QMGreensite_merged
The number of eigenvalues and eigenfunctions,N, is determined by the system of interest and may be finite or infinite. The adjoi ...
vector space can be written as ðtÞ¼ XN n¼ 1 cn (^) n, ½ 2 : 18 in which thecnare complex numbers and may depend upon time. Th ...
In obtaining [2.21], the orthonormality condition [2.16] has been used. The resulting equation forhiA has the following interpre ...
Next, (^) iis expanded in the eigenfuctions ofB: AB (^) i¼aiB (^) i, AB X j cijj¼aiB X j cijj, X j cijbjAj¼ai X j cijbjj, X ...
in which and are the stationary states andE ¼h! andE ¼h! are the energies of the states withm¼1/2 andm¼–1/2, respectively, and ...
in which! 0 ¼! –! ¼ B 0 is the Larmor frequency and¼bais a phase angle. These results utilize the following equations for t ...
Using the Dirac notation, an arbitrary wavefunction,, can be written as a superposition of a set of orthonormal time-independen ...
Now, using [2.33], hiA ¼ X nm cmcnhjmAjin: ½ 2 : 39 In contrast to [2.21], the ketsjin are not necessarily the eigenfunctions ...
The trace of a product of matrices is invariant to cyclic permutations of the matrices. Thus, Trfg¼ABC Trfg¼CAB TrfgBCA: ½ 2 : 4 ...
The ensemble average of coefficients,cncm, forms a matrix that is referred to as the density matrix. The density matrix is the ...
Equation [2.50] can be used to find a differential equation for the matrix elements of the density operator, dkhjjim dt ¼ dckc ...
solution to [2.53] by simple differentiation: dðtÞ dt ¼iHexpðiHtÞð 0 ÞexpðiHtÞþexpðiHtÞð 0 ÞiHexpðiHtÞ ¼ifexpðiHtÞð 0 ÞH ...
Htime independent is the quantum mechanical equivalent of the rotating frame transformation in the Bloch picture. Asimilarity tr ...
Substituting [2.63] and [2.64] into [2.61] yields dr dt ¼iU½,HU^1 þ dU dt U^1 rþrU dU^1 dt ¼ir,UHU^1 U dU^1 dt ...
Although this expression is an infinite series of terms, a compact closed- form solution often can be obtained if recursive rela ...
Arbitrary kets and bras, expressed as a linear combination of the eigenkets or eigenbras, have the representations ji ¼c ji þc ...
in whichUis a unitary operator with matrix elements in the unprimed basis given by Uij¼hijUjji¼hijj^0 i: ½ 2 : 79 The represen ...
in which the prime has been added toIxto emphasize that the eigenbase ofIxis being utilized. Using [2.75] and [2.78], I^0 x^0 ¼ ...
matrix representation of the operator,H^0 , given by H^0 ¼UHU^1 , ½ 2 : 92 is a diagonal matrix. The transformed basis functi ...
the second term in [2.95] is simplified to! 1 (IxcosþIysin). The third term in [2.95] is simplified to !rfIz because an opera ...
«
25
26
27
28
29
30
31
32
33
34
»
Free download pdf