CHM 312 Week 4 The 5 Postulates of QM - CHM 312 Quantum...

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CHM 312 Quantum Mechanics Week 4: The Five Postulates of QM and Some General Principles
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Newton’s Postulates (Laws) Postulates are not provable statements mathematically, but are taken to be true, and then demonstrated as true through experimental verification. As discussed earlier, Newtonian laws are postulates, provable via experiment. 1) Body at rest stays at rest unless acted upon. 2) F = ma 3) For every action there is an equal and opposite reaction. A classical particle’s state is specified completely by the position vector (defined by three coordinates as x , y , and z ), and momentum vector (defined by the three momenta p x , p y , and p z ), at that time. The position, mass, and momentum are all measurable dynamic variables or observables . The time evolution of the classical system is given by: separable into: , , Starting with the particle’s initial position () and momenta () along with the equations above provides a particle’s position in time ( x ( t ) , y ( t ) , and z ( t ) ). This is called the particle’s trajectory . The trajectory provides the complete description of a classical particle’s state. Tells us where it came from and allows prediction where it’s going.
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Special Properties of and ’s All of the properties that we’ve developed so far and are going to develop wrt to and ’s are a result of the eigenvalues having to be real numbers. Remember, all measurements/observables obtained from quantum systems are the result of the wave function that governs the particle. Just as the results that we see for classical systems are the result of the Newtonian equation which govern them. The observable that we see is only one of the eigenvalues that is accessible via the wave function governing the system. Since the result/observable we see is an eigenvalue, it must be a real number. So the and ’s involved in the equations better act in such a way as to produce eigenvalues of real numbers. i.e , it doesn’t make sense to have the result of an experiment be an imaginary number.
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Postulate 1: can describe a system in total Exact trajectories (position and momentum) of QM-particles is not possible according to the uncertainty principle. Due to issues mentioned above, new postulates were devised to describe QM-systems. Postulate 1 (has 3 parts): 1) The state of a quantum system (small system) is completely specified by a position and time dependent wave function, . is the position vector of the particle ( i.e ., the 3 coordinates, x , y , and z of the particle). 2) The t-dependent wave/state function, , contains all possible information about the system. This says in principle you can always determine what is. 3) has the property that is the probability that the particle lies in the volume element located at the position at time t.
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