Postulates of Quantum Mechanics

 

The postulates of quantum mechanics are addressed in greater detail at the end of Chapter 3.  This link provides a quick checklist and a set of leading questions about each postulate to guide your examination of each quantum mechanical system addressed in this book, which include:

Particle in a box (Chapter 4)

Free particle (Chapter 5)

Harmonic oscillator (Chapter 6)

Rigid rotor (Chapter 7)

Hydrogen atom and other single-electron atoms (Chapter 8)

Multielectron atoms (Chapter 9)

Molecules (Chapter 10)

Systematically applying the following seven postulates to each quantum mechanical system will help you identify commonalities and differences among the systems.

 

Postulate 1. The properties (and thus the state) of a quantum mechanical system are determined by a wavefunction Ψ(r,t).

·     What are the spatial coordinates of the system under consideration?

·     Is the wavefunction time-dependent or time-independent?

·     List the specific properties of the system that can be determined:

·     How many states are there for the system?

·     What are the mathematical forms for the wavefunctions describing the three lowest energy states of the system?

·     How do wavefunctions for different states of the system differ from each other?

·     What quantum numbers are used to describe this system?

 

Postulate 2. The quantity Ψ^*(r₀,t)Ψ(r₀,t)dτ is interpreted to be the probability, at time t, of finding a system described by Ψ in a tiny volume element dτ centered at a set of coordinates r₀. Integration allows calculation of probability over a finite volume.  Ψ must be normalized for this interpretation to be valid.

·     What is the mathematical form of the normalized wavefunction for the lowest energy state of the system?

·     Sketch by hand or use Mathcad to represent the probability density functions Ψ^*(r,t)Ψ(r,t) as a function of position for the three lowest-energy states of the system.

·     Identify on each sketch the positions associated with the maximum and minimum values of probability density.

·     For each probability density sketch, shade in a region of your choice that represents approximately a 50% probability of finding the particle(s). 

 

Postulate 3. For every observable property of a system there is a quantum mechanical operator. See the Table of Quantum-Mechanical Operators.

·     What are the interesting physically observable properties of the system under consideration?

·     What operators are associated with the interesting physical observables for this system?

 

Postulate 4. Time-independent (stationary state) wavefunctions of a time-independent Hamiltonian are found by solving the time-independent Schrödinger equation.

                                                                            

·     What is the form of the Hamiltonian operator for the system?

·     Was the Schrödinger equation for the system solved exactly (analytically) or by using approximations or a numerical approach?

·     What is the ground state energy for the system?

·     Sketch or graph an accurate energy level diagram for the system.

·     Are the wavefunctions for the system time-dependent or stationary state functions?

·     What properties of the system are time-independent?

 

Postulate 5. The time evolution or time dependence of a state is found by solving the time-dependent Schrödinger equation.

                            

·     Is the Schrödinger equation for the system under consideration time-dependent or time-independent?

·     Was the time evolution of the system addressed?

·     If so, how does the system evolve with time?

·     What observable properties change with time?                                        

 

Postulate 6. If a system is described by the eigenfunction Ψ of an operator  then the value measured for the observable property corresponding to  will always be the eigenvalue a, which can be calculated from the eigenvalue equation.

                    

·     For what observable properties of the system can exact eigenvalues be calculated?

·     For what operators do the wavefunctions of the system act as eigenfunctions?

·     For what observable properties of the system will repeated measurements always yield the exact same value?

 

Postulate 7. If a system is described by a normalized wavefunction Ψ, which is not an eigenfunction of an operator Â, then a distribution of measured values will be obtained, and the average value or expectation value  of the observable property is given by the expectation value integral over all space:         

                                                                                    

·     For what observable properties of the system can expectation values be calculated?

·     What are the expectation values for the observable properties of the system in its lowest three energy states?

·     For what operators do the wavefunctions of the system NOT act as eigenfunctions?

·     For what observable properties of the system will repeated measurements yield a distribution of values?

·     What is the width of the distribution of measurements for each observable, as characterized by the standard deviation (σ) and the variance (σ²)?