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33

The first paragraph is basically right, but I wouldn't ascribe the uncertainty principle to particles, just to the universe/physics in general. You can no more get arbitrarily good, simultaneous measurements of position and momentum (of anything) than you can construct a function with an arbitrarily narrow peak whose Fourier transform is also arbitrarily ...


31

The question asks about the time dependence of the function $$f(t) := \langle\psi(t)|(\Delta \hat{x})^2|\psi(t)\rangle \langle\psi(t)|(\Delta \hat{p})^2|\psi(t)\rangle,$$ where $$\Delta \hat{x} := \hat{x} - \langle\psi(t)|\hat{x}|\psi(t)\rangle, \qquad \Delta \hat{p} := \hat{p} - \langle\psi(t)|\hat{p}|\psi(t)\rangle, \qquad ...


22

Let a quantum system with Hamiltonian $H$ be given. Suppose the system occupies a pure state $|\psi(t)\rangle$ determined by the Hamiltonian evolution. For any observable $\Omega$ we use the shorthand $$ \langle \Omega \rangle = \langle \psi(t)|\Omega|\psi(t)\rangle. $$ One can show that (see eq. 3.72 in Griffiths QM) $$ ...


21

The uncertainty principle should be understood as follows: The position and momentum of a particle are not well-defined at the same time. Quantum mechanically, this is expressed through the fact that the position and momentum operators don't commute: $[x,p]=i\hbar$. The most intuitive explanation, for me, is to think about it in terms of wave-particle ...


18

This is really a footnote to Chris' answer but it got a bit long for a comment. It sounds odd to claim that a particle has no position, but it's easier to understand if you appreciate that a particle is just an excitation in a quantum field. When Heisenberg was developing his ideas physicists still thought of particles as little billiard balls. With the ...


17

Heisenberg's Uncertainty Principle is, in essence, a consequence of two basic facts: de Broglie's relation between a particle's momentum and its associated wavelength, and a mathematical fact known as the bandwidth theorem, which states an equivalent uncertainty relation between a wave's position and its wavelength. The fundamental physical leap here is of ...


17

There is a fair amount of background mathematics to this question, so it will be a while before the punch line. In quantum mechanics, we aren't working with numbers to represent the state of a system. Instead we use vectors. For the purpose of a simple introduction, you can think of a vector as a list of several numbers. Therefore, a number itself is a ...


16

There is a definine velocity and momentum, we just don't know it. Nope. There is no definite velocity--this was the older interpretation. The particle has all (possible) velocities at once;it is in a wavefunction, a superposition of all of these states. This can actually be verified by stuff like the double-slit experiment with one photon--we cannot ...


15

You are misunderstanding the Uncertainty Principle. The Uncertainty Principle says that a particle cannot simultaneously have a definite momentum and a definite position. This is not due to our incomplete knowledge of parameters. This is a fundamental law of the universe and arises from the fact that the momentum and position operators do not commute in ...


14

The reason for many contradictory statements regarding the nature of virtual particles is that they are often invoked for heuristical explanations of phenomena that arise within the framework of quantum field theory. One then tries to justify those explanations by attributing certain properties to virtual particles they do not actually possess. What ...


13

Manishearth's answer is correct, and this is just a minor extension of it. Manishearth correctly points out that the problem is your statement: There is a definine velocity and momentum, we just don't know it. Your statement is the hidden variables idea, and courtesy of Bell's theorem we currently believe that hidden variables are impossible. Take the ...


12

The Schrodinger equation is time-symmetric. The answer is therefore No. From all of the comments, I feel like I must be oversimplifying or missing something, but I can't see what.


12

We can satisfy your requirement "the photon was emitted at a correct angle" by "the photon was prepared in a momentum eigenstate". If the photon has definite momentum $\bf{k}$, then its direction of travel is well defined, as you have specified. A photon is a discrete excitation of a "mode", i.e. a solution of Maxwell's equations. For a photon in a ...


12

it is the error created by photons striking on elementary particles It's not. Heisenberg's uncertainty principle actually has nothing to do with any particular experiment, or any particular interaction. It's a purely mathematical statement about waves. Its true meaning is explained in detail on the Wikipedia page, but the gist is that if you have a ...


11

If you know its instantaneous velocity at every moment of time in a given interval AND you knew the exact position of a particle at the start of the interval, then you could use integrals to find its exact position at any moment in the interval. That is a logically true statement. There is no contradiction here with the Uncertainty principle because you ...


11

This is an estimation tool not uncommon in theoretical physics. Namely, one knows the value of some quantity for a given problem and therefore assumes that the scale of the problem with regards to that quantity is of the same order of magnitude as the known value. In other words, we assume that the error in our known value must not be too much greater than ...


10

No, the uncertainty principle isn't wrong. The PRL paper doesn't suggest that the original uncertainty principle relating uncertainties of position and momentum fails. It "only" questions a modified interpretation of the principle that says that the momentum is disturbed at least by $\hbar / 2 \Delta x$ for a given precision of the position measurement ...


10

The relation $p={h\over \lambda}$ applies to photons, it has nothing to do with the uncertainty principle. The issue is localizing the photons, finding out where the are at any given time. The position operator for a photon is not well defined in any usual sense, because the photon position does not evolve causally, the photon can go back in time. The same ...


10

Laplace's determinism is not physically correct over long periods of time. That is, it neglects chaos/"sensitive dependence on initial conditions"/exponential growth of microscopic perturbations already in Newtonian dynamics, which was seriously thought about only in the 20th century. Being true, this also will not be overcome. Stochasticity enters ...


10

The uncertainty principle, in the variance formulation, states that in any quantum state $|\rangle$, the quantity $$\langle (p-<p>)^2 \rangle \langle (x-\langle x\rangle)^2\rangle \ge {\hbar^2 \over 4} $$ To understand why shifting p and x by their expected value and squaring gives the squared uncertainty, see this answer. The proof is by noting the ...


10

So, why can't the uncertainty relations be violated in such a case, if I could, say, measure the position of the object with this wave function That's the catch. You can't. Or rather, you can measure the position, but the result you get will vary from one measurement to the next, because the wavefunction $\exp(x^2/2i - cx)$ is not an eigenstate of ...


10

In Physics "nothing" is generally taken to be the lowest energy state of a theory. We wouldn't normally use the word "nothing" but instead describe the lowest energy state as the "vacuum". I can't think of an intuitive way to describe the QM vacuum because all the obvious analogies have "something" instead of nothing "nothing", so I'll do my best but you may ...


10

The uncertainty principle can be seen as a result of space $x$ and momentum $p$ being a Fourier transform pair. The free-particle wave function has, similarly to the exponential $e^{-\frac{i}{\hbar} px}$ an exponential $e^{-\frac{i}{\hbar} E t}$. Thus one could expect a similar uncertainty relation for the variable pair $(E, t)$. An immediate result is that ...


9

Good question! For the properties related by the uncertainty principle, there are two reasons why they come in pairs: Intuitively, the uncertainty principle relates the variance of a function to the variance of its Fourier transform. And, up to a couple of numerical factors, the Fourier transform of a Fourier transform is the original function. ...


9

Your interpretation is not quite right. One sharp interpretation one can give to this "cutting" of phase space into cubes of size $h^{2N}$ (here $N$ is the dimension of the system's configuration space), is that it allows one to use classical phase space to count the number of energy eigenstates of the corresponding quantum hamiltonian. Instead of trying ...


9

You assume that you can instantly measure the momentum to arbitrary precision, and this isn't the case. Let's consider a plane light wave to keep things simple, and suppose you want to measure the momentum so precisely that the position uncertainty becomes exceedingly large. How precisely do we have to measure the momentum? Well the uncertainty principle ...


9

You are right on track. A great way to think about the Hamiltonian is that it's the thing that "causes translation in time". This just jargon for the fact, which you already observed, that the Hamiltonian tells you how the system moves forward in time, as encapsulated by the Schrodinger equation $$i\hbar d_t |\Psi\rangle = H |\Psi \rangle . $$ My guess ...


8

Although the uncertainty principle stems from the mathematical structure of QM, i.e., originates from the noncommutivity of some observable letting them behave as fourier transform pair as explained in another answer, I still think it is a statement on measurements, (i.e., imposes fundamental limits on measurements) since QM itself seems to be a theory of ...


8

First of all, the uncertainty principle is more than just disturbance of observation. From the Wikipedia article "Uncertainty principle": Historically, the uncertainty principle has been confused with a somewhat similar effect in physics, called the observer effect, which notes that measurements of certain systems cannot be made without affecting ...



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