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The result of a measurement of the observable A must yield one of the eigenvalues of A. Thus, we see why A is required to be a hermitian operator: Hermitian operators have real eigenvalues. If we denote the set of eigenvalues of A by tex2html_wrap_inline283 , then each of the eigenvalues tex2html_wrap_inline285 satisfies an eigenvalue equation


where tex2html_wrap_inline287 is the corresponding eigenvector. Since the operator A is hermitian and tex2html_wrap_inline285 is therefore real, we have also the left eigenvalue equation


The probability amplitude that a measurement of A will yield the eigenvalue tex2html_wrap_inline285 is obtained by taking the inner product of the corresponding eigenvector tex2html_wrap_inline287 with the state vector tex2html_wrap_inline241 , tex2html_wrap_inline301 . Thus, the probability that the value tex2html_wrap_inline285 is obtained is given by


Another useful and important property of hermitian operators is that their eigenvectors form a complete orthonormal basis of the Hilbert space, when the eigenvalue spectrum is non-degenerate. That is, they are linearly independent, span the space, satisfy the orthonormality condition


and thus any arbitrary vector tex2html_wrap_inline261 can be expanded as a linear combination of these vectors:


By multiplying both sides of this equation by tex2html_wrap_inline307 and using the orthonormality condition, it can be seen that the expansion coefficients are


The eigenvectors also satisfy a closure relation:


where I is the identity operator.

Averaging over many individual measurements of A gives rise to an average value or expectation value for the observable A, which we denote tex2html_wrap_inline315 and is given by


That this is true can be seen by expanding the state vector tex2html_wrap_inline241 in the eigenvectors of A:


where tex2html_wrap_inline321 are the amplitudes for obtaining the eigenvalue tex2html_wrap_inline285 upon measuring A, i.e., tex2html_wrap_inline327 . Introducing this expansion into the expectation value expression gives


The interpretation of the above result is that the expectation value of A is the sum over possible outcomes of a measurement of A weighted by the probability that each result is obtained. Since tex2html_wrap_inline333 is this probability, the equivalence of the expressions can be seen.

Two observables are said to be compatible if AB=BA. If this is true, then the observables can be diagonalized simultaneously to yield the same set of eigenvectors. To see this, consider the action of BA on an eigenvector tex2html_wrap_inline287 of A. tex2html_wrap_inline343 . But if this must equal tex2html_wrap_inline345 , then the only way this can be true is if tex2html_wrap_inline347 yields a vector proportional to tex2html_wrap_inline287 which means it must also be an eigenvector of B. The condition AB=BA can be expressed as


where, in the second line, the quantity tex2html_wrap_inline355 is know as the commutator between A and B. If [A,B]=0, then A and B are said to commute with each other. That they can be simultaneously diagonalized implies that one can simultaneously predict the observables A and B with the same measurement.

As we have seen, classical observables are functions of position x and momentum p (for a one-particle system). Quantum analogs of classical observables are, therefore, functions of the operators X and P corresponding to position and momentum. Like other observables X and P are linear hermitian operators. The corresponding eigenvalues x and p and eigenvectors tex2html_wrap_inline387 and tex2html_wrap_inline389 satisfy the equations


which, in general, could constitute a continuous spectrum of eigenvalues and eigenvectors. The operators X and P are not compatible. In accordance with the Heisenberg uncertainty principle (to be discussed below), the commutator between X and P is given by


and that the inner product between eigenvectors of X and P is


Since, in general, the eigenvalues and eigenvectors of X and P form a continuous spectrum, we write the orthonormality and closure relations for the eigenvectors as:


The probability that a measurement of the operator X will yield an eigenvalue x in a region dx about some point is


The object tex2html_wrap_inline413 is best represented by a continuous function tex2html_wrap_inline415 often referred to as the wave function. It is a representation of the inner product between eigenvectors of X with the state vector. To determine the action of the operator X on the state vector in the basis set of the operator X, we compute


The action of P on the state vector in the basis of the X operator is consequential of the incompatibility of X and P and is given by


Thus, in general, for any observable A(X,P), its action on the state vector represented in the basis of X is


next up previous
Next: Time evolution of the Up: The fundamental postulates of Previous: Physical Observables

Mark Tuckerman
Wed Mar 10 13:14:21 EST 1999