# Quantum states transform like vectors?

One may define vectors as tuples satisfying the vector transformation law, or as elements of a vector space. On the latter, the superposition of two quantum states is another state, and the product of two states is non-state scalar. The vacuum state $$|\psi\rangle=0$$ exists and, for a scalar $$a$$ and a state $$|\psi\rangle$$, the product $$a|\psi\rangle$$ is still a state vector. Briefly, the axioms of a vector space are satisfied. My question regards the idea that state vectors transform like vectors.

Since states may not be position states, or states with any sort of directionality, I am not sure how to apply the concept of coordiante transformation in an abstract vector space. For instance, if I say,

"Let $$\hat R(\hat n,\phi)$$ be a rotation operator and $$|\psi\rangle=|a\rangle+|b\rangle$$ so that $$\hat R|\psi\rangle=\hat R|a\rangle+\hat R|b\rangle$$ obviously preserves the `angle' between $$a$$ and $$b$$, meaning that $$\hat R|a\rangle$$ and $$\hat R|b\rangle$$ are still orthogonal,"

then I am not satisfied that I have shown the proper vector transformation. Since the states themselves are the spanning basis of the state space, I am not quite getting the gist of what a coordinate transformation would be such that I could make a demonstration like the one below copied from Wolfram. What is the general proof that quantum states transform like vectors even when they are not states directly related to $$\mathbb{R}^3$$ coordinates? Following the form below, if I start with

$$|\psi\rangle=\sum_k a_k|\psi_k\rangle\quad\implies\quad\psi_i=a_{ik}\psi_k$$

then how should I proceed? Namely, what are my $$x_i$$ and $$x_j$$ in the vector space of quantum states?

I wonder if you are confusing several things here. Vectors are defined by the axioms of the vector space.

When you talk about vectors transforming in a certain way, I think you are talking about the vectors in the tangent space of some differentiable manifold. This an additional concept. Not all vectors must have a tangent space in which they 'live'. For example, differential one-forms (e.g. $$dx,\,dy,\,dz$$) are vectors, dual to vectors you are more used to, and they certainly do not transform in the way you describe. By definition, one-forms will do the opposite. A more life-like example is magnetic field. It is an axial vector, and as a result it transforms under reflection in a rather curious way.

As far as state-vectors transforming in a certain way. I feel like the natural way to go here is Wigner's theorem

• When I talk about vectors transforming in certain way, I mean that an N-dimensional state space is spanned by $\hat e_1,\hat e_2,...\hat e_N$ in the same way that $\mathbb{R}^3$ is spanned by $\hat x,\hat y, \hat z$. You can do coordinate transformations on the linearly independent spanning basis of an N-dimensional state space exactly as if it were $\mathbb{R}^n$. I ended up figuring out my question. I just need to write $x_i$ and $x_j$ in terms of the $\hat e_n$ instead of the $\mathbb{R}^3$ basis. Commented Nov 15, 2021 at 5:17
• Usually people don't think of state spaces as geometric spaces, but they are that. It's just weird to think of spin-up, for example, serving the same purpose in state space as $\hat x$ does in $\mathbb{R}^2$. Commented Nov 15, 2021 at 5:17
• @hodopsmith state spaces can be infinite dimensional, in which case, I think your approach to spanning them may need to be more subtle. In reply to "usually people don't think of state spaces as geometric spaces"... perhaps it is a blank in my education but I don't know this term. I get the meaning, I think, but I don't think I know the definition. Topological spaces, metric spaces, manifolds, differentiable manifolds, vector spaces, all make sense. What is a geometric space?
– Cryo
Commented Nov 15, 2021 at 8:53
• @hodopsmith, I am pretty sure that to talk about rotations in the way you talk about them you need some sort of a manifold, i.e. you need a point -set with a defined topology and then a differentiable map that maps this topological space into itself (which then can be rotation under some restrictions). Now each point on that manifold could have a tangent space, and could therefore be the state space you describe, but each tangent space will be completely different, unless you have some connection between them. What I am getting at is that having $N$ basis vectors does not mean you are in R^N
– Cryo
Commented Nov 15, 2021 at 8:58
• You need a lot more structure which seems unnecessary for state vectors in a Hilbert space.
– Cryo
Commented Nov 15, 2021 at 8:59

States live in a Hilbert space which is typically infinite-dimensional where standard coordinate transformations don't apply. However, unitary representations of the rotational group on elements of the Hilbert space can be applied. These representations follow the following rules ($$R$$ -- $$R_1$$ and $$R_2$$ too -- is a rotation from the rotation group SO(3)):

$$U(R) U(R)^\dagger =1 \quad\text{and}\quad U(R_1)U(R_2) = U(R_1 R_2)$$

Trivially one has also (no rotation) : $$U(1) = 1$$

However, it depends strongly on the state what effect they might have. If for instance it acts on a momentum state $$|\mathbf{p} \rangle$$ one gets:

$$U(R)|\mathbf{p} \rangle = |\mathbf{ R p} \rangle$$

This means that if

$$\hat{p}_z |\mathbf{p} \rangle = p_z |\mathbf{p} \rangle$$

then the rotated state will an eigenvector to a different eigenvalue:

$$\hat{p}_z |R\mathbf{p} \rangle = Rp_z |R\mathbf{p} \rangle$$

Of course the action of the rotation $$R$$ on $$p_z$$ is one we are used to in classical physics.

A Hilbert space is a vector space which moreover has a scalar product and is complete. So states indeed live in a vector space, but not the one we used to from classical physics. In order to explain what the vector space means I give a simple example:

One can decompose a member $$f$$ -- we assume for this very simple example $$f$$ as periodic with a period of $$2\pi$$ -- of the Hilbert space (typically a function $$\in\, L_2$$, but other mathematical spaces are possible (with differently defined scalar product)) in a Fourier series:

$$f(\phi) = \sum_{n=0,\ldots,\infty} a_n E_n \equiv \sum_{n=0,\ldots,\infty} a_n e^{in \phi}$$

Then the $$e^{in \phi}$$ are the base vectors $$E_n$$ and the $$(a_n)$$ are the components of the Hilbert space vector $$f$$. (I denoted the base vectors with a capital $$E$$ in order to avoid confusion with the $$e$$ from the exponential function). The components can, as it is well known, computed by projecting the vector $$f$$ on the different base vectors by means of the scalar product:

$$a_n = = < f, e^{in\phi}>$$

which in a Hilbert space is an integral:

$$ = \int f(\phi) \overline{g(\phi)} d\phi$$

Therefore:

$$a_n = \int_0^{2\pi} f(\phi) e^{-in\phi} d\phi$$