Observables eigenvalue equation

This equation is an example of a general class of equations called eigenvalue equations of the form flip = wip where li is an operator and to is the value of an observable corresponding to that operator. (The mathematical expression c is referred to as an eigenfunction of the operator II). 

The second complication is that the equation, as traditionally interpreted, only handles point particles, but produces eigenfunction solutions of more complex geometrical structure. By analogy with electromagnetic theory the square of the amplitude function could be interpreted as matter intensity, but this is at variance with the point-particle assumption. The standard way out is to assume that ip2 represents a probability density rather than intensity. Historical records show that this interpretation of particle density was introduced to serve as a compromise between the rival matrix and differential operator theories of quantum observables, although eigenvalue equations, formulated in either matrix or differential formalism are known to be mathematically equivalent. 

The only possible results from measurements of a physical observable O are a set of eigenvalues o of the eigenvalue equation ... 

Postulate 4. If G is any linear Hermitian operator that represents a physical observable, then the eigenfunctions 4>, of the eigenvalue equation above form a complete set. 

The spaces of interest in physics are spanned by the eigenstates a ) of real (that is self-adjoint) linear operators a, whose eigenvalues a are real. Such operators are called observables. The eigenvalue equation is... 

Since L and Lz are commuting observables they have simultaneous eigenstates m), which obey the eigenvalue equations... 

Suppose and represent eigenstates of an observable A, satisfying the respective eigenvalue equations... 

If two operators commute, there is no restriction on the accuracy of their simultaneous measurement. For example, the x- and y-coordinates of a particle can be known at the same time. An important theorem states that two commuting observables can have simultaneous eigenfunctions. To prove this, write the eigenvalue equation for an operator A ... 

Postulate 3. In any measurement of an observable A, associated with an operator the only possible results are the eigenvalues a , which satisfy an eigenvalue equation... 

The only observable values for a property 2 are one of the eigenvalues to,-of the associated linear Hermitian operator m as defined in the eigenvalue equation... 

The state vector is not physically observable. However, it bears all the information about the system. Such information is withdrawn through the eigenvalue equation. A good question is how the state vectors can be obtained. 

FIGURE 9. Comparison of the observed ionization energies 7 of the C2s bands in the He(IIa) PE spectra of linear, branched and monocyclic hydrocarbons with the lowest eigenvalues (equation 38) obtained by diagonalizing the adjacency matrix A of the corresponding AR-models (cf equation 39). Normal alkanes C , n = 1 to 5, methane to pentane. Branched alkanes i-C = isobutane, 1-C5 = isopentane, neo-Cs = neopentane. Cycloalkanes C , n = 3 to 8, cyclopropane to cyclooctane... 

Postulate 3 In any measurement where an exact solution can be obtained, the only values that will ever be observed are the eigenvalues a that satisfy the eigenvalue equation given by Equation (3.30). This postulate is the one that is responsible for the quantization of energy. 

Observe that Equation (10.37) can be solved for by the method of eigenvectors [2]. It can be seen that nontrivial equations to the set of ordinary differential equations with constant coefficients represented by Equation (10.36) exist only for certain specific values of K called eigenvalues. These are the solutions to... 

In the oblate symmetric top, the rotational Hamiltonian is given by Eq. 5.10. The c axis is denoted the figure axis. According to the commutation rules obtained in Section 5.3, one possible commuting set of observables is J, J, and J. It is then possible to formulate rotational states JKM which simultaneously obey the eigenvalue equations... 

In a prolate symmetric top, the a axis rather than the c axis becomes the figure axis. The body-fixed a, b, c axes depicted in Fig. 5.4 are replaced by the b, c, and a axes, respectively. The mutually commuting set of observables becomes J, Ja and J. With these modifications, the oblate top eigenvalue equations and wave functions (5.21)-(5.24) become applicable to the prolate top as well. The pertinent rotational energies in cm are... 

Another postulate of quantum mechanics states that for every physical observable of interest, there is a corresponding operator. The only values of the observable that will be obtained in a single measurement must be eigenvalues of the eigenvalue equation constructed from the operator and the wavefunction (as shown in equation 10.2). This, too, is a central idea in quantum mechanics. 

There are other common observables in addition to energy. One could operate on the wavefunction with the position operator, x, which is simply multiplication by the coordinate x, but multiplying the sine functions of the particle-in-a-box by the coordinate x does not yield an eigenvalue equation. The " P s of equation 10.11 are not eigenfunctions of the position operator. 

Postulate III. The only values of observables that can be obtained in a single measurement are the eigenvalues of the eigenvalue equation constructed from the corresponding operator and the wavefunction P ... 

This is the same operator we used for 2-D rotation. Because the part of the 3-D rotational wavefunction is exactly the same as for the 2-D rotational wavefunction, it may not surprise you that the eigenvalue equation, and therefore the value of the observable is exactly the same ... 

Construct the complete spherical harmonic for and use the operators for E, L, and to explicitly determine the energy, the total angular momentum, and the z-component angular momentum. Show that the values of these observables are equal to those predicted by the analytic expressions for E, l, and (The objective of this example is to illustrate that the operators do in fact operate on the wavefunction to produce the appropriate eigenvalue equation.)... 

So, invoking the postulate that the value of an observable is equal to the eigenvalue from the corresponding eigenvalue equation ... 

Like the angular momentum of an electron in its orbit, there are two measurables for spin that can be observed simultaneously the square of the total spin and the z component of the spin. Because spin is an angular momentum, there are eigenvalue equations for the spin observables that are the same as for and L, except we use the operators S and to indicate the spin observables. We also introduce the quantum numbers s and m to represent the quantized values of the spin of the particles. (Do not confuse s, the symbol for the spin angular momentum, with s, an orbital that has = 0.) The eigenvalue equations are therefore... 

Close inspection of equation (A 1.1.45) reveals that, under very special circumstances, the expectation value does not change with time for any system properties that correspond to fixed (static) operator representations. Specifically, if tlie spatial part of the time-dependent wavefiinction is the exact eigenfiinction ). of the Hamiltonian, then Cj(0) = 1 (the zero of time can be chosen arbitrarily) and all other (O) = 0. The second tenn clearly vanishes in these cases, which are known as stationary states. As the name implies, all observable properties of these states do not vary with time. In a stationary state, the energy of the system has a precise value (the corresponding eigenvalue of //) as do observables that are associated with operators that connmite with ft. For all other properties (such as the position and momentum). 

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