Summary Professor E. T. Whittaker has recently discovered a Third Quantum-Mechanical Principal Function R(q, Q, t - T) and has worked out the theory of this function in detail when the Hamiltonian is By using the Sturm-Liouville theory of linear differential equations and the properties of Green's function, it is shown that the function is an elementary solution of the adjoint of the Schrodinger wave equation associated with the Hamiltonian H. It is pointed out that the modified Planck constant ħ arises solely from the commutation relation and may, from the analytical view-point, be any constant, real or complex. In particular, if ħ = i, the use of an algebra with the commutation relation leads to an elementary solution of the real equation of parabolic type
Summary Professor E. T. Whittaker has recently discovered a Third Quantum-Mechanical Principal Function R(q, Q, t - T) and has worked out the theory of this function in detail when the Hamiltonian is By using the Sturm-Liouville theory of linear differential equations and the properties of Green's function, it is shown that the function is an elementary solution of the adjoint of the Schrodinger wave equation associated with the Hamiltonian H. It is pointed out that the modified Planck constant ħ arises solely from the commutation relation and may, from the analytical view-point, be any constant, real or complex. In particular, if ħ = i, the use of an algebra with the commutation relation leads to an elementary solution of the real equation of parabolic type
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.
This chapter begins by exploiting the Einstein–de Broglie relation and the notion of symbol, as defined in Eq. (3.6.6). The following steps are the local and global conservation laws associated to the Schrödinger equation, the probabilistic interpretation of the wave function, the spreading of wave packets and transformation properties of wave functions. The solution of the Schrödinger equation for a given initial condition is then studied. For this purpose, one first expands the initial condition in terms of eigenfunctions of the Hamiltonian operator. This is then ‘propagated’ to determine the evolution of the wave function. Thus, one is led to consider the Green kernel of the Schrödinger equation, and the boundary conditions on the wave function that lead to complete knowledge of stationary states. A dynamics involving an isometric non-unitary operator is also studied. The chapter ends with an elementary introduction to harmonic oscillators and to the JWKB method in wave mechanics, plus examples on the Bohr–Sommerfeld quantization. From classical to wave mechanics The correspondence between differential operators and polynomials in k relies on the notion of the symbol defined in Eq. (3.6.6). Now we are aiming to pass from a function on phase space, i.e. the classical Hamiltonian, to an operator which, by analogy, is called the Hamiltonian operator. Such a transition is non-trivial because the map from symbols to operators contains some ambiguity. Such a transition is always possible if polynomials in p have constant coefficients. When this is not the case, i.e. if the coefficients are functions on the configuration space, there is an obvious ambiguity in the association of an operator with xp or px , for instance.
The Schrödinger equation is a linear partial differential equation that describes the temporal evolution of the state of a quantum mechanical system. One of the pillars of quantum mechanics, the Schrödinger equation has been used to describe a wide variety of phenomena such as wave-particle duality, atomic orbitals, quantum oscillations, and even some fluid mechanics problems. Despite its established importance in the development of quantum mechanics, a solution of the Schrödinger equation for an arbitrary distribution of quantum mechanical potential or for an arbitrary number of space dimensions remains unknown. This work seeks to rectify this theoretical roadblock by providing an analytical solution through a recasting of the well-known Cole-Hopf transformation which is used to linearize a broad class of PDEs of which the Schrödinger equation becomes a member after an earlier change of variables. These two successive transformations render the general wave function proportional to the solution of a forced heat equation, allowing an explicit analytical solution applicable to any system in any number of space dimensions. Two practical applications to deriving the explicit wave functions of multielectronic atoms and to the forced Burgers equation of fluid mechanics are discussed, enabling greater mathematical insight into quantum entanglement and nonlinear fluid systems.
We study a recently introduced simple method [S. Jin, N. Liu, and Y. Yu, Quantum simulation of partial differential equations via Schr\"odingerisation, arXiv:2212.13969] for solving general linear partial differential equations with quantum simulation. This method converts linear partial differential equations into a Hamiltonian system, using a simple transformation called the warped phase transformation. Here we provide a more-in-depth technical discussion and expand on this approach in a more detailed and pedagogical way. We apply this to examples of partial differential equations, including heat, convection, Fokker-Planck, linear Boltzmann, and Black-Scholes equations. This approach can also be extended to general linear partial differential equations, including the Vlasov-Fokker-Planck equation and the Liouville representation equation for nonlinear ordinary differential equations. Extension to higher-order time derivatives is also possible.
The form of the complex Hamiltonian evolution equations (linear and nonlinear) is deduced here from that of the classical Hamiltonian systems. Their properties and those of the associated continuous Poisson bracket are studied. It is shown that if any space-localized solution of the above type equations is expanded in any Hilbert space basis the expansion coefficients are canonically conjugate variables and obey Hamiltonian systems of complex equations. The last is proved also for sets of “canonically conjugate” functionals of general type. As an application, commutation relations which are equivalent in form to Dirac’s quantization rules (the constant in place of ℏ is undetermined) are obtained. The question of how a nonlinear field theory may contain quantum mechanics as a special case is discussed in view of this paper’s results.
The solution of a problem in classical dynamics can be described, as Hamilton showed in 1834, by its Principal Function. Considering for simplicity a conservative problem with one degree of freedom, let the co-ordinate at the instant t be q, and let the Lagrangean function be L. Let Q be the value of the co-ordinate at a previous instant T. Let the quantity , after the integration has been performed, be expressed in terms of (q, Q, t — T), and let the function thus obtained, which is Hamilton's Principal Function, be denoted by W.