In My Favorite Equation in Physics we discussed a little bit of classical mechanics, the prevailing view of physics from the time of Galileo Galilei up to the start of the 20th century. Keeping in mind the ideas we introduced in that post, we now move on to one of the most groundbreaking ideas in the history of physics since that time (along with Einstein’s theory of relativity, which we have also discussed a little bit of in From Pythagoras to Einstein), the theory of quantum mechanics (also known as quantum physics).
We recall one of the “guiding” questions of physics that we mentioned in My Favorite Equation in Physics:
“Given the state of a system at a particular time, in what state will it be at some other time?”
This emphasizes the importance of the concept of “states” in physics. We recall that the state of a system (for simplicity, we consider a system made up of only one object whose internal structure we ignore – it may be a stone, a wooden block, a planet – but we may refer to this object as a “particle”) in classical mechanics is given by its position and velocity (or alternatively its position and momentum).
A system consisting of a single particle, whose state is specified by its position and velocity, or its position and momentum, might just be the simplest system that we can study in classical mechanics. But in this post, discussing quantum mechanics, we will start with something even simpler.
Consider a light switch. It can be in a “state” of “on” or “off”. Or perhaps we might consider a coin. This coin can be in a “state” of “heads” or “tails”. We consider a similar system for reasons of simplicity. In real life, there also exist such systems with two states, and they are being studied, for example, in cutting-edge research on quantum computing. In the context of quantum mechanics, such systems are called “qubits“, which is short for “quantum bits”.
Now an ordinary light switch may only be in a state of “on” or “off”, and an ordinary coin may be in a state of “heads” or “tails”, but we cannot have a state that is some sort of “combination” of these states. It would be unthinkable in our daily life. But a quantum mechanical system which can be in any of two states may also be in some combination of these states! This is the idea at the very heart of quantum mechanics, and it is called the principle of quantum superposition. Its basic statement can be expressed as follows:
If a system can exist in any number of classical states, it can also exist in any linear combination of these states.
This means that the space of states of a quantum mechanical system form a vector space. The concept of vector spaces, and the branch of mathematics that studies it, called linear algebra, can be found in Vector Spaces, Modules, and Linear Algebra. Linear algebra (and its infinite-dimensional variant called functional analysis) is the language of quantum mechanics. We have to mention that the field of “scalars” of this vector space is the set of complex numbers .
There is one more mathematical procedure that we have to apply to these states, called “normalization“, which we will learn about later on in this post. First we have to explain what it means if we have a state that is in a “linear combination” of other states.
We write our quantum state in the so-called “Dirac notation”. Consider the “quantum light switch” we described earlier (in real life, we would have something like an electron in “spin up” or “spin down” states, or perhaps a photon in “horizontally polarized” or “vertically polarized” states). We write the “on” state as
and the “off” state as
The principle of quantum superposition states that we may have a state such as
This state is made up of equal parts “on” and “off”. Quantum-mechanically, such a state may exist, but when we, classical beings as we are (in the sense that we are very big) interact or make measurements of this system, we only find it in either in a state of “on” or “off”, and never in a state that is in a linear combination of both. What then, does it mean for it to be in a state that is a linear combination of both “on” and “off”, if we can never even find it in such a state?
If a system is in the state before we make our measurement, then there are equal chances that we will find it in the “on” state or in the “off” state after measurement. We do not know beforehand whether we will get an “on” state or “off” state, which implies that there is a certain kind of “randomness” involved in quantum-mechanical systems.
It is at this point that we reflect on the nature of randomness. Let us consider a process we would ordinarily suppose to be “random”, for example the flipping of a coin, or the throwing of a die. We consider these processes random because we do not know all the factors at play, but if we had all the information, such as the force of the flip or the throw, the air resistance and its effects, and so on, and we make all the necessary calculations, at least “theoretically” we would be able to predict the result. Such a process is not really random; we only consider it random because we lack a certain knowledge that if we only possessed, we could use in order to determine the result with absolute certainty.
The “randomness” in quantum mechanics involves no such knowledge; we could know everything that is possible for us to know about the system, and yet, we could never predict with absolute certainty whether we would get an “on” or an “off” state when we make a measurement on the system. We might perhaps say that this randomness is “truly random”. All we can conclude, from our knowledge that the state of the system before measurement is , is that there are equal chances of finding it in the “on” or “off” state after measurement.
If the state of the system before measurement is , then after measurement it will also be in the state . If we had some state like before measurement, then there will be a much greater chance that it will be in the state after measurement, although there is still a small chance that it will be in the state .
We now introduce the concept of normalization. We have seen that the “coefficients” of the components of our “state vector” correspond to certain probabilities, although we have not been very explicit as to how these coefficients are related to the probabilities. We have a well-developed mathematical language to deal with probabilities. When an event is absolutely certain to occur, for instance, we say that the event has a probability of 100%, or that it has a probability of . We want to use this language in our study of quantum mechanics.
We discussed in Matrices the concept of a linear functional, which assigns a real or complex number (or more generally an element of the field of scalars) to any vector. For vectors expressed as column matrices, the linear functionals were expressed as row matrices. In Dirac notation, we also call our “state vectors”, such as and , “kets”, and we will have special linear functionals and , called “bras” (the words “bra” and “ket” come from the word “bracket”, with the fate of the letter “c” unknown; this notation was developed by the physicist Paul Dirac, who made great contributions to the development of quantum mechanics).
The linear functional assigns to any “state vector” representing the state of the system before measurement a certain number which when squared (or rather “absolute squared” for complex numbers) gives the probability that it will be found in the state after measurement. We have said earlier that if the system is known to be in the state before the measurement, then after the measurement the system will also be in the state . In other words, given that the system is in the state before measurement, the probability of finding it in the state after measurement is equal to . We express this explicitly as
From this observation, we make the requirement that
for any state . This will lead to the requirement that if we have the state , the coefficients and must satisfy the equation
The second expression is to remind the reader that these coefficients are complex. Since we express probabilities as real numbers, it is necessary that we use the “absolute square” of these coefficients, given by multiplying each coefficient by its complex conjugate.
So, in order to express the state where there are equal chances of finding the system in the state or in the state after measurement, we do not write it anymore as , but instead as
The factors of are there to make our notation agree with the notation in use in the mathematical theory of probabilities, where an event which is certain has a probability of . They are called “normalizing factors”, and this process is what is known as normalization.
We may ask, therefore, what is the probability of finding our system in the state after measurement, given that before the measurement it was in the state . We already know the answer; since there are equal chances of finding it in the state or , then we should have a 50% probability of finding it in the state after measurement, or that this result has probability . Nevertheless, we show how we use Dirac notation and normalization to compute this probability:
We know that , and that , which leads to
as we expected. We have used the “linear” property of the linear functionals here, emphasizing once again how important the language of linear algebra is to describing quantum mechanics.
For now that’s about it for this post. We have glossed over many aspects of quantum mechanics in favor of introducing and emphasizing how linear algebra is used as the foundation for its language; and the reason why linear algebra is chosen is because it fits with the principle at the very heart of quantum mechanics, the principle of quantum superposition.
So much of the notorious “weirdness” of quantum mechanics comes from the principle of quantum superposition, and this “weirdness” has found many applications both in explaining why our world is the way it is, and also in improving our quality of life through technological inventions such as semiconductor electronics.
I’ll make an important clarification at this point; we do not really “measure” the state of the system. What we really measure are “observables” which tell us something about the state of the system. These observables are represented by linear transformations, but to understand them better we need the concept of eigenvectors and eigenvalues, which I have not yet discussed in this blog, and did not want to discuss too much in this particular post. In the future perhaps we will discuss it; for now the reader is directed to the references listed at the end of the post. What we have discussed here, the probability of finding the system in a certain state after measurement given that it is in some other state before measurement, is related to the phenomenon known as “collapse“.
Also, despite the fact that we have only tackled two-state (or qubit) systems in this post, it is not too difficult to generalize, at least conceptually, to systems with more states, or even systems with an infinite number of states. The case where the states are given by the position of a particle leads to the famous wave-particle duality. The reader is encouraged once again to read about it in the references below, and at the same time try to think about how one should generalize what we have discussed in here to that case. Such cases will hopefully be tackled in future posts.
(Side remark: I had originally intended to cover quite a bit of ground in at least the basics of quantum mechanics in this post; but before I noticed it had already become quite a hefty post. I have not even gotten to the Schrodinger equation. Well, hopefully I can make more posts on this subject in the future. There’s so much one can make a post about when it comes to quantum mechanics.)
The Feynman Lectures on Physics by Richard P. Feynman
Introduction to Quantum Mechanics by David J. Griffiths
Modern Quantum Mechanics by Jun John Sakurai