Eigenfunctions of commuting operators

The contents of Sections 7.2 and 7.3 can be summarized by the statement that the eigenfunctions of a Hermitian operator form a complete, orthonormal set, and the eigenvalues are real. 

If the state function Ir is simnltaneonsly an eigenfunction of the two operators A and B with eigenvalnes Oj and bj, respectively, then a measurement of the physical property A will yield the resnlt oy and a measurement of B will yield bj. Hence the two properties A and B have definite valnes when is simnltaneonsly an eigenfunction of A and B. 

In Section 5.1, some statanents were made about simultaneous eigenfunctions of two operators. We now prove these statanents. 

we show that if there exists a common complete set of eigenfnnctions for two linear operators then these operators commute. Let A and B denote two linear operators that have a conunon complete set of eigenfunctions g, g2,  

Equation (7.44) is an operator equation. For two operators to be eqnal, the results of operating with either of them on an arbitrary well-behaved function/mnst be the same. Hence 

THEOREM 3. Let the functions g, gi,--- be the complete set of eigenfunctions of the Hermitian operator A, and let the function F be an eigenfunction of A with eigenvalue k (that is, Af = kF) then if F is expanded as F = 2/ fljg,-, the only nonzero coefficients a,- are those for which gj has the eigenvalue k. (Because of degeneracy, several g, s may have the same eigenvalue k.) 

Thus in the expansion of F we include only those eigenfunctions that have the same eigenvalue as F. The proof of Theorem 3 follows at once from fl =/g F dr [Eg. (7.40)] if F and g correspond to different eigenvalues of the Hermitian operator A, they will be orthogonal [Eq. (7.22)] and a will vanish. 

We shall occasionally use a notation (called notation) in which the function/ is denoted by the symbol /). There doesn t seem to be any point to this notation, but in advanced formulations of quantum mechanics, it takes on a special significance. In ket notation, Eq. (7.41) reads 

Ket notation is conveniently used to specify eigenfunctions by listing their eigenvalues. For exan le, the hydrogen-atom wave function with quantum numbers n, I, m is denoted by = n/m). 

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We now prove the converse, namely, that eigenfunctions of commuting operators can always be constructed to be simultaneous eigenfunctions. Suppose that Afi = atfi and that [A, 5] = 0. Since A and B commute, we have... 

The Uncertainty Principle, 21. Wave Mechanics, 23. Functions and Operators, 25. The General Formulation of Quantum Mechanics, 27. Expansion Theorems, 31. Eigenfunctions of Commuting Operators, 34. The Hamiltonian Operator, 37. Angular Momenta, 39. 

In a more favourable case, the wavefiinction ]i might indeed correspond to an eigenfiinction of one of the operators. If = //, then a measurement of A necessarily yields and this is an unambiguous result. Wliat can be said about the measurement of B in this case It has already been said that the eigenfiinctions of two commuting operators are identical, but here the pertinent issue concerns eigenfunctions of two operators that do not conmuite. Suppose / is an eigenfiinction of A. Then, it must be true that... 

It should be mentioned that if two operators do not commute, they may still have some eigenfunctions in common, but they will not have a complete set of simultaneous eigenfunctions. For example, the and Lx components of the angular momentum operator do not commute however, a wavefunction with L=0 (i.e., an S-state) is an eigenfunction of both operators. 

In the first experiment, the fact that they are mutually commutative allowed us to expand the 64 % probable eigenstate with L=1 in terms of functions that were eigenfunctions of the operator for which measurement was about to be made without destroying our knowledge of the value of L. That is, because and can have simultaneous eigenfunctions, the L = 1 function can be expanded in terms of functions that are eigenfunctions of both and L. This in turn, allowed us to find experimentally the... 

We therefore conclude that the act of carrying out an experimental measurement disturbs the system in that it causes the system s wavefunction to become an eigenfunction of the operator whose property is measured. If two properties whose corresponding operators commute are measured, the measurement of the second property does not destroy knowledge of the first property s value gained in the first measurement. 

A. If the two operators act on different coordinates (or, more generally, on different sets of coordinates), then they obviously commute. Moreover, in this case, it is straightforward to find the complete set of eigenfunctions of both operators one simply forms a product of any eigenfunction (say fk) of R and any eigenfunction (say gn) of S. The function fk gn is an eigenfunction of both R and S ... 

The commutator relations (21) show that the true eigenfunctions can be chosen such that they are simultaneously eigenfunctions of the operators H, S, eind Sz. ... 

Since 5 + and both commute with ff and V, both pnp and npn are eigenfunctions of the operator with total spin S and Ms = S. 

For the remainder of this Section, the primary focus is placed on forming proper N-electron wavefunctions by occupying the orbitals available to the system in a manner that guarantees that the resultant N-electron function is an eigenfunction of those operators that commute with the N-electron Hamiltonian. 

Because the total Hamiltonian of a many-electron atom or molecule forms a mutually commutative set of operators with S2, Sz, and A = (V l/N )Ep sp P, the exact eigenfunctions of H must be eigenfunctions of these operators. Being an eigenfunction of A forces the eigenstates to be odd under all Pp. Any acceptable model or trial wavefunction should be constrained to also be an eigenfunction of these symmetry operators. 

In exercise 7 above you determined whether or not many of the angular momentum operators commute. Now, examine the operators below along with an appropriate given function. Determine if the given function is simultaneously an eigenfunction of both operators. Is this what you expected ... 

Symmetry enters the approximate solution of the electronic Schrd-dinger equation in two ways. In the first place, the exact MOs are eigenfunctions of an operator which commutes with all Om of the point group concerned, they therefore generate irreducible representations of that point group (see Chapter 8) and can be classified accordingly. The same is true for the approximate MOs and consequently one constructs them from combinations of atomic orbitals (symmetry orbitals) which generate irreducible representations. 

Thus, in the central field approximation the wave function of the stationary state of an electron in an atom will be the eigenfunction of the operators of total energy, angular and spin momenta squared and one of their projections. These operators will form the full set of commuting operators and the corresponding stationary state of an atomic electron will be characterized by total energy E, quantum numbers of orbital l and spin s momenta as well as by one of their projections. 

The electronic functions ffir, R) are also eigenfunctions of the operator P-, since P-commutes with SQ,... 

An important property of commuting operators is that there is a set of functions that are eigenfunctions for both operators simultaneously (for the demonstration see, for example, ref. 13). This means that a particular measurement of property Oi leading to an eigenvalue oi has also prepared the system to yield a particular value for the property O2 so that a measurement... 

In general, an arbitrary ket is not an eigenfunction of the operator. It may be verified, however, that the three operators H, and all commute with each other. This means that it is possible to construct wavefunctions that are simultaneous eigenfunctions of all three of these operators with suitable linear combinations of expansion kets. 

Consequently, H, V-, and Mz form a complete set of commuting operators with common eigenfunctions ... 

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