The paper considers the sensitivity of quantum dynamics systems with respect to parameters or features in the underlying Hamiltonians. The analysis allows for the calculation of the gradient of the wave function with respect to any well defined system parameters in the Hamiltonian. Differential sensitivity equations are derived from the Schrödinger equation and their solution is expressed in terms of a Green’s function. For scattering problems the necessary asymptotic Green’s function can be exclusively expressed in terms of the available solution to the original Schrödinger equation. Also considered are problems from time-dependent quantum mechanics. In this latter situation sensitivity theory allows for the calculation of gradients of the time evolution operator with respect to system parameters. The scattering formalism is illustrated with a study of the anisotropy coefficient sensitivity in the collision of an atom and a ridgid rotor. The time-dependent theory was applied to a collision-free molecular beam translating through a region of space containing electric and magnetic fields. This case examined the sensitivity of the molecular transition probabilities to field parameters. These brief examples serve to illustrate the potential usefulness of sensitivity analysis in quantum dynamics.
The paper considers the sensitivity of quantum dynamics systems with respect to parameters or features in the underlying Hamiltonians. The analysis allows for the calculation of the gradient of the wave function with respect to any well defined system parameters in the Hamiltonian. Differential sensitivity equations are derived from the Schrödinger equation and their solution is expressed in terms of a Green’s function. For scattering problems the necessary asymptotic Green’s function can be exclusively expressed in terms of the available solution to the original Schrödinger equation. Also considered are problems from time-dependent quantum mechanics. In this latter situation sensitivity theory allows for the calculation of gradients of the time evolution operator with respect to system parameters. The scattering formalism is illustrated with a study of the anisotropy coefficient sensitivity in the collision of an atom and a ridgid rotor. The time-dependent theory was applied to a collision-free molecular beam translating through a region of space containing electric and magnetic fields. This case examined the sensitivity of the molecular transition probabilities to field parameters. These brief examples serve to illustrate the potential usefulness of sensitivity analysis in quantum dynamics.
Introduction Mathematically, given a linear differential operator L x , one encounters the solution to the inhomogeneous differential equation Here ρ ( x ) is a given source function. For a given boundary condition, we assume a solution to exist. The solution can be reduced to a simpler problem. Let G ( x, y ) is the Green's function. This is a function of x with y a parameter. Take G ( x, y ) to satisfy the same boundary conditions as φ ( x ). Then since An example of L x is, of course, the Schrödinger operator We take ρ ( x ) = V ( x ) ψ ( x , t ), ψ ( x , t ) being the wave function and V ( x ) the potential operator. We are interested in Green's functions taken over from the techniques of quantum field theory (Schweber, 1961; Lifshitz and Petaevskii, 1981). We will concern ourselves particularly with one- and two-time Green's functions, since our principal interest is to show a connection to the calculations of linear response theory (Chapter 15) as well as to quantum kinetic equations. Then we wish to compare the methods with those described in Chapter 4. In this we will follow the work of L. P. Kadanoff and G. Baym (1962) and of L. V. Keldysh (1965) and also Zubarev (1974). We will not discuss equilibrium statistical mechanics utilizing Green's function techniques for many-body problems. The literature is exhaustive (see Abrikosov et al. , 1963; Fetter and Walecka, 1971). A good general introduction is the book by G. D. Mahan (2000). One- and two-time quantum Green's functions and their properties Let us introduce the creation operator ψ † ( r, t ) and annihilation operator ψ ( r, t ) of the second quantization formalism (see Schweber, 1961). They have the equal time commutation rules for Bose and Fermi particles: The Hamiltonian operator for the particles is and the number density of particles at r t is the operator Note that r = ( r 1 … r N ) for (1, 2, 3, …, N ) and V (| r − r ′|) is the pair potential depending, as in Chapter 4, on the scalar distance between the particles.
The formalism of sensitivity analysis for stochastic differential equations is adapted and extended to the framework of stochastic mechanics. Thus a methodology is provided for analyzing the dependence of expectations of functionals of the stochastic mechanical diffusion process on parameters defining the system. This should prove useful in numerical simulations employing the latter theory. We take into account the fact that in stochastic mechanics typically a probability distribution of initial positions (rather than the initial position itself) is prescribed. The expression for a sensitivity density with respect to a drift variation depending only on time is shown to follow from the Ito-Girsanov formula, which is also employed to develop new formulas for sensitivity densities with respect to position-dependent drift variations. As an example to verify these techniques, the sensitivity coefficients of the harmonic-oscillator correlation function with respect to the angular frequency and the diffusion constant are calculated; the results coincide with those obtained by direct differentiation.
Abstract This chapter the reviews essential concepts and methods of time-dependent quantum mechanics. (1) Formal solutions of the time-dependent Schrödinger equation. (2) The two-level system. (3) Interacting systems: The time-dependent self-consistent field approximation and the Ehrenfest approximation. (4) The Born–Oppenheimer approximation and nuclear potential surfaces. (5) Time-dependent Hamiltonians: A two-level system in a time-dependent field. (6) Non-adiabatic dynamics: The surface-hopping method. (7) Expressing the time evolution in terms of the Green operator. (8) Representations (the Schrödinger and Heisenberg representations, the interaction representation). (9) Time-dependent perturbation theory. (10) Quantum dynamics of the free particles. (11) Quantum dynamics of the harmonic oscillator. (12) The raising/lowering operators formalism. (13) Tunneling.
In the previous chapter we studied the kinematics of many-body systems, and the form of operators representing the physical properties of a system, all of which were embodied by the quantum field. In this chapter we shall study the quantum dynamics of many-body systems, which can also be embodied by the quantum fields. We shall employ the fact that the quantum dynamics of a system, instead of being described in terms of the dynamics of the states or the evolution operator, i.e. as previously done through the Schrödinger equation, can instead be carried by the quantum fields. The quantum dynamics is therefore expressed in terms of the correlation functions or Green's functions of the quantum fields evaluated with respect to some state of the system. In particular we shall consider the general case of quantum dynamics for arbitrary non-equilibrium states. After introducing various types of Green's functions and relating them to measurable quantities, we will discuss the simplifications reigning for the special case of equilibrium states. Quantum dynamics Quantum dynamics can be described in different ways since quantum mechanics is a linear theory and the dynamics described by a unitary transformation of states. This will come in handy in the next chapter when we study a quantum theory in terms of its perturbative expansion using the so-called interaction picture. Here we first discuss the Schrödinger and Heisenberg pictures. The Schrödinger picture Having the Hamiltonian on the multi-particle space at hand we can consider the dynamics described in the multi-particle space.
Abstract The Green function method is a powerful technique for solving the initial value problem in quantum mechanics. Once the Green function is solved the whole wavefunction evolution is represented in a concise form and can be evaluated conveniently by numerical integration. We present a method for constructing the Green functions systematically which is different from the conventional methods of eigenfunction expansion or path integration. By using variable changing, function substitution, and Fourier transforms, the time dependent Schrödinger equations can be simplified and the solutions for the simplified equations can be easily derived. We then obtain the Green functions for the original equations by the reverse transforms. The method is demonstrated for the linear potential, the harmonic oscillator, the centrifugal potential, and the centripetal barrier oscillator, where the Green function for the centripetal barrier oscillator has not been solved previously by conventional methods. The method and examples illustrated in this paper can be utilised to strengthen undergraduate courses on quantum mechanics and/or partial differential equation.
Quantum systems interacting with a dissipative environment can effectively be described by a nonlinear modification of the Schrodinger equation. For some standard problems that possess analytical solutions, e.g. the free motion and the harmonic oscillator, also the nonlinear equation can be solved in a closed form and has Gaussian-shaped wave packet solutions. The equation determining the time-evolution of the wave-packet width is - already in the case of the usual conservative Schrodinger equation - nonlinear and provides the quantities that are necessary to construct the Green’s function for the respective system. The Green’s functions for the conservative and dissipative cases will be given explicitly in position and momentum space.