The Fundamental Principles Of Quantum Mechanics

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 ($\sum \overrightarrow{F}\propto \frac{\partial \overrightarrow{p}}{\partial t}$) , 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.

Quantum World

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.

Postulate 1.

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 $\mathrm{\Psi }(\overrightarrow{r},t)$.
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 $\mathrm{\Psi }$ 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: $${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\mathrm{\Psi }}^{\ast }\mathrm{\Psi }d\tau =1\equiv <\mathrm{\Psi }|\mathrm{\Psi }>=1$$ 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.

To every observable in classical mechanics there corresponds a hermitian operator in quantum mechanics.
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 ($\overrightarrow{r}$, $\overrightarrow{p}=m\overrightarrow{v}$), and in quantum mechanics by the operators ($\hat{x},\hat{P}=-i\hslash \mathrm{\nabla }$).

Postulate 3.

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. $$\hat{A}{\mathrm{\Psi }}_n=a_n{\mathrm{\Psi }}_n$$

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.
Similarly to the state function, operators $\mathcal{O}$ can viwed mathmatically as a mapping from the states space $\mathcal{V}\mathcal{(}\mathcal{F}\mathcal{)}$ to the real numbers field $\mathcal{R}$.

Postulate 4.

If a system is in a state described by the wave function $\mathrm{\Psi }$, 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: $$<\hat{A}>=\frac{{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\mathrm{\Psi }}^{\ast }\hat{A}\mathrm{\Psi }d\tau }{{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\mathrm{\Psi }}^{\ast }\mathrm{\Psi }d\tau }$$ In the case of a normalized eigenstates, the equation reduces to : $$<\hat{A}>={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}{\mathrm{\Psi }}^{\ast }\hat{A}\mathrm{\Psi }d\tau$$

Postulate 5.

The temporal evolution of the state function is governed by time-dependent Schrödinger equation: $$\hat{H}\mathrm{\Psi }=i\hslash \frac{\partial \mathrm{\Psi }}{\partial t}$$

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 $-\frac{{\hslash }^2}{2m}{\hat{P}}^2+V(\overrightarrow{r})$ where $\hslash$ 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.

Using these five principles one can describe and make probabilistic predictions on the various quantum phenomena.
For further readings check:

• Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë (1977) Quantum Mechanics Volume I, John Wiley & Sons.
• Zetili Nouredine Zettili, (2009) Quantum Mechanics Concepts and Applications, Wiley.
• Albert Messiah, (1999) Quantum Mechanics(Two Volumes Bound as One), Dover Publication Inc.
• Belal E. Baaquie, (2013) The Theoretical Foundations of Quantum Mechanics, Springer.
• David J. Griffiths, (1995) Introduction to Quantum Mechanics, Prentice Hall.
• Leonard I. Schiff, (1955) Quantum Mechanics, McGRAW-HILL BOOK COMPANY, INC.
• Jun John Sakurai, (1994) (Modern Quantum Mechanics, Addison -Wesley Publishing Company.
• Lectures on Quantum Theory by Frederic Schuller

Vectors V:

Introduction:

The concept of vectors plays an important role in physics, computer graphics, probability theory, machine learning, and other fields of science and mathematics.
In this article I will introduce you to the basics of vector calculus, starting from defining vectors then seeing some of the basic operation that can be done to manipulate vectors, in the end you will find a bunch of refrences for further readings on vector calculus

What are Vectors?:

Vectors are mathematical objects that can be represented with numbers as entries, the number of entries required to represent a vector, depends on the dimension of a given vector, for instance the position vector of a ball moving in a 2 dimensional space can be represented by 2 numbers called the x-component, and y-component, each represent how far is the ball's projection on a given direction. The minimum dimensions required to model a problem (aka degrees of freedom) depends on the complexity of the problem, for example in the field of machine learning, data scientists use vectors with millions or even billions of enteries to train their models in order to solve a regression problem for example, in the other hand quantum theorists deal with vectors that have infinite number of components called state vectors which represent the most abstract form of vectors
Vectors can also be viewed as arrows defined by their magnitude and direction, the magnitude is simply the length of the vector, and the direction is is which direction the vector is pointing to, this formulation is quite useful in physics, it gives a visual intuition about the behavior of different vectors, yet this geometrical representation fails to represent vectors that exceed 3 components, because the human mind can only visualize up to only 3 dimensions beyond that it's abstraction

A vector can be represented in different equivalent forms, the following are two basic forms:

Algebraic form of a 2-D vector:

$$\overrightarrow{V}=\overrightarrow{v_x}+\overrightarrow{v_y}\equiv v_x\overrightarrow{e_x}+v_y\overrightarrow{e_y}$$

• $\overrightarrow{v_x}$ the vector's component along the x axis
• $\overrightarrow{v_y}$ the vector's component along the y axis
• $v_y$ the vector's projection magnitude along the x axis
• $v_y$ the vector's projection magnitude along the y axis
• $\overrightarrow{e_x}$ the unit vector along the x axis, it has a magnitude of 1 and directs toward the direction of x direction
• $\overrightarrow{e_y}$ the unit along the y axis, it has a magnitude of 1 and directs toward the direction of x direction

The right hand side notation is called unit vector notation
Matrix form:
$$\overrightarrow{V}=\left(\begin{array}{c}{v}_x\\ v_y\end{array}\right)\text{ or }\overrightarrow{V}=\left(\begin{array}{cc}{v}_x& v_y\end{array}\right)$$ Where the first is called column vector, and the second is the row vector.

Operations on Vectors:

Vectors wouldn't be so useful if we couldn't perform operations on them, here we will see five basic operations that we can use on vectors: addition, subtraction, scaling, dot product, length, and cross product (for curious readers href for refrences)

Addition and Subtraction:

As numbers, vectors too can be added, if we add two vectors $\overrightarrow{V}$ and $\overrightarrow{V}$the resulting vector $\overrightarrow{W}$ will have components, will correspond the sum of the sum of two vecotrs components respectively. $$\overrightarrow{V}=\overrightarrow{v_x}+\overrightarrow{v_y},\overrightarrow{U}=\overrightarrow{u_x}+\overrightarrow{u_y}$$ $$\overrightarrow{W}=\overrightarrow{V}+\overrightarrow{U},\overrightarrow{W}=(v_x+u_x)\overrightarrow{e_x}+(v_y+u_y)\overrightarrow{e_y}\overrightarrow{w_x}=\overrightarrow{v_x}+\overrightarrow{u_x},\overrightarrow{w_y}=\overrightarrow{v_y}+\overrightarrow{u_y}$$

In the same spirit, vectors can be subtracted as follow: $$\overrightarrow{W}=\overrightarrow{V}-\overrightarrow{U}=(v_x-u_x)\overrightarrow{e_x}+(v_y-u_y)\overrightarrow{e_y}$$ note here that the negative of a vector is another vector with the same magnitude, pointing to the opossite direction

Here is an example where the addition of two vectors can come handy, imagine with me two identical balls(billiard balls for instance) with different velocities elastically colliding (elastically means that the mass doesn't change after the collision) what would the velocity of each ball after the collision, the law of conservation of linear momentum(a vector quantity $\overrightarrow{p}=m\overrightarrow{v},\overrightarrow{v}$ is the velocity, m is the mass) states that the sum of intial linear momentums equal the sum of final linear momentums $\overrightarrow{p_1}+\overrightarrow{p_2}={\overrightarrow{p_1}}^{\prime }+{\overrightarrow{p_2}}^{\prime }$, $\overrightarrow{p_1},\overrightarrow{p_1}$ are the linear momentum of the two balls before the collision, and ${\overrightarrow{p_1}}^{\prime },{\overrightarrow{p_1}}^{\prime }$ are the linear momentum of the two balls after the collision.

Scaling:

By scaling either we enlarge or reduce a vector, and we do that by multiplying it by a scalar, a scalar means a number that have a magnitude but no direction, for example if we want to make a vector $\overrightarrow{V}$ larger by a factor $\alpha >1$ the resulting vector $\overrightarrow{W}$ will be : $\overrightarrow{W}=\alpha \overrightarrow{V}=\alpha v_x\overrightarrow{e_x}+\alpha v_y\overrightarrow{e_y}$ which is larger $\alpha$ times in both directions, similarly if we want to make a vector smaller by a factor $\beta$ we do the same thing, such that $0<\beta <1$.

Dot Product:

The dot product can be viewed as the operation that gives us the value of the magnitude of the vector $\overrightarrow{W}$ that is the projection of the vector $\overrightarrow{V}$ on the vector $\overrightarrow{U}$ and it is given by: $$W=\overrightarrow{V}.\overrightarrow{U}=\overrightarrow{v_x}\overrightarrow{u_x}+\overrightarrow{v_y}\overrightarrow{u_y}=v_x{u}_x+v_y{u}_y$$ Or $$\overrightarrow{V}.\overrightarrow{U}=VU\cos (\theta )$$ Where W, V and U the magintudes of the three vectors respectively, and $\theta$ is the angle between the two vectors under dot product

The elementary work done by a force is defined the be dot prodcut of the force vector and the infinitesimal change in position vector: $$dW(F)=\overrightarrow{F}.d\overrightarrow{r}$$ Another example of where dot product is used comes from machine learning's neural networks where the net input z is defined by: $$z=\overrightarrow{w}.\overrightarrow{x}$$ Where $\overrightarrow{w}$ is the weight vector, and $\overrightarrow{x}$ is the input vector.

Length:

The legnth of a vector $\overrightarrow{V}$ represents its magnitude or equivalently how far the head of the arrow is from the origin of the coordinates system, and it is given by: $$V=\sqrt{v_x^2+v_y^2}$$ we can prove this result by taking in account that the angle a between a vector and it self is 0 $$\overrightarrow{V}.\overrightarrow{V}=VVcos(0)=V^2,cos(0)=1$$ We the other definition of dot product we have: $$\overrightarrow{V}.\overrightarrow{V}=\overrightarrow{v_x}.\overrightarrow{v_x}+\overrightarrow{v_y}.\overrightarrow{v_y}={\overrightarrow{v_x}}^2+{\overrightarrow{v_y}}^2$$ Combining the two results we get the previous result: $$V=\sqrt{v_x^2+v_y^2}$$

Cross Product:

The corss product of two vectors $\overrightarrow{V}$ $\overrightarrow{U}$ gives a resulting vector $\overrightarrow{W}$ perpendicular(90 degrees angle) to the two vectors and it is given by: $$\overrightarrow{W}=\overrightarrow{V}\times \overrightarrow{U}=VU\sin (\theta )\overrightarrow{e}$$ Notice that in the dot product the result was scalar whilist in corss product the result is vector and that why we multiplied the result by the unit vector $\overrightarrow{e}$ which is perpendicular to both the vectors $\overrightarrow{V}$, $\overrightarrow{U}$, there is another formula to calculate the cross product which I didn't treat here. A nice illustration of the cross product

Conclusion:

In this article we have defined vectors and showed some of basic operations that we can preform on them.
For further readings:

• Linear Algebra Done Right by Axler, Sheldon Jay
• No Bullshit Guide to Linear Algebra by Ivan Savov
• Problems And Worked Solutions In Vector Analysis by Lewis Richard Shorter
• Vector Calculus by Michael Corral
• Essence of linear algebra by 3Blue1Brown (Youtube Channel)