What is the difference between a state vector and a basis vector in Quantum mechanics?

Question

I searched about the difference between state vector and basis vector in Quantum mechanics but couldn't find any clear explanation. Can someone please give a simple and clear explanation of this?

Answer 1

Basis vectors are a special set of vectors that have two properties:

1. The vectors in the set are linearly independent (meaning you cannot write one vector as the linear combination of other vectors in the set)
2. Every vector in the vector space can be written as a linear combination of these basis vectors

Basis vectors are widely used in linear algebra and are not unique to quantum mechanics.

When we start talking about state vectors in QM, like $|\psi\rangle$, we can choose to express this state vector in terms of any basis we want. In other words, for a discrete basis:

$$|\psi\rangle=\sum_i c_i|a_i\rangle$$

where $|a_i\rangle$ represents basis vector $i$, and $c_i$ is a coefficient saying "how much of $|a_i\rangle$ is in $|\psi\rangle$

Now it could be that $|\psi\rangle$ is equal to one of our basis vectors, say $|a_j\rangle$, so that $c_i=\delta_{i,j}$ and $$|\psi\rangle=\sum_i \delta_{i,j}|a_i\rangle=|a_j\rangle$$

We could even choose express this example in some other basis: $$|\psi\rangle=|a_j\rangle=\sum_id_i|b_i\rangle$$

So to answer the question: basis vectors are just a special set of vectors with the two properties listed above. Each basis vector could be a state vector, if the system is purely in that state, but it does not have to be that way. You can get the entire picture by being more general: state vectors can be expressed as linear combinations of basis vectors of whatever basis we choose to work in. This then covers the case for when our state vector is one of our basis vectors, since this is still the case of a linear combination. The choice of basis is completely subjective though (although some bases are better to work in than others for certain problems).