Operator commutator

The fact that two operators commute is of great importance. It means that once a measurement of one of the properties is carried out, subsequent measurement of that property or of any of the other properties corresponding to mutually commuting operators can be made without altering the system s value of the properties measured earlier. Only subsequent measurement of another property whose operator does not commute with F,... 

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. 

We use an example to illustrate the importance of two operators commuting to quantum mechanics interpretation of experiments. Assume that an experiment has been... 

This analysis can be extended to three or more operators. If three operators A, B, and C have a complete set of simultaneous eigenfunctions, then the argument above shows that A and B commute, B and C eommute, and A and C commute. Furthermore, the converse is also true. If A eommutes with both B and C, and B commutes with C, then the three operators possess simultaneous eigenfunctions. To show this, suppose that the three operators commute with one another. We know that since A and B commute, they possess simultaneous eigenfunctions such that... 

Show that the operator commutes with and with Jy. 

Since the spin operator commutes with the momentum operator, it is possible to speak of states of definite momentum p and spin component /x. The components of the polarization vector may be chosen in such a way that e = XP- The two possible polarizations correspond to only two values of the component of spin angular momentum y,. The third value is excluded by the condition of tranversality. If the z-axis is directed along p, then x0 s excluded. The two vectors Xi and X2> corresponding to circular polarization are equivalent, respectively to Xi and X-i- Thus, the value17 of the spin component y = 1 corresponds to right circular polarization, while /z = — 1 corresponds to left circular polarization. 

Equation (31) is known as Heisenberg s equation of motion and is the quantum-mechanical analogue of the classical equation (17). The commutator of two quantum-mechanical operators multiplied by 2mfh) is the analogue of the classical Poisson bracket. In quantum mechanics a dynamical quantity whose operator commutes with the Hamiltonian, [A, H] = 0, is a constant of the motion. 

If the averages defined by (36) are independent of the time, the ensemble is in statistical equilibrium and dg/dt = 0. As a sufficient condition for equilibrium it is, according to (37) therefore necessary that [g, H] = 0. In general therefore, an ensemble is in statistical equilibrium if the density operator commutes with the Hamiltonian. 

The previous argument is valid for all observables, each represented by a characteristic operator X with experimental uncertainty AX. The problem is to identify an elementary cell within the energy shell, to be consistent with the macroscopic operators. This cell would constitute a linear sub-space over the Hilbert space in which all operators commute with the Hamiltonian. In principle each operator may be diagonalized by unitary transformation and only those elements within a narrow range along the diagonal that represents the minimum uncertainties would differ perceptibly from zero. 

In order to show that the expected-value and derivative operations commute, we begin with the definition of the derivative in terms of a limit 16... 

The expected value on the left-hand side is taken with respect to the entire ensemble of random fields. However, as shown for the velocity derivative starting from (2.82) on p. 45, only two-point information is required to estimate a derivative.14 The first equality then follows from the fact that the expected value and derivative operators commute. In the two integrals after the second equality, only /u,[Pg.264]

In summary, due to the linear nature of the derivative operator, it is possible to express the expected value of a convected derivative of Q in terms of temporal and spatial derivatives of the one-point joint velocity, composition PDF. Two-point information about the random fields U and

expected value and derivative operators commute, and does not appear in the final expression (i.e., (6.9)). 

It is found empirically and of course is predictable theoretically that, when using a model for molecular electronic structure, the set of eigenfunction equations associated with the operators commuting with H are constraints on the action of the variation principle if Et is computed from R subject to symmetry constraints and E2 is computed in the same model with no such constraints then (2)... 

The last problem of general interest in algebraic theory is the evaluation of the eigenvalues of the invariant operators in the basis discussed in Section 2.4. As mentioned before, the invariant operators commute with all the Xs. As a result, they are diagonal in the basis [A,], A, ..., A.v],... 

In the limit N - oo, these commutators vanish, and the corresponding operators commute. By making the identifications... 

Since these operators commute, they admit common eigenstates ... 

Since our sets of boson creation and annihilation operators and fermion creation and annihilation operators commute we can write our unperturbed wavefuntion (po) as the product of the fermion state vector /o) and the boson state vector %o), i.e. 

The term symmetric is used in a variety of ways by mathematicians and in this book. The important point here is that the term implies that for n particles these spin operators commute with any permutation of n objects. 

Because symmetry operators commute with the electronic Hamiltonian, the wavefunctions that are eigenstates of H can be labeled by the symmetry of the point group of the molecule (i.e., those operators that leave H invariant). It is for this reason that one constructs symmetry-adapted atomic basis orbitals to use in forming molecular orbitals. 

In a many-electron system, one must combine the spin functions of the individual electrons to generate eigenfunctions of the total Sz =Li Sz(i) ( expressions for Sx = j Sx(i) and Sy = j Sy(i) also follow from the fact that the total angular momentum of a collection of particles is the sum of the angular momenta, component-by-component, of the individual angular momenta) and total S2 operators because only these operators commute with the full Hamiltonian, H, and with the permutation operators Pjj. No longer are the individual S2(i) and Sz(i) good quantum numbers these operators do not commute with Pjj. 

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 ... 

The column vector is indicated by square brackets, a row vector by round brackets. The quantum numbers may be determined by the complete set of her-mitian operators commuting with the generator of time evolution. Invariance of the quantum state to frame rotation, origin displacement, parity and other symmetry operations determine quantum numbers for the corresponding irreducible representations. Frame related symmetry operations translate into unitary operator acting on Hilbert space (rigged), e.g. Ta. 

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

Since potential and dipole operators commute, we may write... 

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