Anyone can understand quantum mechanics — Part 2

In Part 1 of this series I made the bold claim that, unlike what famous figures in science seemed to suggest, quantum mechanics is a beautiful and simple theory that is accessible to anyone who is enthusiastic about learning it. In Part 2, I am going to put my money where my mouth is and teach you the basics of quantum mechanics in four short lessons. Sounds good?

Before we begin, there are two important points I need to clarify.

  1. I need your help! It would be great if we could learn new things passively and without any effort, like Neo in the first Matrix movie.

kung fu

Unfortunately, the technology for instant learning is not currently available, so we’ll have to rely on old-school methods like reading, thinking, and discussing. For best results, I recommend that you find time during a quiet afternoon, make yourself a coffee or tea, sit down in a comfy chair and go through the material with patience. If you do it together with a friend or loved one, even better. The ideas I will present are not really complicated, but they will be unfamiliar and unlikely to sink in if you don’t help me with your full attention and enthusiasm.

  1. There will be math! I know that the fear of mathematics is a widespread malady that would normally make many of you run away from this post as quickly as possible. My wife, for example, automatically starts yawning when she hears the word “probability”.
Math fear.jpg

There’s math in the post? Noooooo!

I can assure you, there is nothing to fear. Many science writers choose to get completely rid of math when discussing quantum mechanics, probably hoping to increase the sales of their books. This leads to a tragic state of affairs where any explanation of the subject is either full of technical jargon – as in standard university textbooks – or it is carried out with imprecise analogies and hand-wavy arguments. This would be understandable if the mathematics were actually hard – like in general relativity – but in quantum mechanics, we don’t really need to do much more than add and multiply numbers. I won’t insult the intelligence of my students by pretending they will run away just because we’ll use the same type of math they already use when they need to figure out how much money to take on their next vacation. Just stick with me and you’ll be fine.

Lesson 1: What is quantum mechanics?

Surprisingly, quantum mechanics is not a physical theory. It is a framework that is used to build physical theories. In learning and understanding quantum mechanics, we will be learning something that actually doesn’t look like physics at all, but more like a set of abstract rules. My favourite statement of this fact is one that Scott Aaronson, a professor at MIT, makes in his book, Quantum Computing since Democritus:

“Basically, quantum mechanics is the operating system that other physical theories must run on as application software […]. But if quantum mechanics isn’t physics in the usual sense – if it’s not about matter, or energy, or waves, or particles – then what is it about? From my perspective, it’s about information and probabilities and observables, and how they relate to each other.”

This quote is so good, it is even used to sell printers in Australia, as seen in the short ad below.

When physicists talk about a quantum theory, they are referring to a physical theory that follows the framework of quantum mechanics. For example, quantum optics is a theory about the behaviour of light that is constructed according to the rules of quantum mechanics. This is also why it makes sense to talk about quantum computation: it is simply a theory of computation where computers follow the rules of quantum mechanics. Therefore, from now on, instead of talking about electrons and atoms, we will discuss a set of abstract rules that we can later apply to the context of interest.

Lesson 2: Ket notation. An essential starting point in quantum mechanics is the concept of the state of a system, which simply corresponds to one of its possible configurations. For example, when studying a Canadian one dollar coin, we assign one state to the coin showing the face of the queen (Heads) and another state to the coin showing a loonie (Tails). If the coin is in the configuration shown on the left in the image below, we say that it is in state “Tails”.

loonie

We can do the same for any system of interest. For example, we can assign a state to each of the faces of a dice, to the balance of your savings account, or to the energy of an electron in a hydrogen atom. Essentially, all we are really doing is assigning a label to the possible configurations we are considering.

In quantum mechanics we have a particular notation for the state of a system, which we call a “ket”. Although it may seem strange and pointless, it comes in handy when doing advanced calculations and it is a great way of making clear that we are referring to a quantum state. The way this works is that instead of writing a state in quotation marks – like I have done so far – we write inside of an uneven bracket. For example, instead of writing “Heads” or “Tails”, we write |Heads\rangle and |Tails\rangle. Similarly, when listing all the states of a dice we write:

|1\rangle, |2\rangle, |3\rangle, |4\rangle, |5\rangle, |6\rangle.

A possible state of your savings account would be written as |+1,203\rangle . A cat could be in the state |Alive\rangle or the state |Dead\rangle. You get the idea.

I have introduced this notation for two reasons. First, I want to use the notation that cutting-edge researchers employ every day. Second, I want you to be able to recognize, just by glimpsing, that the equation written in a whiteboard has something to do with quantum mechanics.

Big Bang 

Lesson 3: Superposition. In the previous lesson, you were perhaps wondering what was so quantum anyway about writing |Heads> instead of “Heads”. If that crossed your mind, you are right! There’s nothing quantum about that, it’s just notation. Now I am going to tell you one of the two main features that makes quantum mechanics different. In the quantum world, there are more possibilities for the kinds of states that a system can be in. For example, the state:

|Heads\rangle + |Tails\rangle

is also a valid quantum state of a coin. We refer to such a state as being in a superposition of both states. In fact, the state:

|Heads\rangle - |Tails\rangle

is also a valid quantum state, although in a different kind of superposition. We’ll discuss this difference in another lesson. Similarly, the state:

|1\rangle + |2\rangle -|3\rangle -|4\rangle + |5\rangle + |6\rangle

is also a valid state of a quantum dice. In general, we can add and even subtract any set of states in any way we want, or even multiply them by any number, and end up with another valid state[1]. There are as many valid states as there are ways of combing them together by adding and multiplying by a number! For example, the following are also valid states of a quantum coin and a quantum dice, respectively:

-4 |Heads\rangle - 2|Tails\rangle

|2\rangle + 2|3\rangle -|4\rangle +0.5 |5\rangle

In quantum mechanics, there are so many more possibilities. Can you write down other interesting states of a quantum dice or coin?

In lesson 4, you’ll learn that we have to make an important adjustment to this rule, but for now, the important point is that there is an added richness in quantum mechanics in terms of the possible configurations of physical systems, since it is now possible for them to be in superposition.

What does it mean for a coin to be in the state |Heads\rangle + |Tails\rangle? That is a great question. The answer is… we don’t really know. People have come up with different proposals, from multiple universes to subjective views of reality, but none of them is entirely satisfactory. This is precisely the kind of situation that led Feynman to believe that no one understands quantum mechanics. But of course we understand! Quantum mechanics tells us that we can have superpositions, which are just sums of different states. This is not hard to grasp, is it? The difficulty comes when we ask questions regarding a connection of the theory with the natural world. These are very important and interesting questions that we should ask and try to answer and that I am personally very interested in. But strictly speaking, they are not questions about the theory. Remember, quantum mechanics is a framework, a set of rules that any physical theory must obey. Are these rules hard to understand? I don’t think so!

While the message of this lesson sinks in, take some time to admire this beautiful picture of the Milky Way galaxy.

milky way

Lesson 4: Measurements. Speaking of questions, what happens when we ask a quantum system what state it is in? In other words, what happens when we measure a system to determine its state? To provide an answer, we first have to make an adjustment to lesson 3. Remember I claimed that we could add up states in any way we wanted? Strictly speaking, this isn’t true: the state that we end up with must satisfy an additional property. It’s easiest to illustrate with an example. In quantum mechanics, instead of writing:

|Heads\rangle + |Tails\rangle

We must actually write:

\frac{1}{\sqrt{2}}|Heads\rangle + \frac{1}{\sqrt{2}}|Tails\rangle

The numbers \frac{1}{\sqrt{2}} are referred to as the coefficients of the state. For example, what are the coefficients of the state:

\frac{3}{5}|Heads\rangle + \frac{4}{\sqrt{5}}|Tails\rangle ?

That’s right, the coefficients are \frac{3}{5} and \frac{4}{5}. Quantum mechanics requires that, when we square the coefficients and add them together, the result must always equal \(1\). In this case, we say that the state is normalized. For example, since:

(\frac{1}{\sqrt{2}})^2+(\frac{1}{\sqrt{2}})^2 = \frac{1}{2}+\frac{1}{2}=1

then our new state satisfies the rule. Similarly, the state:

is normalized because:

(\frac{3}{5})^2+(\frac{4}{5})^2=\frac{9}{25}+\frac{16}{25}=1.

Fortunately, we can always normalize a state by simply re-scaling its coefficients appropriately, which is why I didn’t bother doing it in the previous lesson.

Good news for all the students who don’t like math: that’s the hardest math we’ll have to use learning quantum mechanics! 

see.that_.wasnt_.so_.bad_

Now, suppose that we have a quantum coin in a given state and we want to measure it. In quantum mechanics, a measurement is equivalent to asking a system: “In which of the following states are you in?” For example, in the case of a quantum coin, a valid measurement is to ask: “Are you in state |Heads\rangle or in state |Tails\rangle?” Quantum mechanics then tells us that the outcome of the measurement – i.e. the answer to our question – will definitely be either |Heads\rangle or |Tails\rangle and the probability that each outcome occurs is equal to the square of the corresponding coefficient.

Let’s look at some examples. Suppose that we have a quantum coin in the state:

\frac{1}{\sqrt{2}}|Heads\rangle + \frac{1}{\sqrt{2}}|Tails\rangle

and we ask: “Are you in state |Heads\rangle or in state |Tails\rangle?”

Then with probability (\frac{1}{\sqrt{2}})^2=\frac{1}{2} the outcome will be |Heads\rangle and with probability (\frac{1}{\sqrt{2}})^2=\frac{1}{2} the outcome will be |Tails\rangle .

Similarly, if we have a quantum coin in the state:

\frac{1}{2}|Heads\rangle + \frac{\sqrt{3}}{2}|Tails\rangle

and ask the same question, with probability (\frac{1}{2})^2=\frac{1}{4} the outcome will be |Heads\rangle and with probability (\frac{\sqrt{3}}{2})^2=\frac{3}{4} the outcome will be |Tails\rangle .

Moreover, if we ask two questions in a row, then quantum mechanics tells us that we always get the same answer. In other words, the theory doesn’t contradict itself. We would be in trouble otherwise! In our first example, it was equally likely to obtain either of the outcomes. But after an outcome has occurred, let’s say |Heads\rangle , if we ask the same question we’ll always obtain |Heads\rangle as the answer, no matter how many times we ask.

This is a good place for a break, so before taking a look at some examples, let’s take a quick break to admire the beautiful building that hosts IQC and from which I wrote these words.

iqc

You may be wondering: how is this quantum coin any different than a regular coin? That’s an excellent question! Remember how quantum mechanics tells us that we can have a richer class of states thanks to superpositions? Well, in quantum mechanics we also have a richer class of measurements! For example, the question:

“Are you in state \frac{1}{\sqrt{2}}|Heads\rangle + \frac{1}{\sqrt{2}}|Tails\rangle or in state \frac{1}{\sqrt{2}}|Heads\rangle -\frac{1}{\sqrt{2}}|Tails\rangle ?”

is a valid measurement in quantum mechanics. This allows something truly unique to happen: A quantum coin in a given state can give probabilistic outcomes with respect to one measurement and deterministic outcomes with respect to another measurement. No regular coin can do this! For example, what happens if we have a coin in the state:

\frac{1}{\sqrt{2}}|Heads\rangle + \frac{1}{\sqrt{2}}|Tails\rangle

and we ask the question:

“Are you in state \frac{1}{\sqrt{2}}|Heads\rangle + \frac{1}{\sqrt{2}}|Tails\rangle or in state \frac{1}{\sqrt{2}}|Heads\rangle -\frac{1}{\sqrt{2}}|Tails\rangle ?”

Well, with certainty, we are going to get \frac{1}{\sqrt{2}}|Heads\rangle + \frac{1}{\sqrt{2}}|Tails\rangle as the answer! This is completely different from what happened with our other question, where we obtained a completely random outcome. John Preskill, a professor at Caltech in the United States, refers to this ability to make different possible measurements by saying that “in the quantum case, there is more than one way to open the box.” Amazing, isn’t it?

The richer class of quantum states and the richer class of quantum measurements that arise because of superposition gives rise to many unique quantum properties – such as the uncertainty principle and entanglement – as well as important implications to the ways in which we can manipulate information, leading to exciting research fields such as quantum computing and quantum cryptography. This will be the topic of the last part of this series, in which we’ll use our new understanding of the rules of quantum mechanics to unravel their implications and potential technological applications.

Thank you for staying until the end of the lesson, I hope it was as much fun for you as it was for me!

giphy

[1] In fact, quantum mechanics allows us to use complex numbers in the superposition.

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