Scientific Posts: Quantum Mechanics- Do we really exist?
We know that all molecules are made of atoms which, in turn, contain nuclei and electrons. The equations that govern the motions of electrons and nuclei are not the familiar Newton equation,
F = ma
But a new set of equations called Schrodinger equations. when scientists first studied the behaviour of electrons and nuclei. They tried to interpret their experimental findings in terms of classical Newtonian motions. But such attempts eventually failed. They found that such small particles behaved in a way that simply is not consistent with the Newton equations. They illustrated some of the experimental data that gave rise to these paradoxes and show you how the scientists of those early times then used these data to suggest new equations that these particles might obey. I want to stress that the Schrcidinger equation was not derived but postulated by these scientists. In fact, to date, to one has been able to derive the Schrcidinger equation.
In conclusion, classical mechanics failed to explain how particles at atomic scale move, which opened a new whole field for physicists to pursue. In classical mechanics, objects exist in a specific place at a specific time. However, in quantum mechanics, objects instead exist in a haze of probability; they have a certain chance of being at point A, another chance of being at point B and so on. Which seriously arises the question, as if we made of atoms and subatomic particles, do we really exist?
Quantum mechanics developed over many decades, beginning as a set of controversial mathematical explanations of experiments that the math of classical mechanics could not explain, as we discussed above. It began at the turn of the 20th century, around the same time that Albert Einstein published his theory of relativity, a separate mathematical revolution in physics that describes the motion of things at high speeds. Unlike relativity, however, the origins of QM cannot be attributed to any one scientist. Rather, multiple scientists contributed to a foundation of three revolutionary principles that gradually gained acceptance and experimental verification between 1900 and 1930. They are:
Quantized properties: Certain properties, such as position, speed and colour, can sometimes only occur in specific, set amounts, much like a dial that "clicks" from number to number. This challenged a fundamental assumption of classical mechanics, which said that such properties should exist on a smooth, continuous spectrum. To describe the idea that some properties "clicked" like a dial with specific settings, scientists coined the word "quantized."
Particles of light: Light can sometimes behave as a particle. This was initially met with harsh criticism, as it ran contrary to 200 years of experiments showing that light behaved as a wave; much like ripples on the surface of a calm lake. Light behaves similarly in that it bounces off walls and bends around corners, and that the crests and troughs of the wave can add up or cancel out. Added wave crests result in brighter light, while waves that cancel out produce darkness. A light source can be thought of as a ball on a stick being rhythmically dipped in the centre of a lake. The colour emitted corresponds to the distance between the crests, which is determined by the speed of the ball's rhythm.
Waves of the matter: Matter can also behave as a wave. This ran counter to the roughly 30 years of experiments showing that matter (such as electrons) exists as particles.
In quantum mechanical contexts, the term ‘observable’ is used interchangeably with ‘physical quantity’ and should be treated as a technical term with the same meaning. It is no accident that the early developers of the theory chose the term, but the choice was made for reasons that are not, nowadays, generally accepted. The state-space of a system is the space formed by the set of its possible states, i.e., the physically possible ways of combining the values of quantities that characterize it internally. In classical theories, a set of quantities which forms a supervenience basis for the rest is typically designated as ‘basic’ or ‘fundamental’, and, since any mathematically possible way of combining their values is a physical possibility, the state-space can be obtained by simply taking these as coordinates. So, for instance, the state-space of a classical mechanical system composed of \(n\) particles, obtained by specifying the values of \(6n\) real-valued quantities — three components of position, and three of momentum for each particle in the system — is a \(6n\)-dimensional coordinate space. Each possible state of such a system corresponds to a point in the space, and each point in the space corresponds to a possible state of such a system. The situation is a little different in quantum mechanics, where there are mathematically describable ways of combining the values of the quantities that don’t represent physically possible states. As we will see, the state-spaces of quantum mechanics are special kinds of vector spaces, known as Hilbert spaces, and they have more internal structure than their classical counterparts.
A structure is a set of elements on which certain operations and relations are defined, a mathematical structure is just a structure in which the elements are mathematical objects (numbers, sets, vectors) and the operations mathematical ones, and a model is a mathematical structure used to represent some physically significant structure in the world.
The heart and soul of quantum mechanics are contained in the Hilbert spaces that represent the state-spaces of quantum mechanical systems. The internal relations among states and quantities, and everything this entails about the ways quantum mechanical systems behave, are all woven into the structure of these spaces, embodied in the relations among the mathematical objects which represent them. This means that understanding what a system is like according to quantum mechanics is inseparable from familiarity with the internal structure of those spaces. Know your way around Hilbert space, and become familiar with the dynamical laws that describe the paths that vectors travel through it, and you know everything there is to know, in the terms provided by the theory, about the systems that it describes.
Question To be continued...