B The Nature of Measurement and Interaction in Quantum Mechanics

[djsourabh]

I am still confused about the difference between measurement and interaction. I mean when electrons are traveling from source to the screen through the slits, there are air molecules in their way. And even if the electron double slit experiment is carried out in total vacuum in a completely closed box, there will still be infrared photons present in that box's volume due to temperature (Stefan's Law & E=hf gives significantly large number of infrared photons per unit volume inside the box). And further the quantum Foam (Heisenberg's uncertainty principle :-E & t pair) will be there to interact with the electrons. Then why these Interactions mentioned above do not collapse the wave function & why does it collapse when we try to "Measure " electrons position??

 

[stevendaryl]

I am still confused about the difference between measurement and interaction. I mean when electrons are traveling from source to the screen through the slits, there are air molecules in their way. And even if the electron double slit experiment is carried out in total vacuum in a completely closed box, there will still be infrared photons present in that box's volume due to temperature (Stefan's Law & E=hf gives significantly large number of infrared photons per unit volume inside the box). And further the quantum Foam (Heisenberg's uncertainty principle :-E & t pair) will be there to interact with the electrons. Then why these Interactions mentioned above do not collapse the wave function & why does it collapse when we try to "Measure " electrons position??

Here's the way that I would define it: A measurement is a special kind of interaction in which a microscopic property of one system is amplified to make a macroscopic difference in another system (the measuring device).

 

[djsourabh]

Thanks for the prompt response. Your answer is general. Can you specify it for electron double slit experiment?

 

[stevendaryl]

Thanks for the prompt response. Your answer is general. Can you specify it for electron double slit experiment?

Well, in a double slit experiment, you only see the diffraction pattern if there is interference between the two possible paths for the electron (through one slit or the other). You only have interference between alternatives if both alternatives are consistent with the same final state.

So taking into account the environment (stray photons, air, etc.), what this means is this:

Let $L$ be the state of the environment caused by the electron going through the left slit. Let $R$ be the state of the environment caused by the electron going through the right slit. Then in order for there to be interference, there must be a neutral state $N$ for the environment such that it is possible for the environment to make the transition from $L$ to $N$ and also from $R$ to $N$. In other words, the effects of the electron on the environment in transit must be reversible. The final state, $N$ has to have no information left as to which way the electron went.

I don't think that there is any definitive calculation that determines whether an interaction will lead to a destruction of the interference pattern. There are certainly cases where you can say that it definitely DOES destroy the interference pattern, such as if the electron collides with a photon, and that photon is captured by a photographic plate, then that would give definite information about where the electron was. If it's a collision with a "soft" low-energy photon, I think it's a little less clear whether it will be enough to destroy the interference pattern.

 

[atyy]

The difference between measurement (when the Born rule is applied) and interaction (when two quantum systems interact as described by the Hamiltonian and the quantum state evolves unitarily according to the Schroedinger equation) is fundamental and subjective.

A measurement occurs when a definite outcome or experimental result is observed. Notionally, we divide the universe into a "classical" (or "macroscopic") part and a "quantum part". A measurement occurs when the classical measuring apparatus interacts with the quantum system to produce a definite outcome.

The boundary between the classical part and the quantum part is (partially) subjective, and can be shifted. If a measuring apparatus is included in the quantum part instead of the classical part, then we need yet another classical measuring apparatus to measure the quantum measuring apparatus. Without a classical measuring apparatus, quantum mechanics makes no predictions.

This divide between classical and quantum parts is obviously unsatisfactory, and for this reason, one says that the quantum state may not necessarily be "real", but it is nonetheless an accurate tool for predicting the probabilities of measurement outcomes, which are real.

The problem of getting rid of the divide between classical and quantum parts is called the measurement problem, which remains unsolved. Prominent approaches to the measurement problem are Bohmian mechanics and the Many Worlds Interpretation.

 

[Fra]

I agree with the responses given but would like to add one more characterisation, that even suggests a path forward.

Instead of thinking of them as different things, the same process can be both an interaction and a measurement dependong on which observer that describes it. And to investigate the disagreement between any two observers, requires a third observer. This way new interactions can be created, as you increase the complexity of the third observer (corresponds to energy scaling), they new inferactions are "explained" as observer-observer disagreements that are found to form new observer equivalence classes as you lower the observational energy scale.

The problem is that this picture does not fully mate with quantum mechanics as it stands, as the notion of "observers" in QM, refers to classical measurement devices, and by assumption there is not observer-observer disagreements in the classical realm, except those relating to the equivalence of inertial and accelrating classical observers, which is partly solved.

So the remaining problem is, what happens when the "observer" is so light, that it no longer can be reasonably considered as a "classical device"? This is from my perspective where we hit the same problems that we face in unification and QG. One also faces a compication of the running energy scale as you scale the observer. This is deeply related to the problem with mixing quantum theory with cosmological theories as well. In effects it proposes a fundamental feedback loop between UV and IR where UV divergences probably shouldt ever happen once the calculation is done the right way. There is a lot of really interesting stuff going on here that i think relates to the measurement problem, and even thinks like possible dependence on dark energy on the observational scale.

/Fredrik

 

[Demystifier]

I am still confused about the difference between measurement and interaction. I mean when electrons are traveling from source to the screen through the slits, there are air molecules in their way. And even if the electron double slit experiment is carried out in total vacuum in a completely closed box, there will still be infrared photons present in that box's volume due to temperature (Stefan's Law & E=hf gives significantly large number of infrared photons per unit volume inside the box). And further the quantum Foam (Heisenberg's uncertainty principle :-E & t pair) will be there to interact with the electrons. Then why these Interactions mentioned above do not collapse the wave function & why does it collapse when we try to "Measure " electrons position??

Let the initial state before interaction be
$$|\Psi\rangle=|\psi\rangle |\phi\rangle$$
where $|\psi\rangle$ is the state of the electron and $|\phi\rangle$ the state of the air. After the interaction the state typically takes the entangled form
$$|\Psi'\rangle=\sum_a c_a |\psi_a\rangle |\phi_a\rangle$$
where $c_a$ are such that the initial electron state is $|\psi\rangle=\sum_a c_a |\psi_a\rangle$. We say that this interaction "measures" the electron if
$$\langle\phi_a|\phi_b\rangle \approx 0 \;\; {\rm for} \;\; a\neq b $$

 

[PeterDonis]

According to me

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