Introduction to the Postulates of Quantum Mechanics:
We have assessed the mathematics (that is, complex linear algebra) essential to understand the quantum mechanics. We will now observe how the physics of quantum mechanics fits into this mathematical frame-work.
A postulate is a somewhat which is supposed to be self-evident, requiring no proof, employed as a basis for the reasoning. The postulates of Quantum Mechanics are the minimum conditions which should be satisfied for Quantum Mechanics to hold. Whenever Quantum mechanics works based on such postulates, it signifies that the postulates are true.
I) Physical meaning of the Wave-function:
Postulate 1: The wave-function tries to explain a quantum mechanical entity (that is, photon, electron, x-ray and so on) via its spatial location and time dependence, that is, the wave-function is in the most general sense based on time and space:
Ψ = Ψ(x,t)
The state of a quantum mechanical system is fully specified through the wave-function Ψ(x,t).
The Probability that a particle will be found out at time to in a spatial interval of width dx centered around xo is found out by the wave-function as:
P (xo,to) dx = Ψ*( xo,to)Ψ(xo,to)dx = |Ψ(xo,to)|2dx
Note: Dissimilar for a classical wave, by well-defined amplitude, the Ψ(x,t) amplitude is not ascribed a meaning.
In order for Ψ(x,t) to symbolize a viable physical state, some of the conditions are needed:
a) The wave-function should be a single-valued function of the spatial coordinates. (That is, single probability for being in the given spatial interval)
b) The first derivative of the wave-function should be continuous in such a way that the second derivative exists in order to satisfy the Schrodinger equation.
c) The wave-function can't encompass infinite amplitude over a finite interval. This would prevent normalization over the interval.
II) Experimental Observables Correspond to Quantum Mechanical Operators
Postulate 2: For each and every measurable property of the system in classical mechanics like position, momentum and energy, there exists a corresponding operator in the quantum mechanics. The experiment in the lab to compute a value for such an observable is simulated in theory via operating the wave-function of the system by the corresponding operator.
Note: Quantum mechanical operators are classified as the Hermitian operators as they are analogs of Hermitian matrices which are stated as having only real Eigen-values. As well, the Eigen functions of the Hermitian operators are orthogonal.
Table: (Engel and Reid): list of classical observables and quantum mechanics operator.
Note: The operators act on a wave-function from the left, and the order of operations is significant (that is, much as in the case of multiplying by matrices-commutativity is significant).
III. Individual Measurements:
Postulate 3: For a single measurement of the observable corresponding to a quantum mechanical operator, only values which Eigen-values are of the operator will be measured.
Whenever measuring energy: one gets Eigen-values of the time-independent Schrodinger equation:
H ˆ Ψn(x,t) = En Ψn(x,t)
Note: The net wave-function stating a given state of a particle requires not is an Eigen-function of the operator (however one can expand the wave-function in terms of the Eigen-functions of the operator as a complete basis).
IV) Expectation Values and Collapse of the Wave-function:
Postulate 4: The average or expectation, value of the observable corresponding to a quantum mechanical operator is represented by:
This is the most common form for the expectation value expression. If the wave-function is normalized, then the denominator is identically 1 (this is supposed to be the case since each and every valid wave-function should be normalized).
V) Time Evolution:
Postulate 5: The time-dependent Schrodinger equation regulates the time evolution of a quantum mechanical system:
Note: The Hamiltonian operator Hˆ comprises the kinetic and potential operators. This equation reflects the deterministic (Newtonian) nature of the particles or waves. It comes out to be in contrast to Postulate 4 (that is, most of the observations lead to the different measured observables, each weighted differently, that is, a probabilistic view of the particle or wave). The reconciliation is in the fact that Postulate 4 relates to the outcomes of measurements at a particular instant in time. Postulate 5 let us to propagate the wave-function in time (that is, we propagate a probabilistic entity). Then, at certain future time, if we make the other measurement, we are again faced by the implications of Postulate 4.
The Correspondence Principle:
The correspondence principle defines that as the quantum number 'n' becomes big, quantum mechanics must approximate the classical mechanics. For a new theory to be acceptable, it should conform to the well-tested existing theories. In this vein, the special theory of relativity, that is significant only if the velocities comprised are large, should conform to Newtonian mechanics if the velocities comprised are small day to day values. As an illustration of this law, we consider the Hydrogen atom as treated by the Bohr. The frequency of the radiation emitted or absorbed in the transition between states n and n' is,
Enn' = RE [(1/n'2) - (1/n2)]
Here 'RE' is the Rydberg energy.
Now, classical mechanics forecasts a continuous spectrum, while Bohr's theory gives mount to discrete lines. Let us take n' as being equivalent to n + 1, then
Enn' = RE [1/(n+1)2 - 1/n2] = RE [(n+1)-2 - n-2]
On applying the Binomial expansion:
RE [(n+1)-2 - n-2] = RE [n-2 - 2n-3 + (2)(3)n-4 - .... -n-2]
Enn' ≈ - 2/n3 RE
This tends to zero as 'n' tends to infinity. We conclude thus, that as 'n' tends to infinity; the spectrum becomes continuous, as predicted through the classical mechanics.
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