Vectors and Column Notation

Quantum mechanics describes the state of a physical system using a state vector — a list of complex numbers arranged in a column. This column notation, inherited directly from linear algebra, is the backbone of every quantum calculation you will ever do.

From arrows to columns

In ordinary 2-D geometry a vector points in some direction. The same vector can be written as an ordered pair of real numbers $(x, y)$ . Column notation just stacks those numbers vertically:

\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}.

Quantum mechanics does exactly the same thing, but the entries are allowed to be complex numbers, and the column has as many rows as there are distinct measurement outcomes for the system.

The two-level quantum system

The simplest quantum system has exactly two distinguishable outcomes when measured. Call them $|0\rangle$ and $|1\rangle$ (the computational basis states). Any state of this system is a linear combination

|\psi\rangle = \alpha\,|0\rangle + \beta\,|1\rangle, \qquad \alpha,\beta \in \mathbb{C}.

In column notation this becomes

|\psi\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix},

where the top entry is the coefficient of $|0\rangle$ and the bottom entry is the coefficient of $|1\rangle$ . The basis vectors themselves are then

|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.

You can verify that substituting these into $\alpha|0\rangle + \beta|1\rangle$ and adding the columns component-by-component recovers $(\alpha,\beta)^T$ , confirming the notation is consistent.

The normalization condition

The entries $\alpha$ and $\beta$ are called probability amplitudes. According to the Born rule, the probability of obtaining outcome $|0\rangle$ is $|\alpha|^2$ and the probability of obtaining $|1\rangle$ is $|\beta|^2$ . Since those are the only two outcomes and probabilities must sum to 1, every valid state vector must satisfy

|\alpha|^2 + |\beta|^2 = 1.

This is the normalization condition. It is not an extra assumption; it follows directly from requiring that probabilities sum to 1.

Higher-dimensional systems

Nothing stops us from having more than two outcomes. An $n$ -level system is described by a column vector with $n$ complex entries $(\alpha_1, \alpha_2, \ldots, \alpha_n)^T$ , subject to

\sum_{k=1}^{n} |\alpha_k|^2 = 1.

For two qubits there are four computational basis states $|00\rangle, |01\rangle, |10\rangle, |11\rangle$ , so the state vector has four entries. Every extra qubit doubles the number of entries — a key reason quantum systems can represent so much information.

Try it

This is a numerical exercise — your code should return a number. Take the equal-superposition state $|\psi\rangle = \tfrac{1}{\sqrt{2}}|0\rangle + \tfrac{1}{\sqrt{2}}|1\rangle$ and verify the normalization condition by computing $|\alpha|^2 + |\beta|^2$ .

Run your code to see the quantum state.