The power of physics theories lays in the fact that they give an adequate description of physical phenomena, and also can make precise predictions about their future, for a long time classical mechanics in its different forms has done a well in both tasks, but when it came to microscopic (angstrom \(10^{-10}m\)) events classical mechanics seemed to be broken, it was not able to describe what was going on on a subatomic level, it was high time for a newer more general theory to emerge, in this article I will present to you the pillars of this newer theory, namely quantum mechanics.
Classical World
We will start from the classical world, assume that I leave my phone on the table home then leave the house alone, we would both agree that the phone will stay unmoved where I left it whther someone was there to observe it or not, we then say that my phone has an intrinsic reality, this is a propriety that defines classical objects (micron \(10^{-4}m\)), their existence does not depend on any external factors, and because my phone stays unmoved we say that its qualities are determined, it wouldn't make sense if I say it is where I left it and also outside the house, would it?
As I have mentioned in the first paragraph a useful physical theory is one which describes well and also gives precise predictions about the future states of phenomena, in classical physics classical entity can be perfectly described using its degrees of freedom (dynamical variables) at each moment of time, and degrees of freedom is just a fancy word to say coordinates, and its evolution in time is governed by Newton's second law of motion: The sum of all forces acting on an object is proportional to the temporal derivative of its momentum (\(\sum \vec{F} \propto \frac{\partial \vec{p}}{\partial t} \ \)) , the first part is what is called the kinematics which is describing the motion using equations, the second part is dynamics, and it gives information about evolution of qualities(position, velocity, energy, ...) of the studied object in time, combining both gives us what we call Classical Mechanics.
Unlike classical objects, quantum entities don't possess an intrinsic reality when they are not being observed, they do acquire an objective reality only when a measurement has taken place, let us go back to my phone example, if we consider my phone to have the dimensions of few angstroms, then when I am not looking at my phone, I can't tell if it is there or not, and not just that, the qualities of phone (position, velocity, energy, ...) are indeterminate when I am looking away, and only when I am looking at that it manifests an objective exitence and a determinate properties, weird huh?
The tools used to describe the classical phenomena won't do any good in describing or predicting the outcome of quantum systems because of the indeterministic nature of quantum objects, then a new set of tools is required, quantum mechanics provides us with such six postulates that serves that.
All information of a quantum system is encoded in the complex-valued wave function that is also called the state function, that depends on time and position, and it is usually denoted with the greek letter Psi \(\Psi(\vec{r}, t)\).
To every observable in classical mechanics there corresponds a hermitian operator in quantum mechanics.
The results of measuring an observable(quality) associated with an operator \(\hat{A}\), the only possible values are the eigenvalues of the operator \(\hat{A}\), the act of measurement is represented by the following eigenvalue equation.
\[\hat{A}\Psi_n =a_n\Psi_n\]
Operators serve a mathmatical tool to represent the act of measuring a propriety of a quantum entity, as we have mentioned ealier, before preforming a measurement on a quantum system, its state is not determined, but as soon as we observe it, it becomes determined, this transition is also refered to as the collapse of the wave function, and it can be modeled by an eigenvalue equation using operators, and the eigenvalues of a certain operator represent all the possible values that can be measured. hermitian operators have an interesting proprietary, that is that their eigenvalues are all real, and we know thatthe result of measurements are always real numbers , hence the Postulate 2.
If a system is in a state described by the wave function \(\Psi\), then the average (expected) value of performing several measurements on of a quality associated with an operator \(\hat{A}\) on identical systems is given by:
\[
<\hat{A}>=\frac{\int_{-\infty}^{+\infty}\Psi^*\hat{A}\Psi d\tau}{\int_{-\infty}^{+\infty}\Psi^*\Psi d\tau}
\]
In the case of a normalized eigenstates, the equation reduces to :
\[
<\hat{A}>=\int_{-\infty}^{+\infty}\Psi^*\hat{A}\Psi d\tau
\]
The temporal evolution of the state function is governed by time-dependent Schrödinger equation:
\[
\hat{H}\Psi = i\hbar \frac{\partial \Psi }{\partial t}
\]
Where \(\hat{H}\) is the Hamiltonian operator obtained by the correspondence principle, as in classical mechanics it represents the total energy of the system and it is equatl to \(-\frac{\hbar^2}{2m}\hat{P}^2+V(\vec{r})\) where \(\hbar\) is the reduced Planck's constant, Schrödinger's equation can be viewed as the equivalent of Newton's second law of motion in the case of a quantum systems.
Quantum World
Postulate 1.
The state function can be understood mathmatically to be the mapping from the degree of freedom space \(\mathcal{F}\) that has all the possible values of qualities of a quantum entity, into the the complex plane \(\mathcal{C}\), bare in mind that one cannot access the degrees of freedom space directly, and hence the use of the wave function
because of the probabilistic interpretation of quantum mechanics, the wave function \(\Psi\) only gives us a probability of getting a certain value when preforming an expriement to measure a certain quality of a quantum entity, unlike classical mechanics we can't know for sure what the result of an expriment will be, which brings up an important property of the state function in the case of a single quantum entity (particle), the state function is normalized, which can be expressed in mathmatical terms as:
\[
\int_{-\infty}^{+\infty}\Psi^*\Psi d\tau=1 \equiv <\Psi|\Psi>=1
\]
This equation means that If I put a quantum object in a box then close it, there is a 100% chance that I will find it in when I open it.
Postulate 2.
Which means that for any given measurable quality in classical mechanics there corresponds a measurable quality in quantum mechanics that belongs to the quantum entity, an operator is just a different mathematical representation of this quality, for in classical mechanics position and momentum are represented respectively by the following vectors (\(\vec{r}\), \(\vec{p}=m\vec{v}\)), and in quantum mechanics by the operators (\(\hat{x}, \hat{P}=-i\hbar\nabla\)).
Postulate 3.
Similarly to the state function, operators \(\mathcal{O}\) can viwed mathmatically as a mapping from the states space \(\mathcal{V(F)}\) to the real numbers field \(\mathcal{R}\).Postulate 4.
Postulate 5.
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