0
votes
1answer
46 views

How can an electron be fired at a target when uncertainty principle says it will spread out around axis of motion?

Consider an electron fired at a target. Taking the axis of motion to be $x$, and position to be $(x,y,z)$ then $\Delta y = \Delta z = 0$ Therefore by the uncertainty principle $\Delta p_y = ...
5
votes
3answers
98 views

Is the ground state closest to the uncertainty relation? [duplicate]

For simplicity, suppose we are only talking about discrete energy levels, ie, bound state case. The energy levels are $E_1, E_2\cdots$, and the corresponding wave functions are $\psi_1, \psi_2 ...
4
votes
1answer
87 views

Which position and momentum distributions arise from some wave function?

Consider a particle in one dimension with wave function $\psi(x)$. The probability density function describing how likely it is to find it in a given position is given by ...
2
votes
0answers
90 views

Momentum representation of a state

I am trying to figure out the momentum representation of the state which has the properties $$\langle \psi |\hat q |\psi \rangle=-q_0,$$ $$\langle\psi|\hat p|\psi \rangle=p_0, $$$$\Delta q\Delta ...
1
vote
2answers
255 views

Gaussian Probability Distribution?

The uncertainty principle states that, $$\sigma _{{x}}\sigma _{{p}}\geq {\frac {\hbar }{2}}.$$ It is mentioned from many sources that the probability distribution of the particle position and ...
0
votes
1answer
123 views

Quantum state with zero standard deviation of position operator

Is any quantum state $|\psi\rangle$ possible such that the standard deviation $\sqrt{\langle\psi|(\Delta\hat{x})^2|\psi\rangle}$ of the position operator $\hat{x}$ is zero? If not, why?
0
votes
4answers
421 views

Why complex functions for explaining wave particle duality?

I have this very bad habit of going to the scratch, discarding all the developments of a theory and worldly knowledge, and ask some fundamental (mostly stupid and naive, as some may say) questions as ...
4
votes
2answers
260 views

A quantum particle which is almost at rest but whose position is random!

Assume a particle is given by a quantum state which is constructed in such a way that it is equally probable to find it anywhere in an fixed interval $(0,L)$ but has arbitrarily low velocity. The ...
5
votes
3answers
343 views

How can particles travel in a straight line?

A particle can be set off in a certain direction by giving them momentum. Momentum is a vector, so the particle heads off in a specific direction. But the wave function of the particle allows it to ...
3
votes
1answer
105 views

Connection between a simple matter wave and Heisenberg's uncertainty relation

When looking at the wave function of a particle, I usually prefer to write $$ \Psi(x,t) = A \exp(i(kx - \omega t)) $$ since it reminds me of classical waves for which I have an intuition ($k$ ...
1
vote
1answer
2k views

Gaussian wave packet

At our QM intro our professor said that we derive uncertainty principle using the integral of plane waves $\psi = \psi_0(k) e^{i(kx - \omega t)}$ over wave numbers $k$. We do it at $t=0$ hence $\psi = ...
8
votes
2answers
498 views

Was uncertainty principle inferred by Fourier analysis?

I would like to know: did Heisenberg chance upon his Uncertainty Principle by performing Fourier analysis of wavepackets, after assuming that electrons can be treated as wavepackets?
1
vote
5answers
2k views

Reason for the Gaussian wave packet spreading

I have recently read how the Gaussian wave packet spreads while propagating. see: http://en.wikipedia.org/wiki/Wave_packet#Gaussian_wavepackets_in_quantum_mechanics Though I understand the ...
5
votes
1answer
273 views

Relativistic contraction for a wave packet and uncertainty on momentum

Consider an electron described by a wave packet of extension $\Delta x$ for experimentalist A in the lab. Now assume experimentalist B is flying at a very high speed with regard to A and observes the ...
3
votes
1answer
488 views

wavefunction collapse and uncertainty principle

We all know that wavefunction collapse when it is observed. Uncertainty principle states that $\sigma_x \sigma_p \geq \frac {\hbar}{2}$. When wavefunction collapse, doesn't $\sigma_x$ become $0$?, as ...
2
votes
1answer
226 views

How do you determine the degree of localization of a wavefunction?

Suppose that there is a wavefunction $\Psi (x,0)$ where 0 is referring to $t$. Let us also say that $a(k) = \frac{C\alpha}{\sqrt{\pi}}\exp(-\alpha^2k^2)$ is the spectral contents (spectral amplitudes) ...
2
votes
2answers
203 views

Measurement and uncertainty principle in QM

The Wikipedia says on the page for the uncertainty principle: Mathematically, the uncertainty relation between position and momentum arises because the expressions of the wave function in the two ...
1
vote
3answers
851 views

Why can't we know the speed, $\vec{v}(t)$, and position, $\vec{r}(t)$, of an electron (the two) at the same time $t$?

I've read something about this and I conclude that it happens because of the uncertainty principle. But I don't understand very well the meaning of that. I mean, it's very abstract that the speed, ...