# What does the notation $|\psi(0)\rangle = |0\rangle|n=0\rangle$ mean?

Let us take the initial state with the particle located at the origin $$|n=0\rangle$$ and the coin state with spin up $$|0\rangle$$. So, $$|\psi(0)\rangle = |0\rangle|n=0\rangle,$$ where $$|\psi(0)\rangle$$ denotes the at the initial time and $$|\psi(t)\rangle$$ denotes the state of the quantum walk at time $$t$$.

Please, help me understand that, if on RHS we have a matrix product of two column vectors?

This is common shorthand for the tensor product. That is, you should read it as $$|0 \rangle \otimes | n=0\rangle$$.
• I would doubt it. What kind of matrix product are you thinking of? If written $|a \rangle \langle b|$ then it denotes the outer product usually but this would not make sense given the LHS of your equation which is a vector $|\psi(0)\rangle$. The only other possibility I could imagine (but which I very very much doubt is the case) is that it denotes an elementwise product but this only makes sense if $|\psi(0)\rangle$, $|0\rangle$ and $|n=0\rangle$ are all vectors from the same dimensional vector space. If this notation is not specified explicitly then I am confident it is the tensor product. Nov 5 '20 at 16:14