Are Hilbert spaces in Quantum Mechanics time-dependent?

Question

I'm a mathematician, and am trying to grasp, at least on its surface, the mathematical foundations of quantum mechanics... but there is something that seems just odd with the mathematics.

Suppose i want to model the motion of a classical system through its configuration space $K$ in terms of the real parameter $t\in\mathbb{R}$, where the system makes a path $\lambda\subset K$ between the points $p_i$ and $p_f$ in the configuration space. To correctly model the motion, i can just look for the unique function $\vec{x}\in \Omega_\lambda$ that solves a given set of differential equations: $$\Phi_k(\vec{x},\frac{\mathrm{d}\vec{x}}{\mathrm{d}t},t)=0$$ Where $\Omega_\lambda$ is the set of all smooth functions $f:\mathbb{R}\rightarrow K$ such that $$\lambda=\left\{f(t)| f(t_i)=p_i\text{ and }f(t_f)=p_f; \quad t_i,t_f\in \mathbb{R}\right\}$$ and $\Phi_k$ depends on whether the formulation is lagrangian, newtonian, hamiltonian, relativistic, etc., So it all comes down to finding $\vec{x}$.

I get that this description is as short as it is vague, but, up to some point, and up to my knowledge, it is very accurate. However, in trying to do the same for QM, a difficulty arises: no function $\vec{x}$ is ever well-defined, as stated by Heisenberg's Principle . I know, by the postuates of QM, that there are roundabouts of this problem that rely on finding solutions to the Schrödinger equation, that can be formulated as an equation $$\Pi(\Psi,\hat{H})=0$$ where $\Psi$ is the wave function of the system, and $\hat{H}$ its hamiltonian operator... But some questions arise to mind: First, the wave function is an element in a complex hilbert space $\mathcal{H}$, so we need, at the very least, a correspondence between subsets of the flat euclidean space $S\subseteq \mathbb{R}^4$ and $\mathcal{H}$, given by $$\Psi:S\rightarrow \mathcal{H}$$ For these equations to be macroscopically useful or instructive. Moreover, this correspondence must be well-defined... however, i don't see these correspondence even vaguely defined anywhere. Also, if the wave function can be time-dependant, can't $\mathcal{H}$ be time-dependant too? After all, the wave function is what defines the quantum state of a system, and hence, can also define the basis of the hilbert space. How can the configuration space of a system be time-dependant?

In general, i can't manage to find a consistent way of modeling quantum phenomena that can be made into a consistent pseudo-algorithm, like the one i descibed above for classical physics. Am i misunderstanding something? It can't be a normal perk of QM!

Answer 1

What you describe is not how QM works. The basic axioms is that states are (unit) vectors in a Hilbert space $\mathcal{H}$. The "trajectory" of a system is then a time-dependent vector $\psi(t)$, whose time evolution is given by a self-adjoint operator $H: \mathcal{H} \to \mathcal{H}$ through the Schrödinger equation

$$H \psi = i \frac{\partial\psi}{\partial t}.$$

There is no concept of a path through space; instead, observable quantities are represented by self-adjoint operators. The possible values for an operator $A$ are its eigenvalues $a_i$, and given a state $\psi$ the probability of observing the value $a_i$ is $P_i = |(\alpha_i, \psi)|^2$, where $\alpha_i$ is the eigenvector corresponding to $a_i$.

In particular, when the system is a particle moving through space, there is a position observable with eigenvectors $\{\alpha_x | x \in \mathbb{R}\}$, and the wavefunction is $\psi(x,t) = (\alpha_x, \psi(t))$; that is, the representation of the state vector in this basis.

Of course, this is by necessity a very brief description. But since your question is "what's the mathematical formalism of quantum mechanics?", a complete answer would just be a QM textbook.

Answer 2

Quantum mechanics doesn't describe systems as having a single classical path. Rather, it describes them in terms of states and observables. The state can be described by a vector or operator on a Hilbert space. The state and observables can be used to make predictions about finding a system to have a particular value or range of values of an observable. There are books about Hilbert spaces in quantum mechanics such as "Quantum Mechanics in Hilbert Space" by Eduard Prugovecki.

If a quantum system interacts with the environment it undergoes a process called decoherence and as a result the expectation values of the position and momentum act approximately like a classical trajectory:

https://arxiv.org/abs/0903.1802

https://arxiv.org/abs/1111.2189