<aside> ❓ A barrier to studying Quantum Physics properly is its need for advanced mathematics to formulate the theory! Quantum Mechanics utilises Functional Analysis, and infinite-dimensional complex Hilbert spaces, in particular. Although for Quantum Computing finite dimensional complex Hilbert spaces suffices, it still requires advanced concepts in Linear Algebra like self-adjoint linear operators, eigenvalues and eigenvectors.
There is no other way around it then to overcome that barrier by learning and understanding these mathematical prerequisites step-by-step. Here, we shall give you a more detailed breakdown what topics you should cover, what aspects you should focus on, in particular, and what resources we recommend for you to look at so you can grasp the math.
Note that we provide you with a wealth of resources on this page up to post-graduate level. They are not independent resources, though, and there is a lot of overlap. Check out and combine several of the resources. Find the one that suits your current level of understanding the best, and then check out others to get a better well-rounded presentation of a topic.
</aside>
<aside> 🌱
Depending on whether you are a high schooler or an undergrad, you will have different needs. We will provide resources for both, but there will be naturally more for high schoolers, as they have more to cover. What should be your focus? 🤔
<aside> 🤔 Example 1 (Linear Algebra)
Let’s assume you have a basic intuition and understanding of vectors and matrices already; note that the algebra of real numbers is very different from the algebra of vectors. On the other hand, the algebra of vectors is similar to the algebra of matrices, but there is an important difference: matrices can be multiplied under certain conditions; but this multiplication is different from the multiplication of numbers, as the order matters, and you cannot divide by matrices as freely as you can divide by numbers. (What does it even mean to divide by a matrix?) A high schooler should be questioning and thinking up to this point, and then focus on actually mastering the algebra and some of the applications of vectors and matrices. They should also start thinking about how they can implement vectors and matrices and their algebra in code.
An undergrad should think further: how can we make precise the similarities and differences in the algebra of those different objects? We are dealing with different algebraic structures here. What are those structures? The real numbers form a field (in fact a Dedekind-complete ordered field), so do complex numbers (but without an order, why?), vectors form an (abstract) vector space. Matrices also form an (abstract) vector space, but the multiplication (of $n\times n$ matrices, say) and addition of matrices make it also a ring. But what if we wish consider the multiplication between arbitrary matrices? This is a bit more tricky as the multiplication is not defined for an arbitrary pair of matrices; it is only defined partially. After thinking about this for a while (and learning about Category Theory) you might discover that the matrices with matrix multiplication form an example of a category…
But there are also other, more imminent questions you should ask yourself as an undergrad:
</aside>
<aside> 💡
Once you explore what is sketched in this example in full detail, ask many questions, seek answers to them, and understand what is sketched in the above example, you have completed a deep dive into abstract algebra, abstract algebraic thinking and what it means to think and learn as a researcher! Well done! 👏 **
</aside>
<aside> 🤔 Example 2 (Complex vector spaces)
Let’s assume you have a basic intuition and understanding of vectors and matrices, once again. Multiplying a 2D vector with a real scalar is just re-scaling said vector, and , based on the sign of the scalar, flip it’s direction if the scalar is negative. Now, what does it mean to multiply a vector with a complex number? It is not hard to define complex scalar multiplication for column or row vectors algebraically by simply mimicking the definition of a scalar multiplication for real column or row vectors. (Note how we are building on existing prior knowledge and generalise this knowledge to new situations by following the same pattern within a new context. This is the power of abstraction!)
Now, a high-schooler should focus on the basics, making sure they can handle the algebra of complex numbers confidently. They should also get some geometric intuition by studying what complex addition and complex multiplication mean when representing complex numbers as 2D vectors in a pointed plane. However, how could we visualise multiplying a vector with a complex number? What do you think it should mean?
An undergrad should go further and ask: why are we using complex vector spaces in Quantum Mechanics in the first place? (Go and investigate this question. You will get a multitude of answers; but which one is the best, and why?)
</aside>
<aside> 🧑🔬
Note how in both examples we ask questions that naturally occur while engaging with and thinking about vectors. Some of these questions do not have easy, or straight forward answers, and you you should not accept the answers you find, or the ones you come up your self blindly. Learning to ask and trying to answer such questions allows us to develop and build expert knowledge. This is what we mean when saying to approach studying a topic from a Researcher’s Point of View.
</aside>
<aside> 📌
Tips
<aside> ❓ The math of Quantum Mechanics is is very different from the math used for modelling Classical Physics. We shall start by sharing a playlist with you that provides a light and inquisitive introduction to the math of Quantum Mechanics, which is Linear Algebra and Functional Analysis. It’s suitable for non-mathematicians, and starts you off with some good questions about the math used in Quantum to which it provides some initial answers (albeit those answers are largely incomplete).
</aside>
<aside> 💡
The point of this series is to help you understand and provide some motivation why you are looking at rather abstract mathematics, which seems to have nothing to do physics at first glance.
</aside>
<aside> 📌 We recommend you start by watching the first video of the playlist shared below. There are different reactions you might have.
Note that even when you are familiar with the math you might find it confusing why Quantum Physics is modelled this way, mathematically. Well done if you are! We will tackle this question in the Foundations of Quantum Mechanics session in the programme…
</aside>
<aside> ❓ The math of Quantum Mechanics is based on complex rather than real numbers. It is therefore important to get familiar with the algebra of complex numbers and their geometric representation in the plane.
</aside>
<aside> 📌 Here is the minimum you need to know and be able to apply from complex numbers:
<aside> ⚙ Recommended resources to learn complex numbers for a beginner:
<aside> 🌱 If you are familiar with complex numbers already, we suggest
<aside> ❓ Every time we wish to calculate something concretely in math it essentially boils down to do linear algebra (unless it is a very abstract calculation). This makes Linear Algebra the calculation tool number one, and it is not surprising to see it being used everywhere. Linear Algebra over complex numbers is also the mathematical language used for Quantum Mechanics and Quantum Computing. Note that (multivariable) Calculus and Linear Algebra together form the backbone of applied math.
</aside>