# What is a term for a basis state along with its corresponding complex amplitude?

For some arbitrary state $$|\psi\rangle = c_0|x_0\rangle + c_1|x_1\rangle + c_2|x_2\rangle ... + c_{2^n}|x_{2^n}\rangle$$, where each of $$|x_i\rangle$$ is a basis state, and each of $$c_i$$ is the corresponding complex amplitude, what term would someone use to describe one of the $$c_i|x_i\rangle$$ terms? It is not just a basis state, and it is not just a complex amplitude. It is both of them combined. What is this called? How would it be denoted? Would it be denoted as something like $$|\psi^i\rangle$$, or $$|\psi\rangle_i$$?

• While "term" is generally appropriate (and quite generic), what you want to use may vary with what you want to emphasize. You can distinguish between scalar components $c_i$ and component states $c_i|x_i\rangle$ of $|\psi\rangle$, and then proceed to point out that component states are scaled basis states, to emphasize the basis state aspect, as $c_i|x_i\rangle$ is just $|x_i\rangle$ scaled by $c_i$. Aug 7, 2021 at 6:32

I would call it a "term", same as a term in different kinds of mathematical expressions like sums or products.

I don't think I've seen a special math notation for terms. People usually spell out the complex amplitudes separately from the basis states, since the states will usually be part of some transformation, and their transformation will depend on $$c_i$$ and $$x_i$$ separately, rather than combined together.

• I think 'term' is most appropriate. Aug 6, 2021 at 17:45

I typically refer to it loosely as the "contribution" of the state $$|x_i\rangle$$, or verbosely the "projection" of $$|\psi\rangle$$ onto $$|x_i\rangle$$. As for denoting it, I'll often write out the "projection" explicitly: $$|x_i\rangle\langle x_i|\psi\rangle$$. I'm sure that's more verbose than you'd prefer.

I'm a big fan of inventing notations on the spot! I think both of your suggestions probably work fine, as long as they're not easily confused with something else in context.

• I appreciate the feedback, thank you. Aug 6, 2021 at 17:43

When a vector is written as $$\sum c_i \vec b$$, that's referred to as a decomposition into components. In traditional linear algebra, in which vector spaces are generally modelled in terms of physical space, $$c_i \vec b_i$$ is referred to as the $$\vec b_i$$ component, the component along the $$\vec b_i$$ axis, or the component along the $$\vec b_i$$ direction, but quantum mechanics generally distinguishes more between physical space and state space, and thus the latter two, especially "direction", are less common than just "the $$\vec b_i$$ component" or "the $$\vec b_i$$ contribution". Also, as linear combinations are often written in terms of eigenstates of a particular operator, the components, being scale multiples of eigenstates, will also be eigenstates, so the components can be referred as "the eigenstates of the solution".