Blog 2: Some basic Quantum Mechanics by Professor Mandaar Pande

This is the second in a series of blogs that I will be writing for everyone to get a basic understanding of this immensely important research field which is poised to become mainstream in a few years and significantly impact our daily lives.

Before I venture into explaining the concepts required to understand quantum computing, let me first talk a bit about classical and quantum mechanics. An understanding of quantum mechanics is important because every concept of quantum computing has its roots in this fundamental field of physics.

 

I’m sure all of us have heard of Newton’s laws of motion. These were the laws that laid the foundation for classical mechanics. Classical Mechanics is a field through which the motion of everything that we can see and feel is based. For example, Lionel Messi’s curling football kick into the net, the route of the sixer that Virat Kohli hits that is visually recreated by the IPL TV replays, the overhead badminton shuttle drop shot that P.V. Sindhu uses to wrong-foot her opponent, or the graceful lob that Roger Federer uses to keep his adversaries off balance can all be explained through the principles of classical mechanics. In addition to these easily understood events, the trajectory of a missile, the flight path of an airplane or rocket, the elliptic or circular path taken by an ordinary and a geostationary satellite, respectively around the earth, tides in our oceans, motions of the planets, the solar system, galaxies, can all be explained by using the principles of classical mechanics. So, in essence, classical mechanics describes nature at the macroscopic scale.

 

Towards the end of the 19th century, some extremely respected physicists thought that there was nothing new left to discover in physics. For example, Lord Kelvin, (a Scottish Mathematician and Physicist, also known as Sir William Thomson, who was instrumental in formulation of the first and second laws of thermodynamics, the transatlantic telegraph project and a number of critical contributions in electricity. The unit “kelvin” of the absolute temperature is in his honor) stated in an address to a gathering of physicists in 1900 that, “There is nothing new to be discovered in physics now. All that remains is more and more precise measurement” [Ref 2].  A similar statement is attributed to the American physicist Albert Michelson.

 

However, at the turn of the century, things were to rapidly change in the arena of physics. All the phenomena that I just described above are of systems that are visible to the naked eye (or macroscopic, as I mentioned earlier). Some physicists were working on phenomena in systems at a microscopic scale. For example, the black-body radiation problem by Max Planck in 1900, photo-electric effect proposed by Einstein in 1905, the Rutherford gold foil experiments (through which scientists discovered that every atom contains a nucleus where all its positive charge and almost all of its mass is concentrated) and Compton scattering (in which light is scattered by a free charged particle resulting in a change in wavelength of the scattered light) are some of these microscopic phenomena. All these were fundamental concepts and experiments, and none of them could be explained through classical mechanics. This is when physicists realized that there was a need to generalize the theory of classical mechanics. Physicists including Max Born, Niels Bohr, Louis de Broglie, Werner Heisenberg, Erwin Schrodinger and Paul Dirac were the scientists who developed the mathematical formulations which helped in explaining these and a host of new phenomena which kept coming in through the first quarter of the 20th century.  This was the theory of quantum mechanics. Thus, quantum mechanics is essential to understanding the behavior of systems at atomic length scales and smaller, also called the microscopic scale.

 

The beauty and test of any new theory is that under the old conditions, it should be able to yield all results of the old theory, since the original theory has been explaining the experimental results in its own domain. And indeed, all results of classical mechanics were proved by quantum mechanics when the systems being considered became macroscopic [Ref 3].

 

The Schrodinger Wave Equation: While the quantum phenomena that I mentioned above were important landmarks in the evolution of physics, physicists wanted a mathematical equation, which would help in explaining the experimental results and consequently govern the behavior of mechanical phenomena. In 1926, Erwin Schrodinger came up with a differential equation (a partial differential equation, to be precise) which was to replace Newton’s second law of physics, F = Ma as the basic law of nature in mechanics. The mechanics which was based on this equation came to be called as wave mechanics and which is now also called as quantum mechanics [Ref 3]. The equation is the following:

 

 

In this equation, H is the total energy of the system under consideration.  is a solution of the time dependent Schrodinger equation and called as a wave function. It is a mathematical function containing all the information that can be obtained about the system it represents [Ref 3]. It is important to understand that whenever we solve a physical problem in physics, we will be using this equation, just as we use the second law of motion while solving problems in classical mechanics.

 

Now that we have a 35,000 feet view of some of the basics of quantum mechanics, let us define some terms which will come in handy while explaining the concepts of quantum computing.

 

Quantum system: This is the system that is being studied for quantum mechanics. For example, a system under study could be the hydrogen atom and the various energy states in which it can exist. You can immediately see that the system we have considered is microscopic.

 

Observable: In physics, an observable is any quantity that can be measured. For example, position, momentum, displacement, angular momentum, energy are all measurable quantities [Ref 2].

 

Measurement: Once a quantum system has been prepared in the laboratory, some measurable quantity (observable) such as the observables mentioned above (position, momentum, energy etc.) is measured [Ref 2].

 

Quantum State: A quantum state refers to the state of a quantum system. A quantum state provides a probability distribution for the value of each observable, that is for the outcome of each possible measurement on the system. In other words, any solution of the Schrodinger equation which provides information about the position, momentum, energy, or any other observable as defined above, constitutes a quantum state of the system. The classical equivalent of this is the intuitive understanding that we have of the state of a classical system which can be described by its position or momentum (for example the state of a cricket ball after being hit by Kohli can be described by its position and momentum or velocity at any instant of time). A mixture of quantum states is again a quantum state (which leads to the definition of the superposition given below). Quantum states that cannot be written as a mixture of other states are called pure quantum states while all other states are called as mixed quantum states [Ref 2, 3].

 

Quantum superposition: Also called the principle of superposition, it is one of the fundamental principles of quantum mechanics. In classical mechanics, we can add two or more waves. In a similar fashion in quantum mechanics, we can add (superpose) two or more quantum states. And conversely, every quantum state can be represented as a sum or superposition of two or more distinct states. From a perspective of mathematics, superposition refers to a property of the solutions to the Schrodinger equation. Since the Schrodinger equation is linear (that is the wave function  and its derivatives appear only in the first degree and not any other higher power for example 2 or 3 or higher) every linear combination of its solutions is also a solution of this equation [Ref 2, 3].

 

Quantum Entanglement: One of the most unusual, counterintuitive and fascinating aspects of quantum mechanics is the fact that particles or systems can become entangled. Let’s take the simplest case of two quantum systems. Let’s denote the two systems as A and B. If these systems are entangled, it means that values of certain properties of system A are correlated with the values that those properties will assume for system B. Most importantly, the properties can become correlated even when the two systems are far apart in space – leading to the phrase spooky action at a distance [Ref 4].

 

In my next blog, I will elaborate a bit more on the Quantum Entanglement as there is a lot of history involving the greatest Physicist of the 20th century, Albert Einstein. I will also briefly touch upon about the quantum computing architecture and its building blocks (quantum gates and quantum circuits).

 

References:

  1. Images, courtesy the Internet
  2. https://en.wikipedia.org/wiki/Quantum_mechanics.
  3. http://iopscience.iop.org/book/978-0-7503-1206-6/chapter/bk978-0-7503-1206-6ch1
  4. Quantum Computing Explained, David McMahon, page 147, Wiley, 2007.
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