114

What you cannot see by drawing the picture is the velocity of the individual points of the string. Even if the string is flat at the moment of "cancellation", the string is still moving in that instant. It doesn't stop moving just because it looked flat for one instant. Your "extra" or "hidden" energy here is plain old kinetic energy. Mathematically, the ...


78

The state \begin{equation} |\Psi \rangle = \frac{1}{\sqrt{2}}\left(|\psi_1\rangle +|\psi_2\rangle \right) \end{equation} is a pure state. Meaning, there's not a 50% chance the system is in the state $|\psi_1\rangle$ and a 50% it is in the state $|\psi_2\rangle$. There is a 0% chance that the system is in either of those states, and a 100% chance the system ...


64

ACuriousMind's excellent description was missing a picture. Here it is: This clearly shows that for the wave moving to the right, the front is moving up and the rear is moving down. For the opposite wave traveling to the left, the front (now on the left) is moving down and the rear is moving up. Summing them, you get a straight line with significant ...


43

We treated this a while back at University... First of all, I assume you mean global cancellation, since otherwise the energy that is missing at the cancelled point simply is what is added to points of constructive interference: Conservation of Energy is only global. The thing is, if multiple waves globally cancel out, there are actually only two possible ...


43

Waves always travel. Even standing waves can always be interpreted as two traveling waves that are moving in opposite directions (more on that below). Keeping the idea that waves must travel in mind, here's what happens whenever you figure out a way to build a region in which the energy of such a moving wave cancels out fully: If you look closely, you will ...


41

Before reading this answer (and to those who are downvoting), I am addressing if the cat is both alive and dead. I don't think the question is asking for a complete explanation of the Schrodinger's cat experiment, nor is it asking how this links to all of the deeper mysteries of quantum mechanics and how we should think of them. Therefore, while there is ...


37

Just to complement the other excellent answers, here's an animation showing what two wave pulses with opposite amplitude passing through each other actually look like: You can clearly see that, at the instant when the string is momentarily flat, it's not stationary but rather moving quite rapidly, and thus will not stay flat for long. (Obviously, the ...


36

If a wave $f(x,t)$ is something that satisfies the wave equation $Lf=0$ where $L$ is the differential operator $\partial_t^2-c^2\nabla^2$ then, because $L$ is linear, any linear combination $\lambda f+\mu g$ of solutions $f$ and $g$ is again a solution: $L(\lambda f + \mu g)=\lambda Lf+\mu Lg=0$. In general, there might be things that propagate (not exactly ...


35

This is is known as the Wigner's friend thought experiment. According to the many World's interpretation, the superpositions are not a problem. The whole universe ends up in a superposition where all experimental outcomes are realized, but such a superposition is entangled with the environment, from a macroscopic point of view it takes the form of a ...


29

Yes, always. I would like to disagree with stafusa's answer here, expanding on Rod's comment. Interference will not occur, since for whispering the sources of sound will be statistically independent. For demonstration, let us look at two people. Person 1 produces a whisper that can be characterized by a propagating sound field $E_1(\vec{r},t)$, where $\vec{...


28

In a bubble chamber experiment, film was the detecting medium, but film was taken automatically, by the thousands of frames. These bobbins of film went to the various laboratories involved in the experiment, and were scanned for interesting events which were measured and the cross sections for the interactions recorded. This is a clear example of an ...


26

First, a historical subtlety: Schrödinger has actually stolen the idea of the cat from Einstein. Second, both men – Einstein and Schrödinger – used the thought experiment to "explain" a point that was wrong. They thought it was absurd for quantum mechanics to say that the state $a|{\rm alive}\rangle+b|{\rm dead}\rangle$ was possible in Nature (it was ...


26

As coconut wrote, the superposition principle comes from the linearity of the operator involved. This is the case for electromagnetic radiation in vacuum. Approximations to water waves are also linear (since it is an approximation) but probably will have small non-linear parts. Free quantum field theory is also linear, therefore you have a superposition ...


25

Basically the answer is yes, the cat is both dead and alive. People used to discuss this sort of thing in terms of the Copenhagen interpretation (CI) and the Many-Worlds interpretation (MWI), but those discussions tend not to be satisfying, because both CI and MWI are designed so that in almost all real-world measurements, they give the same predictions. A ...


24

Interference requires exactly same frequency in both the sources and also needs them to be coherent i.e. their phase relation must remain same throughout. It's very hard to create such things for macroscopic water bodies. Nevertheless in laboratory environment, you can see perfect interference in water waves.


22

One of the common misconceptions that people starting out with QM often have is to think that a system is either in a superposition state or it is not. Actually superposition is only defined relative to a particular basis (such as the eigenstates of some observable). If a system is in a state of superposition relative to one basis it is always possible to ...


20

The amplitude of the sum of $1000$ equally loud uncorrelated noises will be about $\surd1000$, or approximately $32$, times the amplitude of a single noise. That might be enough to make an inaudible whisper just audible. Consider the practicalities, though. The people cannot all occupy the same spot. If dispersed, most of them will be too far away to hear. ...


20

A particle isn't located at a point. Particles are always delocalised over a non-zero volume of space, so while the expectation value of a particle's position is a point, the particle is not located at that point. For a particle to be located at a point its wavefunction would have to be an eigenfunction of the position operator, i.e. a Dirac delta, but ...


19

In this case it's probably best to be pragmatic. A pulse can be described as a superposition of sine waves that extend infinitely into space and time. But it's just that: a mathematical description that is useful for your purposes. There is not necessarily a physical meaning connected to it. Nevertheless, in quantum mechanics the wave-description of ...


19

To answer the question, you have to specify what observable’s eigenstates the “superpositions” you’re talking about are superpositions of. If you measure the energy, you always get one single energy eigenstate, not a superposition of energy eigenstates. The same applies for any other observable quantity. But that one single energy eigenstate might be a ...


17

Just in case anyone (e.g. student) would be interested in the simple answer for mechanical waves: CASE 1 (global cancellation): Imagine that you have crest pulse moving right and equally large though pulse moving left. For a moment they "cancel", e.g. there is no net displacement at all, because two opposite displacements cancel out. However, velocities ...


17

The simple answer is that we don't know because we have no theory of quantum gravity. If I interpret your question correctly you're thinking about semiclassical gravity, where matter is quantised and gravity isn't. The Einstein equation becomes: $$ \mathbf G = \langle \mathbf T \rangle $$ where $\mathbf G$ is the classical curvature, i.e. gravity, and $\...


17

No. Despite what several answers on this thread will tell you, there are plenty of phenomena which are perfectly deserving of the term "wave" which do not satisfy the superposition principle. In technical language, the superposition principle is obeyed whenever the underlying dynamics are linear. However, there are plenty of situations that do not obey this ...


17

I'm not sure I would have phrased it exactly that way, but I think his statement is by-and-large defensible. The crux of the issue is that, in liquid water, there is no sharp line between intramolecular and intermolecular H-O bonding. The intramolecular H-O bond is closer than the intermolecular H-O hydrogen bond, but it's a difference of degree not kind, ...


16

The sentence of Wikipedia : "For example, there may be a 50% probability that the state vector is $| \psi_1 \rangle$ and a 50% chance that the state vector is $| \psi_2 \rangle$ . This system would be in a mixed state." is false. The difference between pure states and partially or completely mixed states, is only a difference of structure of the ...


16

You are starting from the incorrect point. The argument follows by linearity of the equation. Suppose $\Psi_k(x,t)$ is solution of the time dependent Schr$\ddot{\hbox{o}}$dinger equation: $$ i\hbar \frac{\partial }{\partial t}\Psi_k(x,t)=-\frac{\hbar^2}{2m}\frac{\partial^2\Psi_k(x,t)}{\partial x^2}+U(x)\Psi_k(x,t)\, . $$ Then: $$ \Phi(x,t)=a_1\Psi_1(x,t)+...


16

I feel like all the answers here are missing the point. The cat is not both alive and dead at the same time. That would be, as you put it, ludicrous. The truth is that the cat is in a superposition state of the states "alive" and "dead". The problem is that there is no way to make sense of this statement without studying the underlying mathematics. Humans ...


15

Apart from the already mathematically detailed answers given above, perhaps it would be useful to have a physical picture in mind -- the double slit experiment. The classical 50:50 picture corresponds to the case where you send, at random i.e. 50% chance, through either one of the slits. This will result in no interference pattern on the receiving screen. ...


15

I think it's tosh. A very small fraction of liquid $\rm H_2O$ will spontaneously decompose and recombine with very small amount of $\rm H_3O^+ $ (hydronium ion) and $\rm OH^-$ (hydroxyl) ions. Weak hydrogen bonds may form between neighboring water molecules, but that doesn't really count as $\rm H_4O_2$ or higher order molecules. While I have a great deal ...


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