# Tag Info

14

What you describe in your question is the "Copenhagen interpretation" of quantum mechanics. There are more nuanced views of this nowadays that don't treat "measurements" quite so asymmetrically, see e.g. sources that talk about decoherence. I recommend watching the classic lecture "Quantum Mechanics in your face" by Sidney Coleman for a nice take on this ...

12

Well, I think you said the answer yourself when you used the words "projection operator." In the Heisenberg picture the operators get projected down to a subspace at the time of the collapse. In other words, the operator 'collapses' by picking up a projection piece that kills the unphysical part of the state. Forget about pictures for a second, the physical ...

11

Assuming wave-function collapse is correct (which can be a relatively hefty philosophical claim in some circles), then think of measurement this way: When you measure an observable on a system, you collapse the wave-function of the system into a Dirac delta function in the eigenbasis for that observable. If you measure position, you get a delta function in ...

10

Dear Jack, there is no physical phenomenon that could be called the collapse. The collapse of the wave function, as first emphasized by Werner Heisenberg and then many others, is just the event when we learn something about a physical property of a physical system. When we learn that Osama bin Laden is located in a building in Pakistan, his wave function - ...

8

Interactions merely involve a correlation developing. For example, if an electron is put through a Stern-Gerlach apparatus, a correlation develops between the distance travelled in the x direction and the distance deviated in the y direction. It is reversible. The measurement which occurs when the particle hits the photographic plate is irreversible. It ...

8

There are two different issues. One of them is the sign of the momentum; the other one is whether the momentum is spread (it's not because of the unnatural boundary conditions). Concerning the first point, the standing wave (sine) is a real function and every real wave function has the same probability to carry momentum $+p$ and $-p$. So indeed, both of ...

7

Assuming that the incoming "first" particle is prepared in a pure state, interaction with another particle does seem necessary. Such an interaction might simply be the spontaneous emission of a photon or other particle by the original incoming particle, however. Most importantly, such an interaction is not itself sufficient. For a measurement event to ...

7

There are currently two different accounts that give a larger picture of what happens when a quantum system is measured. One of them is the fact that many random interactions between the system (which might be a 1-body or N-body quantum system) and the environment (which is considered for most purposes a pseudo-classical system with infinite degrees of ...

7

An observation is an act by which one finds some information – the value of a physical observable (quantity). Observables are associated with linear Hermitian operators. The previous sentences tautologically imply that an observation is what "collapses" the wave function. The "collapse" of the wave function isn't a material process in any classical sense ...

6

I'll start with the second one. $\int\phi^\ast\psi\,\mathrm{d}x$ is, as Chris says in the comments, the scalar (or dot) product of $\phi$ and $\psi$. In the Dirac notation, it is written as $\langle\phi|\psi\rangle$ and it gives the overlap of the two wavefunctions. In other words, it gives the probability amplitude (i.e., what you call square root of ...

5

The collapse of the wavefunction is generally attributed to decoherence. This is time asymmetric in the same way the second law of thermodynamics is time asymmetric. I suppose it's theoretically possible for a wavefunction to uncollapse, but this is like saying it's theoretically possible for a broken egg to reassemble itself.

5

Unitary operators are operators that satisfy some conditions. Among other things, they have to be linear: http://en.wikipedia.org/wiki/Unitary_operator The operation (or "an operation") that maps any $\psi(x)$ to $\delta(x-x_0)$ where $x_0$ is the random position resulting from a measurement can't be associated with any linear operator. It's easy to ...

5

Answer Rigorous adherence to the liturgical rituals of the "Church of the Larger Hilbert Space" is feasible in principle yet exponentially inefficient in practice. Exercise One way to answer this question is by reference to a feasible numerical computation. So fire-up MatLab; specify the dynamical system as (say) $n\sim 10$ interacting qubits; specify ...

4

Much has been covered in these answers, but one aspect has been left out. The actual physics going on in any measurement process includes amplification. Feynman thought this was significant. Here is a perhaps little-known quotation of his: We and our measuring instruments are part of nature and so are, in principle, described by an amplitude function ...

4

1) Let there be given a Hilbert space $H$ and a mixed state described by a density operator $\hat{\rho}:H\to H$, which is a positive operator $\hat{\rho}\geq 0$, and with trace ${\rm Tr}(\hat{\rho})=1$. 2) Let $V\subseteq H$ be an eigenspace of states for an Hermitian observable $\hat{A}:H\to H$ with eigenvalue $\lambda\in\mathbb{R}$. (We will ignore ...

4

I'll take a stab at this though my answer may be incomplete / fuzzy: The double slit experiment demonstrates wave-particle duality, not entanglement. It shows that a "particle" can interfere with itself, demonstrating that it really acts as a wave in this instance. Entanglement is correlation of measurements of particles (most commonly) that were generated ...

4

An electron, indeed any particle, is neither a particle nor a wave. Describing the electron as a particle is a mathematical model that works well in some circumstances while describing it as a wave is a different mathematical model that works well in other circumstances. When you choose to do some calculation of the electron's behaviour that treats it either ...

4

In the following answer I am going to refer to the unitary evolution of a quantum state vector (basically Schrodinger's Equation which provide the rate of change with respect to time of the quantum state or wave function) as $\mathbf{U}$. I am going to refer to the state vector reduction (collapse of the wave function) as $\mathbf{R}$. It is important to ...

4

Let me take a slightly more "pop science" approach to this than Luboš, though I'm basically saying the same thing. Suppose you have some system in a superposition of states: a spin in a mix of up/down states is probably the simplest example. If we "measure" the spin by allowing some other particle to interact with it we end up with our original spin and the ...

4

I think you've misconstrued the Schrodinger picture. The Schrodinger and Heisenberg pictures are physical theories that make testable predictions, are strictly mathematically equivalent to each other, and are unitary. Neither theory says anything about wavefunction collapse. Collapse (of the wavefunction or of an operator) is a feature of a particular ...

4

Leaving aside the quantum measurement problem (i.e. whether or not there is a "collapse" of quantum state to an eigenstate of an observable on measurement) and talking wholly about quantum state between "measurements" and its unitary evolution, I'd say that the transition is definitely a smooth shifting from one "eigenstate" to another, so that the ...

4

You already have some very erudite and good answers. I will give an experimentalist's point of view: I'm wondering, when an electron changes state, does it move from one state to another over some (very small) time period? Or does it change from one state to another in no time? An electron is par excellence an elementary particle and it is quantum ...

3

An often good on-line source for the interpretation of quantum theory is the Stanford Encyclopedia of Philosophy, which has a page on "collapse theories". There is a lot of literature on whether one needs collapse if one takes the wave function seriously, as opposed to the mainline Physicist's approach of taking a more empiricist view, as outlined well by ...

3

"Collapse the waveform" is a loaded term, that would not be agreed to by all physicists. There are a great many "no-collapse" interpretations out there in which there is no special role for measurement that directly alters the wavefunction. There are also collapse-type interpretations in which the collapse happens more or less spontaneously, as in Roger ...

3

As suggested in the answer above, in general, decoherence increases the entropy associated with a quantum system and as such has the same type of time-reversal asymmetry that appears in thermodynamics. The question, however, is also concerned with how an "uncollapse" would look like. Here I want to illustrate how this can be done in principle. The net ...

3

In terms of quantum field theory, an electron is an elementary excitation of the electron field, in a similar way as a water wavelet is an excitation of a water surface. There is a slight difference, though, as excitations of a quantum field are quantized, hence come in discrete bunches (1,2,3,... electrons) describing their size, while water wavelets are ...

3

The double slit experiment actually has nothing to do with entanglement by itself. Photons are generated in processes that are quantum in nature, and their direction that they come out in is subject to some randomness. Photons will then be in a state of superposition with respect to their direction. When a photon goes through the double slit, it goes ...

3

I have been looking at what is available of the book online, and believe that you are right that the Postulate 3 is (in a sense discussed below) weaker than the usual QM Projection Postulate. Firstly there are some notational issues here. The $M_m$ are a family of operators, called Measurement Operators, indexed by $m$ which is a label for the outcome ...

3

Much of how you answer this question comes down to your view of the wavefunction or state. If you think that the quantum state is a state of reality (that is, an ontic state), then you must either reproduce the predictions of orthodox (Copenhagen) QM without the measurement postulate or you must explain why nature provides two forms of evolution. The former ...

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