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Energy eigenstates are states where x and t can be separated in the time-dependent equation, so the solutions can be factored into two terms, as f(x)exp(iwt). Solutions with no potential-energy function (V(x,t) = 0) are traveling plane waves, and solutions with an infinite potential well are not dispersive. So calling the Schrödinger equation a "diffusion equation" seems misleading to me, and "wave equation" seems reasonable. I think the linear time derivative in the Schrödinger equation is misleading because its coefficient is imaginary. I think it has more in common with a real-valued second derivative from a physical or dynamical point of view, even though it superficially looks like a diffusion term. Traveling waves aren't defined in terms of oscillation, but standing waves are certainly associated with it, and classical diffusion is a completely different phenomenon, with no oscillation at all. PS: How do you edit equations here?

No. That's one good reason, and a pretty important one, but not the only reason. Any initial-condition wave function of any shape you like --not necessarily a function for which Ax+By+Cz=K (plane) is a surface of constant phase, and let it propagate freely. Eventually, it will get close to a plane wave if you leave it alone, but it never reaches that profile. It's curved and contorted for a long, long while, ever so slightly less so as time goes by, but never totally plane. It takes infinite time to do so, and then the multiplicative constant must become zero. "Plane" is what they tend to be, given enough time, but not what they are. Plane waves are extreme simplifications. Their localisation probabilities produce an infinity, so they're not the actual representation of a physical state. They're toy models. Plane waves are, eg, what the amplitude looks like in some region when you prepare the state having it go through infinitely many collimating screens, and then let it "relax" until it reaches this situation in some region of interest. OTOH, there has been extensive study of states which propagate in one direction, but package orbital angular momentum in the directions perpendicular to the propagation direction, so they're not plane waves. Look up for Bessel and Airy packets. They're very interesting, and quite a surprise when you're used to this simplifying idea that free waves are plane waves. Many people say it, but it's very old, sloppy, non-rigorous QM. We understand it better now. Another more realistic approach to a free Schrödinger wave is a Gaussian wave packet. Another one is the wave function of a particle coming out of a slit. It's never plane, although once it's got out of the slit, it's totally free. So V=0. But even more simply. Take the free Schrödinger equation: \[ i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi \] Now suppose you know, for some reason, that the momentum is in the z-direction. So you can do the separation \( \psi\left(x,y,z,t\right)=e^{-iEt/\hbar}e^{ip_{z}z/\hbar}\varphi\left(x,y\right) \). Now plug it into the time-independent Schrödinger equation: \[ -\frac{\hbar^{2}}{2m}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}\right)\psi=\frac{p_{z}^{2}}{2m}\psi \] So your Schrödinger equation splits into, \[ \frac{\hbar}{i}\frac{\partial}{\partial z}\psi=p_{z}\psi \] and, \[ \left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right)\varphi=0 \] The second one is the Laplace equation, so any harmonic function in the variables perpendicular to the selected momentum will do as a perfectly valid --and actually much more realistic-- solution to the Schrödinger equation. This is why people have been studying for some time now these very interesting states with orbital angular momentum packaged in them that I like to call --privately-- fusilli or tagliatelle electrons. They are free particles, and they are not plane waves.

You're absolutely right. +1. An image is worth a thousand words: The image is not mine, of course. It's from Wikipedia, and it represents the time evolution of a free quantum-mechanical wave packet. You can actually see how dispersive the non-relativistic regime is. The low-down is: Even empty space somehow operates as a dispersive medium for Schrödinger waves. Although you can get a similar behaviour for waves that are actually waves --same order in time and space derivatives-- by having them propagate through a dispersive material. The diffusion equation is, \[ \frac{\partial n}{\partial t}=-D\nabla^{2}n \] with D being what we call the diffusion coefficient. The Schrödinger equation, OTOH, is, \[ i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi \] So it's exactly mathematically equivalent to the evolution of a complex space-time valued function with complex values and purely imaginary diffusion coefficient, \[ D\rightarrow\frac{i\hbar}{2m} \] Whether something is a wave or not is, of course, a matter of definition. I would be happy enough with an equation that's linear in the field variable and admits travelling solutions being in some sense a wave. Travelling solutions meaning, \[ \psi\left(x,t\right)=u\left(\omega t-kx\right) \] If the equation is linear, we can do a Fourier analysis of the wave, and an arbitrary solution is a linear superposition of infinitely many travelling solutions like these. But the problem of whether our equation is dispersive or not is coded in the relation, \[ \omega\left(k\right) \] That's why it's called dispersion relation. Fourier components with different frequencies have different velocities. The velocity of propagation for each component of wave number k depends on that particular value of k. That's why the wave spreads out as it evolves. In the case of the Schrödinger equation, the dispersion relation is, \[ \hbar\omega=\frac{\left(\hbar k\right)^{2}}{2m}\Rightarrow \] so that, \[ \omega\left(k\right)=\frac{\hbar k^{2}}{2m} \] The phase velocity for Schrödinger waves being, \[ v_{p}=\frac{\omega}{k}=\frac{\hbar k}{2m} \] And their group velocity being, \[ v_{g}=\frac{d\omega}{dk}=\frac{\hbar k}{m} \] For light in a vacuum, there's no dispersion, or the dispersion relation is linear, so group velocity and phase velocity coincide. If we enter a medium, then we have dispersion. For relativistic (massive, matter) waves, the dispersion relation is very interesting, giving a group velocity that's subluminal, and a phase velocity that's superluminal, the product of both giving exactly c2. The problem with relativistic equations is that they cannot be consistently interpreted in terms of one particle. They are multi-particle systems from the get go.

Environmental concerns should always be at the forefront, of course, but I don't see any reason why tidal generators can't be made safe. The avoidance method is probably more expensive, and obviously more complicated, than simply building a fence to keep sea-creatures out of harm's way. It depends on how the turbines are located with respect to the shore, and whether fishing is an issue, etc. Whatever safeguards are implemented, they will add to the initial cost and maintenance costs. That makes private investors wary, especially as the returns have not been that great, so far. New technology - growing pains. A good, accessible overview, designed for students. The one in the Netherlands seems to be working satisfactorily. It's the biggest, I think, to date.

It doesn't matter if they aren't spotted. The interesting question is which side of a zebra has the most stripes? And of course, it's the outside. Last time I bought zebra, the clerk scanned the wrong barcode and I got charged for rhino.

Maybe not intended but I see this as yet an analogy for entanglement. Assume a couple is married and then separated by some (great) distance. When one (random) individual of the married couple dies we immediately know that the other party has become a widower or a widow. The immediate change from wife to widow (or husband to widower) does not need a signal.

You have a point, I think. As I recall, Schrödinger's equation is not a true wave equation because it only has a single rather than double derivative with respect to time. I think I remember Peter Atkins telling us it is more properly a diffusion equation, rather than a strict wave equation. He went on say, rather enigmatically, that it might be thought more appropriate for a description of the behaviour of matter to be governed by a diffusion equation.............

Haha, sorry just search galactosedestroyer on the forums and you'll see The fun part is, you never really know whether or not I'm pitching the idea next Friday mwhahahahaha btw what if blood vessels were replaced with cybernetic tubes that could carry these particles faster-hmmm still trying to figure out how it could really work I'm just imagining infinite running without ever running out of breath The fun thing to calculate would be the amount of energy if you could control the electrochemical gradient in the intermembrane space galactosedestroyer

After WWII, the West did 6 things. 1) If you talk about the real GDP - the buying power, China is No 1, India will be No 2, the USA will be No. 3. 2) If you talk about the real GDP - the buying power, the top 5 will be China, India, the USA, Japan/Indonesia. There will be 0 European country. 3) England will be a second country. 4) The rest of the world consider Australia as Nothing. --The Malaysian PM Dr Mahathir said "Australia is a small country, and should act as a small country." --Australia has been described as “gum stuck to the bottom of China’s shoe.” https://www.theguardian.com/world/2020/apr/28/australia-called-gum-stuck-to-chinas-shoe-by-state-media-in-coronavirus-investigation-stoush 5) Even Australian consider the Chinese President is the most power man in the world. https://www.skynews.com.au/world-news/china/xi-jinping-is-the-most-powerful-man-in-the-world/video/c673dfb1075ffd1a1aca86acb2b0410e 6) The West believe the rest of the world would support them to fight with Russia, they just find out now the world do not care about the West.