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, both the wave function and the hamiltonian are elements 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 $$\Psi:S\rightarrow \mathcal{H}$$ or alternatively, $$\hat{H}:S\rightarrow \mathcal{H}$$ For these equations to be macroscopically useful or instructive. Moreover, these correspondences must be well-defined... however, i don't see these correspondences even vaguely defined anywhere. Also, if both the wave function and the hamiltonian 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!

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!

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 $$\lambda=\left\{f(t)| f(t_i)=p_i ans f(t_f)=p_f; \quad t_i,t_f\in \mathbb{R}\right\}$$ 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, both the wave function and the hamiltonian are elements 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 $$\Psi:S\rightarrow \mathcal{H}$$ or alternatively, $$\hat{H}:S\rightarrow \mathcal{H}$$ For these equations to be macroscopically useful or instructive. Moreover, these correspondences must be well-defined... however, i don't see these correspondences even vaguely defined anywhere. Also, if both the wave function and the hamiltonian 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!

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, both the wave function and the hamiltonian are elements 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 $$\Psi:S\rightarrow \mathcal{H}$$ or alternatively, $$\hat{H}:S\rightarrow \mathcal{H}$$ For these equations to be macroscopically useful or instructive. Moreover, these correspondences must be well-defined... however, i don't see these correspondences even vaguely defined anywhere. Also, if both the wave function and the hamiltonian 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!