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I am currently learning quantum mechanics and I am trying to understand the connection between the wave function and the state vector. Is it correct to say that the components of the state vector of a quantum system are all possible wavefunctions in that state?

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    $\begingroup$ Could you clarify how (if at all) your sources define/describe what state vectors and wavefunctions are? $\endgroup$ Jul 24 at 22:42

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A state vector is just a wavefunction where the domain doesn't need to be $\mathbb{R}^d$. Similarly, you can call wavefunctions vectors because the functions used in quantum mechanics (square integrable ones) are elements of a space which is closed under addition and scalar multiplication. I.e. a vector space.

A simple example is a large atom with a magnetic moment. In the Stern-Gerlach experiment, quantum fluctuations of its position are negligible but its spin (which can be up or down) is highly probabilistic. So instead of a wavefunction $\psi: \mathbb{R}^3 \to \mathbb{C}$, we just need $\psi: \{0,1\} \to \mathbb{C}$. A function from a two element set to $\mathbb{C}$ can be regarded as a two component vector so we can write this object as \begin{align} \left | \psi \right > = \begin{pmatrix} a \\ b \end{pmatrix} \end{align} where $|a|^2$ is the spin-up probability and $|b|^2$ is the spin-down probability. This is also written \begin{align} \left < + | \psi \right > = a, \quad \left < - | \psi \right > = b. \end{align} Depending on how formal you want to be, the normalization condition $|a|^2 + |b|^2 = 1$ can also be seen as an integral of $|\psi|^2$ except using the counting measure instead of the Lebesgue measure.

We can also consider the opposite case. A particle with no spin but probabilistic position on a line. Position is a continuous degree of freedom so our $\psi: \mathbb{R} \to \mathbb{C}$ will be a vector with "a continuum of components". Instead of being able to project $\left | \psi \right >$ onto just two basis states ($\left < + \right |$ or $\left < - \right |$), we will be able to project it onto infinitely many position eigenstates. One for each position we can imagine the particle to have. The eigenstates can be labelled by $\left < x \right |$ and the projection by $\left < x | \psi \right >$ which is nothing more than $\psi(x)$ in the usual function notation.

Since we now have an all-encompassing framework that accounts for both continuous and discrete degrees of freedom, we can combine the two and consider things like $\psi: \mathbb{R} \times \{0, 1\} \to \mathbb{C}$ for particles with uncertain positions and spins. This allows us to consider projections like $\left < +, x | \psi \right >$ or $\left < -, y | \psi \right >$. In this situation, textbooks often use a hybrid notation where the $2 \times \infty$ components of the state vector are written as two functions worth of components. Something like \begin{align} \left | \psi \right > = \begin{pmatrix} \psi_+(x) \\ \psi_-(x) \end{pmatrix}. \end{align} This may have given rise to your notion of a vector consisting of many wavefunctions.

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    $\begingroup$ Great answer, but a small note: The notation in your last equation is a bit weird, since an $x$ appears on the RHS but not on the LHS. I guess this is a common abuse of notation, but, in my experience, also a common source of confusion when first encountering Dirac notation/ intro quantum mechanics. $\endgroup$ Jul 25 at 10:37
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The relationship between wave functions and state vectors is the exact same relationship as between vector components and vectors in Euclidean vector space.

When you have an "ordinary" vector in 3D Euclidean space, given in terms of components, such as

$$\mathbf{v} = \langle 1, -2, 5 \rangle$$

you've likely seen them written with a subscript notation, e.g. $v_x = 1$, $v_y = -2$, and $v_z = 5$. You can think of this as a function from the label, here $x$, $y$, or $z$, to the corresponding component number, here $1$, $-2$, and $5$. The function, remember, is the association between the two, not how it's written.

In the case of quantum mechanics, though, the vectors can have infinitely many - even continuously many - "components". Thus, it becomes more natural to use function notation directly, and we call such a function a "wave function". In particular, $\psi(x)$ is the "$x$th component" of the vector $|\psi\rangle$ in some basis (usually the position basis given the use of $x$ as the variable [note this $x$ is not the same as the label-$x$ from before], but other bases are possible).

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