues a j . The probability for getting a j in a state | ψ i is Prob ( A = a j ) = D X i =1 |h i, a j | ψ i| 2 , where the sum ranges over the D -dimensional space of eigenvectors | i, a j i , i = 1 , . . . , D , with the given eigenvalue a j . Note that if the eigenvalue a k is non-degenerate then Prob ( A = a j ) = |h j | ψ i| 2 . Note also that h j | ψ i is the component of | ψ i along | j i . Let us prove the probability formula in the case where there is no degeneracy. Allowing for degeneracy is no problem; think about it as an exercise. Let f ( x ) be the characteristic function of a j . We have (exercise) f ( A ) = | j ih j | . Then, in the state | ψ i we have Prob ( A = a j ) = h f ( A i ) = h ψ | ( | j ih j | ) | ψ i = |h j | ψ i| 2 . Because the state vectors have unit norm, we have 1 = h ψ | ψ i = X j h ψ | j ih j | ψ i = X j |h j | ψ i| 2 , * For now we state it in a form that is applicable to finite-dimensional Hilbert spaces. We will show how to generalize it to infinite-dimensional ones (such as occur with a “particle”) in a little while. 1
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Physics 6210/Spring 2007/Lecture 5 that is, the probabilities Prob ( A = a j ) add up to unity when summed over all the eigen- values. This indicates that the probability vanishes for finding a value for A which is not one of its eigenvalues. We can write the expectation value of an observable so that the probabilities feature explicitly. As usual, we have A | i i = a i | i i , h i | j i = δ ij . If a system is in the state | ψ i , we compute (by inserting the identity operator twice – good exercise) h A i = h ψ | A | ψ i = X ij h ψ | i ih i | A | j ih j | ψ i = X i a i |h i | ψ i| 2 .
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05 - Physics 6210/Spring 2007/Lecture 5 Lecture 5 Relevant...

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