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MODULE 5. HERMITICITY, ORTHOGONALITY, AND THE SPECIFICATION OF STATES we have stated that we need Hermitian operators because their eigenvalues are real This is so they can be related to experimentally determined observables (always real )
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MODULE 5 HERMITICITY, ORTHOGONALITY, AND THE SPECIFICATION OF STATES we have stated that we need Hermitian operators because their eigenvalues are real This is so they can be related to experimentally determined observables (always real) A definition of Hermiticity (see Barrante, chap 10) is To prove the real property consider the eigenvalue equation with the eigenket normalized
MODULE 5 Now form the complex conjugate of both sides From the Hermiticity condition the two LHS are equal and therefore the two RHS are equal i.e. w = w* which is only possible if w is real
MODULE 5 More about Orthogonality We have stated, and used a symmetry argument to show, that “eigenfunctions corresponding to different eigenvalues of the same operator are orthogonal.“ Now we can be a little more rigorous and prove the condition According to the orthogonality statement, if we have two eigenkets of the Hermitian operator W^ having eigenvalues w1and w2 where w1 =/= w2,then
MODULE 5 Suppose we have two eigenstates That satisfy the two EV equations Now form the complex conjugate of the RH equation and subtract it from the LH one LH is zero (Hermitian condition)
MODULE 5 Our initial condition was that the two eigenvalues are different. The only way to satisfy the last equation is for And since then Thus different eigenfunctions of a Hermitian operator having different eigenvalues are orthogonal.
MODULE 5 The Specification of States Three questions: • Can a state be simultaneously an eigenstate of all possible observables, A, B, C, … ? • Are there restrictions in the number or type of variables that can be simultaneously specified? • And if so, how can we identify such things? If simultaneity is possible then if we measure the observable represented by the operator A^ we shall get exactly a as the outcome (P 4) and likewise for the other observables.
MODULE 5 We first find the conditions under which two observables may be specified simultaneously with arbitrary precision. we need to establish the conditions whereby a given ket can be simultaneously an eigenket of two Hermitian operators We assume that the property is true and find conditions that allow it to be so. Thus we assume that ket Iq> is an eigenket of A^ and B^
MODULE 5 Write the following chain: Thus the two operators commute (Barrante, chap 10) This is the condition that is necessary for Iq> to be a simultaneous eigenstate of the two operators.
MODULE 5 However, we need to find out whether commutation of the operators is a sufficient condition for simultaneity. Or, if is it certain that the ket is also an eigenstate of B^?
MODULE 5 This is an eigenvalue equation (with eigenvalue a) and by comparing it with where b is a constant of proportionality. Thus, is an eigenstate of B^, as we set out to prove.
MODULE 5 Thus if we wish to know whether a pair of observables can be specified simultaneously we need to inspect whether the corresponding operators commute. The converse is also true, i.e., if a pair of operators does not commute, then their corresponding observables will not have simultaneously precisely defined values. We can extend this to more than two observables by taking them successively in pairs in which one of the pair is common to all pairs.
MODULE 5 For example Is the simultaneous specification of the position and linear momentum of a particle allowed? The three operators for the x, y, and z coordinates commute with each other (these operators tell us to multiply by the variable and multiplication is always commutative). Also the three operators for the components of momentum commute with each other.
MODULE 5 There are no constraints on the complete specification of position or of momentum as individual observables. What about the position and momentum pairs? To proceed we need to find the commutator of
MODULE 5 Similarly And we see that momentum and position cannot be simultaneously specified with arbitrary precision
x pz pz py py z y z px MODULE 5 In general we can see that the operator for any component of momentum does not commute with its own position operator, But it does with the other position operators. Lines link those observables that can be specified simultaneously; those that cannot be so specified are not linked.
MODULE 5 Observables that cannot be simultaneously specified are said to becomplementary. This is completely against the tenets of classical physics which presumed that no restrictions existed on the simultaneous determination of observables (no complementarity). Quantum mechanics tells us there can be restrictions on the extent that we can specify a state. Refer to the final segment of Module 5 about Complementarity and Uncertainty
