If you’re losing your grip on quantum mechanics, you may want to read up on this

By | January 8, 2016
To many students, quantum mechanics is an academically imposed phantom that comes and goes as quickly as the semesters start and end. A primary reason, I believe, is that they find it so far removed from their familiarities. They approach quantum mechanics with the Newtonian mindset which is in accord with their everyday experience. When they first step into the quantum territory, they find themselves in an alien world where the Newtonian notion of particle as a localized object is now replaced with a wave of undulating complex probability amplitude called the wavefunction, and where the certainty inherent in the Newtonian description of the physical universe is replaced with the inherent uncertainty in the probabilistic nature of quantum mechanics. They find the quantum landscape surreal and bereft of the solidity of the Newtonian landscape they are familiar with.

Their shock is exacerbated by the much higher level of mathematical sophistication needed to solve problems in quantum mechanics, not to mention the possible role of an equally confused instructor or an overly assuming accomplished instructor. No wonder many just want to get over with it. I suspect that many students just managed to survive quantum mechanics and went on to consign their “quantum awakening” to the forgettable category. If such indeed is the case, then many students stand to loss much. In this post, I attempt to salvage whatever is left there and bring into focus what are the essential principles of quantum mechanics without the clutter of mathematics. In so doing I wish to reignite the curiosity of those students on quantum anythings, and perhaps lead them to ask astute questions in quantum mechanics.

Here we go. Just to make sure we have a common ground, let me define some terms. When we refer to a system we mean any object or a collection of objects that is our subject of scrutiny or measurement. Also when we say the state of a system we mean the minimum collection of information that completely describes the system. Moreover, by state space we mean the collection of all possible states of a given system. Now by a quantum system we mean a system whose description is governed by laws of quantum mechanics, which I am attempting to summarize in this post.

Despite the voluminosity of the textbooks expounding quantum mechanics, it can, in fact, be summarized in three principles: First is on the structure of the state space of a quantum system (the quantum superposition principle); second is on the dynamics of quantum systems that do not interact with their environments (the quantum unitarity principle); and third is on the dynamics of quantum systems under measurements (the quantum randomness principle). These three principles are to quantum mechanics as Newton’s laws of motion are to classical mechanics.

To have a grasp of the first principle, let us consider a system that is within our everyday experience, a marble in a two compartment box. If you put the marble inside the box, our experience tells us that the marble can only be in either of the two compartments, either in the left or right compartment. Our everyday marble has two possible states: the state of being in the left compartment and the state of being in the right compartment. We denote the former by $|L \rangle$ (read as “ket-L”) and the later by $|R\rangle$. These are the only two possible states; and their collection, the set containing $|L\rangle$ and $|R\rangle$, constitutes the state space of our everyday marble, which is referred to in physics as a classical marble.

On the other hand, quantum mechanics has a completely different description of the possible states of the marble. Quantum mechanics asserts that the states $|L\rangle$ and $|R\rangle$ are still possible states of the marble. However, quantum mechanics now claims that they are not the only available states, but also everything in between them. That is a quantum marble can also be in a state such that we can consistently describe it as in the state of being in the state $|L\rangle\mathbf{and}|R\rangle$, which roughly means that the marble is simultaneously in the left and right compartments. This is known as the quantum superposition principle which states that if $|\psi\rangle$ and $|\phi\rangle$ are states, then the state of being simultaneously in these states is again a legitimate state. The quantum superposition principle is the heart of quantum mechanics (sounds cliched but it’s just what it is).

The state space of a quantum marble is then much larger than the state space of its classical counterpart. The classical marble has only two possible states, but the quantum marble has infinitely or uncountably many possible states. It is this vastness of the available states to a quantum object that introduces many of the weird behavior of a quantum object. For example, there is nothing that prohibits us from considering instead a cat. The classical cat has $|\mathbf{dead}\rangle$ and $|\mathbf{alive}\rangle$ as the only possible states. But applying the quantum superposition principle to the cat leads to the morbid and bizarre state that the cat is simultaneously in the state of being dead and alive states.

The second principle of quantum mechanics is a statement about a quantum system that does not interact with the rest of the universe. Such a system is called a closed quantum system. Quantum mechanics says that a closed quantum system, left on its own, will evolve in time such that, starting from some initial state, it will have a state such that it is a linear superposition of all possible classical states in accordance with the superposition principle. This is what is known as the quantum unitarity principle. What this principle means, specifically, is that if we started with the marble in the left compartment, then later, in the most general case, the marble would already be in a state of being at the two compartments at once, then in the state of being in the right compartment, then back in its initial state of being in the left compartment. This goes on indefinitely, repeating the same dynamics with periodicity. It is only under a special condition that the state will not change in time.

Again there is nothing special about the marble that what is true to it is also true to a cat. So if we started with an alive cat and place in a box to isolate it from the rest of the universe, then the quantum unitarity principle says that the cat will evolve in a superposition of dead and alive state, then to a dead cat, then back to an alive cat. And the process goes on indefinitely periodically. The cat then goes on its life from living, to living and dead simultaneously, then dead, and then to resurrection. That is not the kind of life that a classical cat is consigned to live: When we put the cat alive and well in the box, the cat at most stay alive for a while and then eventually starve to death, with no hope of resurrection.

Finally, the third principle is on what happens when the quantum object is observed or when it is subjected to measurement. This is where quantum mechanics diverges from classical mechanics in all aspects. For a classical object, such as a classical marble and a classical cat, if one knows the initial state of the object and every detail of the interaction of the object with its environment and the measuring instrument, the outcome of any measurement is completely predictable. Moreover, the measurement can be made in such a way that the measurement will not in any disturb the state of the object. We are not surprised that when we open the box, our knowing where the marble is will not in anyway change the fact that the marble is in the left compartment. We are not surprised either that looking at the cat will kill it.

But the behavior of a quantum object under measurement is completely different from the classical measurement. It is only under one exceptional condition that the measurement will have a predictable outcome and does not disturb the state. Let us consider the marble again. For example, I place the marble in the left compartment. I then ask a friend to open the box and determine where the marble is. My friend writes down his result. I can confidently predict with certainty what my friend found. He found the marble in the left compartment. Not only that, after my friend determines that the marble is in the left compartment, the marble remains in the left compartment. That is the measurement has not disturbed the state of the marble. The same thing happens when I place the marble in the right compartment, then close the box and ask my friend again to open the box. Under this condition, the result of quantum measurement is in complete accord with classical measurement.

However, quantum mechanics gives a completely different description of measurement when the marble is prepared in a linear superposition of left and right states. Now let us say that I prepared the marble in such a state. I close the box and my friend immediately opens the box to determine where the marble is. My friend jots down his observation. This time I can no longer predict with certainty the outcome of my friend’s measurement. My friend will now either find the left or the right compartment as a result of his measurement. Quantum mechanics asserts that the outcome of my friend’s measurement is completely random. That is his outcome is not determined by anything, not even how I prepared the marble in the given initial state or how carefully my friend made the measurement. The outcome of my friend is completely unpredictable. At most I can only guess the outcome of my friend’s measurement. And quantum mechanics can only predict the probability of finding the marble to be in the left or right compartment. This known as the quantum randomness principle.

Moreover, the measurement will cause an unpredictable change in the state of the marble. If the outcome of the measurement is left, then the state of the marble collapses to the state $|L\rangle$; if the marble is found in the state right, the marble collapses to the state $|R\rangle$. This disturbance cannot be minimized as much as it can in classical measurement. There is no way of performing measurement on the marble on its location that the state is not disturbed when it is in a superposition of being in the left and the right compartments. The outcome of the measurement is not repeatable as well. That is even if I prepare the marble in the same state every time my friend performs exactly the same measurement procedure, the outcome of his result is not necessarily the outcome of his previous measurement.

That’s the whole of quantum mechanics. The rest are mere details and specific applications of the above principles. You may forget how to solve the Schrodinger equation or the energy eigenvalue problem or the scattering amplitude; but if you remember these principles and are “shocked” (said Bohr) by their implications, then you have a full grip on quantum mechanics and you are on your way to asking astute questions on quantum anythings.

(A related earlier post is found here.)

5 thoughts on “If you’re losing your grip on quantum mechanics, you may want to read up on this

  1. anong

    where is the postulate on observables that says quantum observables are hermitian operators? is not supposed to be part of the core principles of quantum mechanics?

    Reply
    1. Eric A. Galapon Post author

      The postulate is embedded in the third principle. However, it is not part of the core principles of quantum mechanics in the sense that the statement “quantum observables are hermitian operators” is not generally true. For finite dimensional quantum systems, such as spin systems, hermitian operators representing quantum observables have orthogonal spectral decompositions. This implies that there are as many possible number of observable values (at most) as the dimension of the system Hilbert space. However, for example, in quantum optics there are cases where the number of observable values exceeds the dimension of the system.

      Reply
  2. Lemuel John Sese

    Dr. Galapon this is a wonderful article for such a student like me for having a grasp of curiosity in Quantum Mechanics as the three important principles of QM explained so much, likely friendly and can be understood by many. Thank you Sir for having an opportunity to read this kind of article, also to write some comments and ask some questions here. Sir, I have a question about the third principle, is the perturbation can be classified as an example of measurement? and how the perturbation affects the randomness of states in open quantum system?

    Reply
    1. Eric A. Galapon Post author

      Hi Lemuel. A measurement is always a perturbation on the system under observation, but a perturbation is not always a measurement. A measurement is characterized by the extraction of outcomes, recorded or unrecorded, which, according to the rules of quantum mechanics, perturb the state of the system irreversibly. On the other hand, a perturbation can be applied to a closed quantum system, such as a perturbing potential, with the sole purpose of modifying the dynamics of the system without eliciting any measurement outcome.

      Reply

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