As I understand it forces depending on velocity and position are allowed in classical physics, so a force $F{\propto}bvr+r$ or $F{\propto}\sin(brv)$ would be allowed in classical physics. As I understand it anything that could be modeled using equations in classical physics could be modeled using the Schrodinger Equation in Quantum Mechanics. Looking at the potential function in the Schrodinger equation for the hydrogen atom it looks like the $e^2$ is actually a $Qq$ in disguise, seeing as how the protons charge is exactly opposite the electrons charge, and the potential operator is inversely proportional to r, so it looks like the potential operator is the negative of the integral of the classical coulomb force. This makes it look like if in classical physics there is a bound system of two bodies interacting via a central force then the analog in the time independent Schrodinger equation is as simple as taking the negative integral of the classical central force with respect to r and then inserting that negative integral in as the potential operator.

If the classical force is something like $F{\propto}bvr+r$ or $F{\propto}\sin(bvr)$ with $r$ being distance, and $v$ being speed, then it's a bit harder to see what the quantum analog would be as with the Schrodinger equation either only depending on position, or only depending on momentum finding an analogous potential function would not be as simple as taking the integral of the classical force with respect to position when the classical force is a function of position and velocity. I thought maybe there would be some variant of the Schrodinger equation that depends both on position and momentum that I didn't know about, making finding the quantum analog of a classical force that is a function of position and velocity easier, but it looks like there's only a Schrodinger equation for position and a Schrodinger equation for momentum. It seems like if the classical forces $F{\propto}bvr+r$ and $F{\propto}\sin(brv)$ have a quantum analog it must be in the position Schrodinger equation and in the momentum Schrodinger equation and involve position operators and momentum operators.

My question is, is there a quantum analog to any classical force, even classical forces that depend on position and velocity, and if so how do you find the quantum analog to any classical force? What is the quantum analog to $F{\propto}bvr+r$ for instance?


1 Answer 1


One of the foundations of quantum mechanics is that observables are represented by Hermitian operators. So expressions for forces that depend on $x$ and $\dot x$, need to be expressed in the corresponding quantum operators for position and velocity$^1$, namely $\hat x$ and $\hat {\dot x}$

The quantum mechanical analog of velocity, can be obtained from the classical Poisson bracket $$\dot{x} = \{x, H\}$$ which becomes$^2$ $$\tag 1\hat {\dot x}=\frac{i}{\hbar}[\hat H,\hat x]$$ where $\hat H$ is the Hamiltonian and $\hat x$ is the position operator in standard quantum mechanics.

In fact, it should be okay to express any classical force $f(x,\dot x)$ in terms of the operators $\hat x$ and $\hat{\dot x}$

So the quantum analog of your force $$F\propto bvr+r$$ can be written as the operator equation $$\hat F\rightarrow \frac{bi}{\hbar}[\hat H,\hat x]\hat x + \hat x$$

$^1$ One could also use the classical formulation of momentum $$p=m\dot x$$ to define a corresponding velocity operator $${\hat {\dot x}} =\frac{\hat p}{m}$$ and so you could write the forces according to this and the position $\hat x$, but the most general definition for the velocity operator would be given by equation (1).

$^2$From the Schrodinger equation $$\dot \Psi=-\frac{i}{\hbar}\hat H\Psi$$ where $\dot \Psi$ is the time-derivative of the position space wave function.

  • $\begingroup$ I am not sure this answers the question: as I understand, the question is about a QM formulation of the EOM $m\ddot{ \mathbf x} = \mathbf F(\mathbf x,\dot{\mathbf x})$. This is equivalent to asking, if there is a Hamiltonian formulation of this problem. The case of the Lorentz-force is very well known which derives from the Hamiltonian $H= (\mathbf p - \mathbf A)^2/2m$. $\endgroup$
    – Fabian
    Commented Aug 19, 2021 at 4:33
  • $\begingroup$ Actually, the OP is asking "is there a quantum analog to any classical force" that is expressed classically in terms of x and $\dot x$, and how would one express any such force in QM. Thanks. $\endgroup$
    – joseph h
    Commented Aug 19, 2021 at 4:45

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