6-1 Introduction
Chapter 6
Postulates and Theorems
of Quantum Mechanics6-1 Introduction
The first part of this book has treated a number of systems from a fairly physical viewpoint, using intuition as much as possible. Now, armed with the concepts already developed, the reader should be in a better position to understand the more formal foundation to be described in this chapter. This foundation is presented as a set of postulates.
From these follow proofs of various theorems. The ultimate test of the validity of the postulates comes in comparing the theoretical predictions with experimental data. The extra effort required to master the postulates and theorems is repaid many times over when we seek to solve problems of chemical interest.
6-2 The Wavefunction Postulate
We have already described most of the requirements that a wavefunction must satisfy: ψ must be acceptable (i.e., single-valued, nowhere infinite, continuous, with a piecewise continuous first derivative). For bound states (i.e., states in which the particles lack the energy to achieve infinite separation classically) we require that ψ be square integrable.
So far we have considered only cases where the state of the system does not vary with time. For much of quantum chemistry, these are the cases of interest, but, in general, a state may change with time, and ψ will be a function of t in order to follow the evolution of the system.
Gathering all this together, we arrive at Postulate I Any bound state of a dynamical system of n particles is described ascompletely as possible by an acceptable, square-integrable function
(q1, q2, . . . q3n, ω1, ω2, . . . , ωn, t), where the q’s are spatial coordinates, ω’s are spin coordinates, and t is the time coordinate. ∗ dτ is
the probability that the space-spin coordinates lie in the volume element
dτ (≡ dτ1dτ2 · · · dτn) at time t, if is normalized.
For example, suppose we have a two-electron system in a time-dependent state described by the wavefunction (x1, y1, z1, ω1, x2, y2, z2, ω2, t). The spin coordinates ω would each be some combination of spin functions α and β. If we integrate ∗ over the spin coordinates of both electrons, we are left with a spin-free density function.
Call it ρ(x1, y1, z1, x2, y2, z2, t) ≡ ρ(v1, v2, t). We interpret ρ(v1, v2, t) dv1 dv2 as
166
Postulates and Theorems
of Quantum Mechanics6-1 Introduction
The first part of this book has treated a number of systems from a fairly physical viewpoint, using intuition as much as possible. Now, armed with the concepts already developed, the reader should be in a better position to understand the more formal foundation to be described in this chapter. This foundation is presented as a set of postulates.
From these follow proofs of various theorems. The ultimate test of the validity of the postulates comes in comparing the theoretical predictions with experimental data. The extra effort required to master the postulates and theorems is repaid many times over when we seek to solve problems of chemical interest.
6-2 The Wavefunction Postulate
We have already described most of the requirements that a wavefunction must satisfy: ψ must be acceptable (i.e., single-valued, nowhere infinite, continuous, with a piecewise continuous first derivative). For bound states (i.e., states in which the particles lack the energy to achieve infinite separation classically) we require that ψ be square integrable.
So far we have considered only cases where the state of the system does not vary with time. For much of quantum chemistry, these are the cases of interest, but, in general, a state may change with time, and ψ will be a function of t in order to follow the evolution of the system.
Gathering all this together, we arrive at Postulate I Any bound state of a dynamical system of n particles is described ascompletely as possible by an acceptable, square-integrable function
(q1, q2, . . . q3n, ω1, ω2, . . . , ωn, t), where the q’s are spatial coordinates, ω’s are spin coordinates, and t is the time coordinate. ∗ dτ is
the probability that the space-spin coordinates lie in the volume element
dτ (≡ dτ1dτ2 · · · dτn) at time t, if is normalized.
For example, suppose we have a two-electron system in a time-dependent state described by the wavefunction (x1, y1, z1, ω1, x2, y2, z2, ω2, t). The spin coordinates ω would each be some combination of spin functions α and β. If we integrate ∗ over the spin coordinates of both electrons, we are left with a spin-free density function.
Call it ρ(x1, y1, z1, x2, y2, z2, t) ≡ ρ(v1, v2, t). We interpret ρ(v1, v2, t) dv1 dv2 as
166