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CRC Press eBooks

The Equations of Quantum Theory

Jonathan Allday

Published Sep 21, 2022• 0 citations
Publisher

Abstract

In classical physics, if you want to calculate what’s going to happen in a given situation, you basically have two options open to you. You can carefully specify the positions and momenta of all the particles involved, list the forces acting, and then plug the lot into Newton’s laws of motion and calculate away. Alternatively, you can inspect all the kinetic energies, potential energies, and any other form of energy involved and work through a calculation based on energy conservation. As it happens, the second approach, using energy, is the more amenable to a straightforward quantum development. The first Ehrenfest relationship should be compared with the classical formulation px=mdxdt. Moving on from Ehrenfest’s theorem, let us revisit Heisenberg’s equation of motion and explore one of its significant consequences. The left-hand side of this relationship is the time rate of change of the expectation value of the operator. Let’s think carefully about what that means.

DOI
10.1201/9781003225997-16

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In classical physics, if you want to calculate what’s going to happen in a given situation, you basically have two options open to you. You can carefully specify the positions and momenta of all the particles involved, list the forces acting, and then plug the lot into Newton’s laws of motion and calculate away. Alternatively, you can inspect all the kinetic energies, potential energies, and any other form of energy involved and work through a calculation based on energy conservation. As it happens, the second approach, using energy, is the more amenable to a straightforward quantum development. The first Ehrenfest relationship should be compared with the classical formulation px=mdxdt. Moving on from Ehrenfest’s theorem, let us revisit Heisenberg’s equation of motion and explore one of its significant consequences. The left-hand side of this relationship is the time rate of change of the expectation value of the operator. Let’s think carefully about what that means.
Schrödinger’s approach to the equations of quantum theory is not unique. Heisenberg worked out an alternative formulation. We already know how to calculate expectation values in principle, even if we don’t have the mathematical skill to do it in practice. If a system is changing and developing over time, then it makes sense to suppose that the expectation values of its physical variables will also change over time. en, the expression ∇t〈O〉 will tell us the rate at which this change is happening (look back at the de nition of ∇t in Chapter 12).
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dPdtF=\frac{dP}{dt}F=dtdP​, P=mdVdtP=m\frac{dV}{dt}P=mdtdV​, and [X,P]=i\left[X,P\right]=i[X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.
In a physical theory we typically describe a system in terms of its state . We specify the relevant physical quantities or variables and then use dynamical laws to find the time evolution of these variables to predict their values in the future. For example, in classical particle mechanics, the state of a system is specified by the positions and velocities (or, equivalently for us, the momenta) of all of the particles in the system (that is, r ( t ) and v ( t ) for each particle). Newton's second law, F = ma , then determines the time evolution of the variables, or of the state, of the system. For electrodynamics, the state variables are the electric and magnetic fields, E and B , and Maxwell's equations govern the time evolution of these. In classical physics (including relativity here), the state variables that are the central entities of the dynamical equations (typically the positions and momenta of the particles) are also the directly observable physical quantities. That is, the state itself is specified in terms of the observables of the theory. The essential features of the classical world view (as modified by relativity) are that the present state of a system in principle determines its future state (as in the flight of a baseball under the influence of gravity) and that the agency responsible for this event-by-event causal structure must propagate from the cause to the effect at some rate no greater than the speed of light.
A Brief Survey of Analytical Dynamics, Discovery of Matrix Mechanics, Mathematical Preliminaries Postulates of Quantum Theory Equations of Motion, Hamiltonian Operator and the Commutation Relations Symmetries and Conservation Laws, Classical and Quantum Integrability Classical and Quantum Mechanical Degeneracies Bound State Energies for One-Dimensional Problems Exactly Solvable Potentials, Supersymmetry and Shape Invariance Supersymmetry and Superpotential Shape Invariant Potentials Two-Body Problem Determination of the Angular Momentum Eigenvalues Methods of Integration of the Heisenberg Equations of Motion Discrete-Time Formulation of the Heisenberg's Equations of Motion Quantum Tunneling Using Discrete-Time Formulation Integration of the Operator Differential Equations Perturbation Theory Other Methods of Approximation Heisenberg Correspondence Principle, Quantization of the Equations of Motion with Higher Derivatives Determinantal Method in Potential Scattering Time-Dependent Scattering Theory The Scattering Matrix Impact Parameter Representation of the Scattering Amplitude Transition Probability Quantum Diffraction, Motion of a Charged Particle in Electromagnetic Field The Aharanov-Bohm Effect Neutron Interferometry Quantum Many-Body Problem Hartree and Hartree-Fock Approximations Second Quantization Bogoliubov Transformation for a Many-Boson System Bogoliubov Transformation for Fermions Interacting Through Pairing Forces Quantum Theory of Free Electromagnetic Field Casimir Force Interaction of Radiation with Matter Heisenberg Equations for Interaction of an Atom with Radiation Bell's Inequalities
Addendum: Dec. 15, 2020. The title of the first reference is: Phase Shift of exp[i delta(x)] for Bound Quantum Systems In classical mechanics, interactions follow Newton’s second law which is equivalent to a conservation of energy equation with a potential V(x) for conservative forces. The Schrodinger equation, we argue, allows for two kinds of interactions. The first is like the classical situation and involves average kinetic energy as part of a classical conservation equation with V(x). This is also associated with a (p rms) i.e. root mean square momentum, which is identical to classical momentum. There exists, however, also = Sum over p p P(p/x) with P(p/x)=a(p)exp(ipx)/W(x) ( W(x) is the wavefunction). This momentum is responsible for dW/dx and in turn the change in spatial density W(x)W(x) i.e. quantum humps. , like prms follows from interactions of the particle with V(x) in a bound system and possibly other reasons, but does not follow Newton’s law. In this note, we consider the more general case of which appears in the Aharonov-Bohm problem. It seems one may form a generalized = A(x) + i.e. calculate various pieces of which govern the spatial density, i.e. interference, without necessarily contributing to KEave(x) in KEave(x)+V(x)=E ((1)). In particular, A(x), the magnetic vector potential behaves like a portion of and is involved in interference, but plays no role in ((1)). We suggest two different kinds of interactions may occur in a quantum system. The first is the usual Newton type interaction applied to averages, but a second involves interactions that create which does not behave according to Newtonian mechanics. For a bound state problem with V(x), both types of interactions exist, thus one has two “momenta”, p rms and , with only the latter obeying Newtonian mechanics. For the time-independent Schrodinger equation, Newtonian conservation of energy applies only to average values such as KEave(x) and V(x) which is itself an average, we argue. Given that there exists microscopic motion and a distribution a(p)exp(ipx), these may give rise to non-Newtonian behaviour and interactions other than the usual -dV/dx in ((1)).