Hermitian operators proof

The matrices in equation (35) for a system of n spins of 1/2 have dimensions of 22n. This means that, for example, a four-spin system must be considered within a space of 256 dimensions. If we deal with the motion of a spin system in a static magnetic field (as in pulse-type experiments), significant simplifications are possible owing to the rules of commutation. Namely, if the Hermitian operators A and 6 commute in Hilbert space, then all the corresponding superoperators AL, AR, AD, BL, BR, and BD in Liouville space also commute. The proof of this is given in reference (12). In Hilbert space, the following commutation takes place ... 

It is generally true that the normalized eigenfunctions of an Hermitian operator such as the Schrodinger Ti constitute a complete orthonormal set in the relevant Hilbert space. A completeness theorem is required in principle for each particular choice of v(r) and of boundary conditions. To exemplify such a proof, it is helpful to review classical Sturm-Liouville theory [74] as applied to a homogeneous differential equation of the form... 

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. 

Extension of the above proofs to the case of more than two operators shows that for a set of Hermitian operators A,B,C,... there exists a common complete set of eigenfunctions if and only if every operator commutes with every other operator. 

This postulate is more a mathematical than a physical postulate. Since there is no mathematical proof (except in various special cases) of the completeness of the eigenfunctions of a linear Hermitian operator, we must assume completeness. Postulate 4 allows us to expand the wave function for any state as a superposition of the orthonormal eigenfunctions of any quantum-mechanical operator ... 

Proof That Eigenvalues of Hermitian Operators Are Real... 

The point of the above example is that all of our proofs about eigenvalues or eigenfunctions of hermitian operators refer to cases where the eigenfunctions satisfy the requirement that /i/t = 0. Square-integrability guarantees this, but some... 

Proofs exist that certain hermitian operators corresponding to observable properties have eigenfunctions forming a complete set in the space of well-behaved (continuous. 

This theorem shows that every eigenfunction of a Hermitian operator is orthogonal to every other eigenfunction with a different eigenvalue. Furthermore, it is possible to prove that eigenfunctions of a Hermitian operator can be constructed to be orthogonal even if some have like eigenvalues. This is a corollary to the prior theorem which we consider without a detailed proof. 

In general ViAfi is not equal to MiVi but is its Hermitian conjugate, since (p7r)t = Trtpt. Therefore, it should be reasonably obvious that the ViAfi operators are also linearly independent. We note that an alternative, but very similar, proof that all a, = 0 in Eq. (5.42) could be constructed by multiplying on the left by Vj, j = 1,2,...,/ sequentially. 

A full proof of this can be found in Taylor [15]. The factor p is +1 if the operator is Hermitian and —1 if it is anti-Hermitian. We can see an immediate complication relative to our earlier formula Eq. 5.11 in that a full representation matrix for irrep T is required. This is considered in more detail below. Additional redundancies in the P2 list that arise from the form of particular operators axe also treated by Taylor. 

A simple but nonrigorous version of this proof is the following. Since the Liouville operator is Hermitian its eigenvalues A are real and its eigenfunctions x(r) are orthogonal where... 

If an operator is Hermitian, the expectation values are real, and this can be proved by following steps similar to those in the proof of Theorem 8.1. Consequently, observables always have associated operators that are Hermitian. We do not, for instance, measure length as having a real and imaginary component. 

In what sense is it possible to say that 3/9/ behaves like a 1-electron operator in the derivation of (12.6.1)7 Continue the derivation of the time-dependent analogue (12.6.7) of the HP equations, giving a proof that the matrix of Lagrangian multipliers is Hermitian. [Hint Write down a complex-conjugate equation, take appropriate scalar products, and eliminate the time.]... 

---