Does wave function get updated instantly as environment changes?

[Happiness]

TL;DR Summary

Does the wave function get updated instantly, or is there a lag?

Suppose we have in a box a particle that is travelling left and right at some speed, bouncing off the walls of the box. The wall on the right is then removed such that the particle would be free to escape the box.

Does the wave function of the particle get "updated" instantly the moment the wall is removed, or does it take some time for the wave function to be updated (eg, after the time for light to travel from the right wall to the particle's position has elapsed)?

 

[Dale]

TL;DR Summary: Does the wave function get updated instantly, or is there a lag?

eg, after the time for light to travel from the right wall to the particle's position has elapsed

The particle doesn’t have a definite position unless you measure it.

 

[Happiness]

The particle doesn’t have a definite position unless you measure it.

Suppose I measured it and it now has a definite position. Then the environment changes.

Would the wave function change instantly? Or does it take some time for the wave function to change (eg, the time for light to reach the particle from the location of the environmental change)?

 

[Demystifier]

It takes time.

 

[PeterDonis]

Does the wave function of the particle get "updated" instantly the moment the wall is removed, or does it take some time for the wave function to be updated (eg, after the time for light to travel from the right wall to the particle's position has elapsed)?

This question is interpretation dependent. On some interpretations, the wave function is a physical thing and it has to get physically updated; an answer like the one given by @Demystifier would be reasonable for such an interpretation.

On other interpretations, the wave function is just a mathematical description and there is no physical process of "updating the wave function", so the question you are asking would not even be well defined.

 

[PeterDonis]

Moderator's note: Thread moved to interpretations subforum.

 

[Demystifier]

This question is interpretation dependent. On some interpretations, the wave function is a physical thing and it has to get physically updated; an answer like the one given by @Demystifier would be reasonable for such an interpretation.

I think it doesn't depend on interpretation. The "update" that OP talks about is not caused by an observation, but by a physical removal of the wall. All interpretations agree that in this case you just need to solve the time-dependent Schrodinger equation with initial condition given by the wave function before the removal.

 

[PeterDonis]

The "update" that OP talks about is not caused by an observation, but by a physical removal of the wall.

The removal of the wall is a physical process independent of any interpretation, but that doesn't mean "wave function update" is a physical process independent of any interpretation. In a statistical interpretation (e.g., the one used in Ballentine), the wave function doesn't even describe an individual quantum system to begin with, so asking what happens to it when something is done to that individual quantum system makes no sense.

 

[PeterDonis]

All interpretations agree that in this case you just need to solve the time-dependent Schrodinger equation with initial condition given by the wave function before the removal.

Yes, but this is math and does not make any claims that whatever "wave function update" is described by the math must be a physical process. Only certain interpretations do that.

 

[Demystifier]

The OP just uses the wrong word "update".

 

[PeterDonis]

The OP just uses the wrong word "update".

I don't think there is any right word that describes a purported physical process that happens to the wave function independent of any interpretation. There can't possibly be one since not all interpretations even treat the wave function as a thing that a physical process can happen to at all.

 

[Happiness]

This question is interpretation dependent. On some interpretations, the wave function is a physical thing and it has to get physically updated; an answer like the one given by @Demystifier would be reasonable for such an interpretation.

On other interpretations, the wave function is just a mathematical description and there is no physical process of "updating the wave function", so the question you are asking would not even be well defined.

Suppose a particle is in an enclosed box for 10 seconds. At the 10th second (t=10s), the right wall of the box is removed. Suppose it takes 5s for light to travel from the right wall to the particle.

Then for the period between t=10s and t=15s, if we want to get the particle’s position probability density function, do we use the old wave function or the new one? (The new one is the wave function where the right wall is not there.)

 

[Arjan82]

If you insist that the wall is removed instantly, then you're introducing a nonphysical event into your problem. You cannot expect the answer to be physical, so in that case the update of the wave function is instantaneous. That is nonphysical, as is the instantaneous removal of the wall.

However, if you are removing the wall in some physical way then the wave function just smoothly adjust to the new boundary conditions as it itself is a time dependent function.

That being said, if a measurement is performed, the wave function does need to be updated instantaneously. According to Sabine Hossenfelder this is what Einstein called 'spooky action on a distance'. So according to her, this has noting to do with entanglement per se, but with the need to instantaneously update the wave function after a measurement.

 

[PeterDonis]

Suppose a particle is in an enclosed box for 10 seconds. At the 10th second (t=10s), the right wall of the box is removed. Suppose it takes 5s for light to travel from the right wall to the particle.

Then for the period between t=10s and t=15s, if we want to get the particle’s position probability density function, do we use the old wave function or the new one? (The new one is the wave function where the right wall is not there.)

Note that this question is different from the question you asked in your OP, because it is clear that it is only asking what math we should use to make predictions about probabilities.

The answer @Arjan82 gave in post #13 is fine (although it is non-relativistic--I'll make a separate post about that shortly); however, that answer does not imply anything about the wave function being an actual physical thing, or the "update" of the wave function being an actual physical process. It's just an answer about what math to use to make predictions about probabilities.

 

[PeterDonis]

the wave function just smoothly adjust to the new boundary conditions as it itself is a time dependent function.

In this way of describing it, you are assuming non-relativistic QM, and in that framework, the wave function throughout the box adjusts instantly to a change in the boundary conditions; there is no "light speed time lag".

In relativistic QM, i.e., quantum field theory, things are more complicated because there is no wave function; you have quantum field operators at different spacetime events, and you have to consider the entire spacetime configuration when computing probabilities using those operators. That computation should show a "light speed time lag" in how the effects of removing the wall propagate (although I have not done the computation), but those results will not necessarily be easily describable in terms like "adjusting to new boundary conditions".

 

[Happiness]

In ... non-relativistic QM, the wave function throughout the box adjusts instantly to a change in the boundary conditions; there is no "light speed time lag".

In relativistic QM, ... computation should show a "light speed time lag" in how the effects of removing the wall propagate.

So non-relativistic QM and relativistic QM give different predictions even when the particle is not moving at relativistic speeds? I thought their predictions would agree at low speeds.

Anyway, for what really happens in the real world, there would be a "light-speed time lag" (in the particle's probability density function or in the wave function to be used for calculating probabilities) when the environment changes (since relativistic QM is the more accurate model for the real world)?

 

[PeterDonis]

So non-relativistic QM and relativistic QM give different predictions even when the particle is not moving at relativistic speeds?

The predictions will be close enough to be indistinguishable in practice for low speeds; but the fact that your question mentioned light speed time delay implied, to me, that you wanted to consider cases where the particle is not moving at low speed. For low enough particle speeds, the light speed time delay is negligible, which is why the non-relativistic model gives predictions close enough to the relativistic model to be indistinguishable in practice.

for what really happens in the real world, there would be a "light-speed time lag" (in the particle's probability density function or in the wave function to be used for calculating probabilities) when the environment changes (since relativistic QM is the more accurate model for the real world)?

How accurately we can measure "what really happens in the real world", and whether such measurements will show any "light speed time lag", will depend on the specific scenario. As I said above, for low enough particle speeds the "light speed time lag" will be negligible in practice.

 

[Morbert]

TL;DR Summary: Does the wave function get updated instantly, or is there a lag?

Suppose we have in a box a particle that is travelling left and right at some speed, bouncing off the walls of the box. The wall on the right is then removed such that the particle would be free to escape the box.

Does the wave function of the particle get "updated" instantly the moment the wall is removed, or does it take some time for the wave function to be updated (eg, after the time for light to travel from the right wall to the particle's position has elapsed)?

This sounds like a scenario where, after some time T, the Hamiltonian (not the wavefunction) is updated and the wavefunction proceeds to evolve under this new Hamiltonian. If you have python installed, the attached script is my interpretation of your Q. It simulates a particle in a finite well, and plots the density associated with the wavefunction. After a fixed time, one side of the well is instantaneously shifted to the right. Note that this does not induce an instantaneous change in the wavefunction. Instead, the wavefunction smoothly evolves into the new space available.

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation


def solve_schrodinger_dynamic_step():
    hbar = 1.0
    m = 1.0
    L = 10.0

    V0 = 10.0
    step_left = 1.0
    step_right = 5.0
    step_right_initial = 5.0
    step_right_final = 9.0

    N = 8192
    dx = L / N
    x = np.linspace(0, L, N)

    dt = 0.005
    t_steps = 10000
    transition_point = int(t_steps * 0.05)

    k = 2.0 * np.pi * np.fft.fftfreq(N, dx)

    sigma = 0.7
    x0 = (step_left + step_right) / 2
    p0 = 0.0
    psi = np.exp(-((x - x0) ** 2) / (4 * sigma**2)) * np.exp(1j * p0 * x / hbar)
    psi = psi / np.sqrt(np.sum(np.abs(psi) ** 2) * dx)

    T = np.exp(-1j * hbar * k**2 * dt / (2 * m))

    psi_t = []
    V_t = []

    for t in range(t_steps):
        if t < transition_point:
            step_right = step_right_initial
        else:
            step_right = step_right_final

        V = V0 * (
            1 - 0.5 * (np.tanh((x - step_left) * 5) - np.tanh((x - step_right) * 5))
        )

        psi = np.fft.ifft(T * np.fft.fft(psi))
        psi = np.exp(-1j * V * dt / hbar) * psi

        if t % 100 == 0:
            psi = psi / np.sqrt(np.sum(np.abs(psi) ** 2) * dx)

        if t % 10 == 0:
            psi_t.append(np.abs(psi) ** 2)
            V_t.append(V)

    return x, np.array(psi_t), np.array(V_t)


x, psi_t, V_t = solve_schrodinger_dynamic_step()

fig, ax1 = plt.subplots(1, 1, figsize=(10, 8))
fig.tight_layout(pad=3.0)

(line1,) = ax1.plot([], [], "b-", label="|ψ|²", linewidth=2)
(potential_line,) = ax1.plot([], [], "r--", label="V(x)/20", alpha=0.5)
ax1.set_xlim(0, 10)
ax1.set_ylim(0, 0.8)
ax1.set_xlabel("Position", fontsize=12)
ax1.set_ylabel("Probability Density", fontsize=12)
ax1.set_title("Dynamic Finite Well: Time Evolution", fontsize=14)
ax1.legend(fontsize=10)
ax1.grid(True, alpha=0.3)

potential_height = []


def animate(frame):
    line1.set_data(x, psi_t[frame])
    potential_line.set_data(x, V_t[frame] / 20)
    potential_height.append(np.max(V_t[frame]))

    return line1, potential_line


ani = FuncAnimation(fig, animate, frames=len(psi_t), interval=50, blit=True)

plt.show()

 

[PeterDonis]

the Hamiltonian (not the wavefunction) is updated and the wavefunction proceeds to evolve under this new Hamiltonian

That is what would happen in a non-relativistic model, yes; the Hamiltonian would change because the potential energy changes when the wall is removed, and that would change the time evolution of the wave function.

 

[Fra]

I think the statistical layer here is also essential, although this is also an interpretational issue, but at least some interpretations to associate the wavefunction to statistical information relative some "observational context", wether that is the macroscopical environment or agent or something else is further interpretational dependent.

But regardless of the details, one can also ask, even if it's just a hamiltonian update; at what "speed" is the "observational context" updated on the hamiltonian? Does the context KNOW on beforehand that we have a time dependent hamiltonian? or is it a "surprise"; if so how is the observer informed? Consider that this "wall" is something at subatomic level. Then some kind of quantum process tomography is the normal method do infer this in the statistical interpretaton. This definitely is not instant. Even in som qbist inspired information processing agent interpretaion, this takes time. Ie. how quickly is the agents "expectation" updated (via internal processes?) in the light of NEW unexpected information?

So to answer the question I think we need to qualify: HOW is the "observational context" (in SOME interpretation, there are many) "informed" about the change of the hamiltonian?

IF the premise is just a hypothetical consideration from some "mathematics perspective", then the question is not physical and can't be answerd in any interpretation I think? As there is no real "time" involved in such mathematical perturbations.

/Fredrik