# Newtonian mechanics as an approximation of Quantum Mechanics

It is said that Newtonian mechanics is an approximation of Quantum Mechanics. I am trying to understand this with below example. Is this the right way of thinking.?

A particle at rest is applied some force. As per Newtonian mechanics, it will move in the direction of force and if I have to find its location after say 5 secs, I can find that. For the same particle if we apply Quantum mechanics and try to find its location after 5 secs, I am not going to get exact its location. It will tell me it has probability of being found everywhere including on moon (of course the probability will be different at different places) but the maximum probability would be where Newtonian mechanics said It would be after 5 sec. Is this how Newtonian mechanics is approximation of Quantum mechanics

• There are more in-depth ways to understand that, but your example is close. The average (not maximum probability) position in QM follows Newtonian trajectories. That is from something called Ehrenfest's theorem. – doublefelix Nov 28 '18 at 18:49
• Thanks. You are right. I missed the average bit. But what will Quantum mechanics say about the probability at the place where Newtonian mechanics predicted it should be there.? Will it have more probability than anywhere else? – user31058 Nov 28 '18 at 19:03
• Not necessarily, and not even usually. That would require the peak to always be the average but you can have stuff like probability density proportional to $cos^2(x)$ for a particle which has no probability of being outside the interval $[0,\pi]$ Then the average position is $x=\pi /2$, but this is actually the position of least probability: it has zero probability! See graph of this function here if you want: wolframalpha.com/input/?i=cos%5E2(x) – doublefelix Nov 28 '18 at 20:50

## 1 Answer

It is said that Newtonian mechanics is an approximation of Quantum Mechanics. I am trying to understand this with [the example described in the question]. Is this the right way of thinking?

This question might look simple, but it is surprisingly deep when studied carefully. A couple of key points were already mentioned in the comments posted earlier by @doublefelix. I'll elaborate on those points here, adding more context and quantitative detail.

Ehrenfest's theorem was already mentioned in a comment. To illustrate Ehrenfest's theorem and its limitations, consider a typical single-object model in which the observables $$X_k$$ and $$P_k$$ correspond to the $$k$$-th component of the object's location and momentum, respectively, Ehrenfest's theorem says $$\frac{d}{dt}\langle X_k\rangle = \frac{\langle P_k\rangle}{m} \tag{1}$$ and $$\frac{d}{dt}\langle P_k\rangle = -\big\langle \nabla_k V(\mathbf{X})\big\rangle \tag{2}$$ where $$\mathbf{X}=(X_1,X_2,X_3)$$, where $$m$$ is the object's mass, and where $$V(\mathbf{x})$$ is the "potential" that, in the classical version, would give the force on the object as $$-\nabla V$$. The notation $$\langle\cdots\rangle$$ is an abbreviation for $$\langle \cdots\rangle \equiv \frac{\langle\psi|\cdots|\psi\rangle}{\langle\psi|\psi\rangle} \tag{3}$$ where $$|\psi\rangle$$ represents the state of the system.

Equations (1)-(2) look similar to the equations describing a classical object with location $$\langle \mathbf{X}\rangle$$ and momentum $$\langle \mathbf{P}\rangle$$, but with an important difference on the right-hand side of equation (2). For the classical approximation to work well, this condition needs to be satisfied: $$\big\langle \nabla_k V(\mathbf{X})\big\rangle \approx \nabla_k V\big(\langle \mathbf{X}\rangle\big). \tag{4}$$ The approximation (4) can be justified if the object's location remains relatively sharply defined compared to the scale over which $$V(\mathbf{x})$$ varies — that is, if the quantity $$(\Delta X_k)^2\equiv \Big\langle\big(X_k-\langle X_k\rangle\big)^2\Big\rangle \tag{5}$$ remains small enough. The uncertainty principle $$(\Delta X_k)(\Delta P_k)\geq\frac{\hbar}{2} \tag{6}$$ puts a limit on how tightly the object can remain localized for how long. We can make $$\Delta X_k$$ arbitrarily small initially, but then (6) says that $$\Delta P_k$$ must be large, which means that the object's state will necessarily involve a superposition of a large spread of momenta, which causes it to become delocalized quickly. For the simplest case $$V=0$$, the following table gives some numeric examples for "Gaussian states," which saturate the inequality (6), making the left-hand side as small as possible and with the smallest possible rate of delocalization. The last column in the table indicates how long (order-of-magnitude only) the object remains localized with $$\Delta X< 1$$ millimeter, given that it was initially localized to the width of an atom: $$\begin{array}{l|l|l|l} \text{Mass }m & \text{Initial width} & \text{Final width} & \text{Elapsed time} \cr \text{(kg)} & \text{(meters)} & \text{(meters)} & \text{(seconds)}\cr \hline 10^{-3} \text{ (pill)} & 10^{-10} \text{ (atom)} & 10^{-3} & 10^{18} \cr 10^{-9} \text{ (speck)} & 10^{-10} \text{ (atom)} & 10^{-3} & 10^{12}\cr 10^{-30} \text{ (electron)} & 10^{-10} \text{ (atom)} & 10^{-3} & 10^{-9}\cr \end{array}$$ Note that $$10^{12}$$ seconds is more than 10,000 years. Another example where the object is a person was considered in this fun post:

Uncertainty principle for a sitting person

Here's the point: For macroscopic objects in suitably-prepared states, the approximation (4) can be excellent, and then equations (1)-(2) do indeed say that the object behaves according to classical Newtonian mechanics.

However, most states don't behave that way. (By "most states", I mean "most of the states that quantum theory allows us to construct.") One example was described in another comment by @doublefelix. Another fun example where the condition (4) necessarily breaks down is described in this paper:

That paper considers a real-world example of a chaotic system, namely the tumbling motion of Hyperion, one of Saturn's moons. The defining feature of a chaotic system is that its behavior is extremely sensitive to the initial conditions, so that even tiny changes in the object's initial location (and orientation, etc) can lead to drastic differences in its future location (or orientations, etc) in a relatively short time. For Hyperion, this timescale was estimated to be roughly 20 days. For a quantum object like the one described above, this means that no matter how we choose the object's initial state, it will inevitably become extremely delocalized on a timescale that is much shorter than how long we've been observing Saturn's moons.

That raises an obvious question: Why does Hyperion still appear to remain tightly localized as it moves, just like any other familiar macroscopic object? The answer, as explained in the paper, is that Hyperion is not an isolated object. It is constantly interacting with interplanetary molecules, cosmic background radiation, and so on. Even if we somehow managed to produce a state in which a macroscopic object like Hyperion were in a quantum superposition of two very different locations, quantum theory predicts that its location would almost immediately become "entangled" with all of those other things. This physical phenomenon, in which the object induces widespread and practically-irreversible location-dependent changes in its surroundings (interplanetery molecules, cosmic background radiation, etc) has all of the essential features of a measurement, so we can treat it just like we treat any other measurement in quantum theory. As a result of this unavoidable and perpetual measurement, the object remains effectively localized. (This phenomenon, often called decoherence, does not solve quantum theory's infamous "measurement problem." It only says that we should treat this situation the same way we treat any other measurement in quantum theory, and it tells us which observable — location in this case — has effectively been measured.)

By the way, the rate at which the object effectively becomes localized is analyzed in this post:

Planetary-sized pure quantum states

The conclusion is that it happens very, very quickly. This helps explain why there is no conflict between quantum theory and the empirical fact that Hyperion remians tightly localized as it tumbles around chaotically in its orbit.

Altogether, with caveats, Newtonian mechanics is indeed an approximation to quantum mechanics, at least for typical macroscopic objects under typical macroscopic conditions. The caveats are part of what make this question so interesting.