- 11.3 Obtainment of the Dynamics 325. - 14.5 Quantization of the Electromagnetic Fields 423. - 6.3 Stabilization of the energy of the eight lowest eigenvalues E k (n. - 9.2 Progressive stabilization of the eigenvalues appearing in Eq. - 9.4 Five lowest wavefunctions k (ξ) of the Morse Hamiltonian compared to the five symmetric or antisymmetric lowest wavefunctions n (ξ) of the. - 9.5 The 40 lowest energy levels of the Morse oscillator. - (9.103) in terms of the geometric parameters V 1. - 9.13 Schematic representation of the two wavefunctions (9.120). - ε γ is the width of the energy cells given by Eq. - The initial excitation energy of the site k = 1 is α 2 1 = N. - The initial excitation energy of the site k = 1 is α 2 1 = N . - 13.3 Temperature evolution of the elongation Q(T ) (in Q. - The intensities are normalized to the maximum of the curve. - Figure 9.4 Five lowest wavefunctions k (ξ) of the Morse Hamiltonian compared to the five symmetric or antisymmetric lowest wavefunctions n (ξ) of the harmonic Hamiltonian.. - Consider a linear transformation of the form (1.5). - Now, the Hermitian conjugate of the linear transformation (1.7) is. - Consider now an eigenvalue equation of the form (1.6). - ξ | ξ or, in view of the inequality (1.20). - of the Hermitian conjugate A † of the linear operator A. - Moreover, the Hermitian conjugate of the linear transformations (1.40) is |B. - where A i is the scalar eigenvalue of the operator A. - In addition, owing to the nullity of the commutator (1.61), we have. - Next, assume that the inverse U − 1 of the linear operator U is the Hermitian conjugate of U:. - Invariance of the trace:. - where | n are the corresponding eigenkets of the eigenvectors of A. - Owing to the Hermiticity of the observables, their eigenvalues are real:. - so that due to the orthonormality of the eigenkets of a Hermitian operator A(P k , Q k ) P μη. - where A 2 is the average of the square of A. - Let us write the eigenvalue equation of the Hamiltonian H:. - i of the Hamiltonian, that is,. - (2.37) is the dispersion of the operator A averaged on the ket. - If it is normalized, the norm of the ket| (t. - i (t)|AH| (t) or, in term of the commutator of H and A. - (2.56) to the product QP of the coordinate and momentum operators Q and P. - 2.4.4.2 Applications of the virial theorem. - where k is the force constant of the potential, which is a scalar. - 2.5.1 Eigenvalue equations of the position and momentum operators. - Consider the eigenvalue equation of the coordinate operator Q:. - P|{P} and {P}|P = {P}|P (2.93) Here, |{P} 2 is an eigenket of the momentum operator P with the eigenvalue P.. - Now, calculate the commutator of the operator (2.95) with Q. - Now, consider the eigenvalue equation of the momentum operator corresponding to the zero eigenvalue, that is,. - 3.1.4.1 Simple linear transformations Consider the eigenvalue equation of the Hermitian operator B:. - leading to the following matrix representation of the inverse linear transformation:. - which, owing to the orthonormality properties (3.28) of the basis. - l } yielding a matrix representation of the Hamiltonian.. - j u } of the kets | l belonging. - the action of the operator Q over some ket | reads. - This representation of the operator is therefore J. - H| (t) (3.76) where H is the total Hamiltonian of the system. - First consider that of the Q(t) coordinate. - H| (t) SP Thus, the time derivative of the IP ket becomes. - (3.139), the square of the density operator reduces to ρ 2. - or, because of the orthonormality properties (3.138), to ρ 2. - where | l(r) is the lth ket | l of the rth particle.. - Finally, owing to the orthonormality properties (3.138) of the basis. - where C k are the coefficients involved in the expansion of the logarithm. - the matrix elements of the density operator (3.158) read {Q}| ρ |{Q. - 3.4.5.2 Momentum representation Now, write the eigenvalue equation of the momentum P as. - Next, consider a matrix element of the density operator in the basis. - Thus, write the eigenvalue equation of the Hamiltonian:. - the matrix representation of the Liouville Eq. - (4.5) is of the form. - Then, in terms of the energy units E. - the diagonal matrix elements of the full Hamiltonian are. - Now, the eigenvalue equation of the full Hamiltonian (4.37) is. - the matrix representation of this eigenvalue equation in the basis of the eigenkets of H ◦ being. - where the square root appears to be of the form. - 4.3 of the two interacting energy level systems with β negative. - C where the C k ± are the components of the eigenvectors. - E − )t (4.84) Furthermore, by aid of the trigonometric relations. - ω 2 kl (1 − cos ω kl t) Moreover, by aid of the usual trigonometric relations. - i where the reduced mass m of the oscillator is given by. - (5.8) leads to the following fundamental expression for the Hamiltonian of the quantum harmonic oscillator:. - 5.1.2 Resolution of the Hamiltonian eigenvalue. - Again, consider the action of the commutator (5.15). - Consider again the action of the commutator (5.16) on |a † {n}:. - {n}|a † (5.32) Then, owing to the property of the norm, requiring. - which, with the help of the eigenvalue equation (5.17), transforms to { n } a. - Now, suppose that the energy of the ground state |{0} is zero. - n Now, consider the corresponding average value of the momentum, which, according to Eq. - This is the purpose of the present section.. - (5.9), the average value of the harmonic Hamiltonian is. - Then, the Hamiltonian of the oscillator yields. - (3.60), the Hamiltonian of the harmonic oscillator is, therefore, H ˆ. - Search for a solution of the form. - C 0 (e − ξ where C 0 is the normalization constant of the wavefunction.. - and where C 0 is the normalization constant of the wavefunction. - (5.154), that of the position operator reads {n}|Q(t)|{n}. - the average value of the position coordinate becomes {n}|Q(t)|{n. - (5.164), the average value of the potential energy is {n}|V(t)|{n. - (5.181), the matrix representation of the Hermitian operator N f defined by Eq. - Eigenvalue equation of the harmonic Hamiltonian:. - Action of the ladder operators on the harmonic Hamiltonian eigenkets:. - Time dependence of the Boson operators:. - α } are the eigenkets of the lowering a operator, obeying therefore. - (6.11), to get the norm of the coherent state { α. - which, due to the expansion properties of the exponential, yields. - β } obeying an expression of the same form as Eq. - Next, consider the corresponding average value of the squared Hamiltonian, that is, H 2 α. - so that the average value of the squared Hamiltonian given by Eq