A complex scalar field that describes a quantum mechanical system. The square of the modulus of the wave function gives the probability of the system to be found in a particular state.
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9answers
4k views
About the complex nature of the wave function?
1.
Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an ...
8
votes
2answers
254 views
Wavefunction collapse and gravity
If gravity can be thought of as both a wave (the gravitational wave, as predicted to exist by Albert Einstein and certain calculations) and a particle (the graviton), would it make sense to apply ...
7
votes
3answers
255 views
If superposition is possible in QM, why do we often assume systems are already in their eigenstates?
My understanding is that an arbitrary quantum-mechanical wavefunction can be written as a linear combination of eigenfunctions of some Hermitian operator, most commonly the Hamiltonian; when a ...
7
votes
2answers
251 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?
7
votes
2answers
227 views
Superconducting Wavefunction Phase (Feynman Lectures)
In Volume 3, Section 21-5 of the Feynman lectures (superconductivity), Feynman makes a step that I can't quite follow. To start, he writes the wavefunction of the ground state in the following form ...
6
votes
7answers
452 views
Is it wrong to talk about wave functions of macroscopic bodies?
Does a real macroscopic body, like table, human or a cup permits description as a wave function? When is it possible and when not?
For example in the "Statistical Physics, Part I" by Landau & ...
6
votes
1answer
410 views
Must the derivative of the wave function at infinity be zero?
I came across a problem in Griffiths where the derivative of the wave function (with respect to position in one dimension) evaluated at $\pm\infty$ is zero. Why is this? Is it true for any function ...
5
votes
3answers
364 views
Meaning of $\int \phi^\dagger \hat A \psi \:\mathrm dx$
While analysing a problem in quantum Mechanics, I realized that I don't fully understand the physical meanings of certain integrals. I have been interpreting:
$\int \phi^\dagger \hat A \psi ...
5
votes
3answers
531 views
Electrons - What is Waving?
If an electron is a wave, what is waving?
So many answers on the internet say "the probability that a particle will be at a particular location"... so... the electron is a physical manifestation of ...
5
votes
3answers
145 views
Time Varying Potential, series solution
Suppose we have a time varying potential $$\left( -\frac{1}{2m}\nabla^2+ V(\vec{r},t)\right)\psi = i\partial_t \psi$$ then I want to know why is the general solution written as $\psi = ...
5
votes
2answers
307 views
Does quantum mechanics allow faster than light (FTL) travel?
Let's suppose I initially have a particle with a nice and narrow wave function[1] (I will leave these unnormed):
$$e^{-\frac{x^2}{a}}$$
where $a$ is some small number (to make it narrow). Let's also ...
5
votes
1answer
197 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 ...
4
votes
3answers
998 views
What is the relation between position and momentum wavefunctions in quantum physics?
I have read in a couple of places that $\psi(p)$ and $\psi(q)$ are Fourier transforms of one another (e.g. Penrose). But isn't a Fourier transform simply a decomposition of a function into a sum or ...
4
votes
2answers
322 views
Amplitude of Probability amplitude. Which one is it?
QM begins with a Born's rule which states that probability $P$ is equal to a modulus square of probability amplitude $\psi$:
$$P = \left|\psi\right|^2.$$
If I write down a wave function like this ...
4
votes
2answers
307 views
Exactly how is the constant measured velocity of light deduced from Maxwell's equation?
For electromagnetic radiation the velocity of propagation is $c = 1/\sqrt{\mu_0 \epsilon_0}$. Since both $\mu_0$ and $\epsilon_0$ do not vary in any inertial frame, then $c$ must be constant in any ...
4
votes
1answer
34 views
Tip of a spreading wave-packet: asymptotics beyond all orders of a saddle point expansion
This is a technical question coming from mapping of an unrelated problem onto dynamics of a non-relativistic massive particle in 1+1 dimensions. This issue is with asymptotics dominated by a term ...
4
votes
1answer
176 views
Boundary conditions from single-valuedness of spherical wavefunctions
This question is a follow-up to David Bar Moshe's answer to my earlier question on the Aharanov-Bohm effect and flux-quantization. What I forgot was that it is not the wavefunction that must be ...
4
votes
1answer
205 views
Young's double slit
Am I right to think the (general) probability distribution of photon in a double slit experiment at the screen has the form $|\psi|^2 = c e^{\alpha x^2}\cos^2(\beta x)$? (Due to the superposition of ...
3
votes
3answers
223 views
Can a wavefunction be solved to any arbitrary precision, given enough computer time?
I learned that the wavefunction for the hydrogen atom can be solved analytically (we did the derivation in class), but that for more complicated atoms it is "impossible" to solve and that only ...
3
votes
2answers
364 views
Is the free electron wavefunction stable?
The wavefunction of a free electrons is variously described as a plane wave or a wave packet. I am fairly happy with the wave packet, as it is localised.
But if we change to the electron's rest ...
3
votes
4answers
251 views
Does the wave nature of a particle refer to the wave function?
In quantum mechanics when we talk about the wave nature of particles are we referring in fact to the wave function? Does the wave function describes the probability of finding a particle (ex: ...
3
votes
3answers
260 views
How to compute the expectation value $\langle x^2 \rangle$ in quantum mechanics?
$$\langle x^2 \rangle = \int_{-\infty}^\infty x^2 |\psi(x)|^2 \text d x$$
What is the meaning of $|\psi(x)|^2$? Does that just mean one has to multiply the wave function with itself?
3
votes
1answer
254 views
Help me understand the first equation in Landau & Lifshitz's Quantum Mechanics
While I've covered a basic course in Quantum Mechanics, I'm self-studying Landau & Lifshitz's book to help me understand what's going on.
Unfortunately, I'm stuck on the very first equation in ...
3
votes
1answer
142 views
Expected value inequality
Why is $\langle p^2\rangle >0$ where $p=-i\hbar{d\over dx}$, (noting the strict inequality) for all normalized wavefunctions? I would have argued that because we can't have $\psi=$constant, but ...
3
votes
3answers
346 views
Smoothness constraint of wave function
Is there anything in the physics that enforces the wave function to be $C^2$? Are weak solutions to the Schroedinger equation physical? I am reading the beginning chapters of Griffiths and he doesn't ...
3
votes
3answers
588 views
Historical background of wave function collapse
I wonder what were the main experiments that led people to develop the concept of wave function collapse? (I think I am correct in including the Born Rule within the general umbrella of the collapse ...
3
votes
1answer
33 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 ($k$ ...
3
votes
1answer
122 views
What does the appearance of a classical particle fundamentally reduce to?
I've been reading an article that describes what seems to be a classical particle as a regularity in the global wavefunction over a quantum configuration space:
When you actually see an electron ...
3
votes
1answer
663 views
Confusion between the de Broglie wavelength of a particle and wave packets
So I learned that the de Broglie wavelength of a particle, lambda = h/p, where h is Planck's constant and p is the momentum of the particle. I also learned that a quantum mechanics description of a ...
3
votes
1answer
309 views
Expectation values-Wavefunction
I'm a bit puzzled about an excercise in which I have to find the expectation values for position and momentum. Normally this should be pretty easy but in this case I just don't get the point.
...
3
votes
2answers
145 views
Interpretation of $e|\psi|^2$ as electron density
In solid state physics the electron density is often equated to $e|\psi|^2$. However, the Sakurai says (Chapter 2.4, Interpretation of the Wave Function, p. 101) that adopting such a view leads "to ...
3
votes
1answer
324 views
Even and Odd States of a 1D finite potential well
Is it possible for a particle trapped in a 1D finite potential well to evolve from a even state to an odd state and vice-versa? Why?
3
votes
1answer
261 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 ...
3
votes
1answer
345 views
Where does the wave function of the universe live? Please describe its home
Where does the wave function of the universe live?
Please describe its home.
I think this is the Hilbert space of the universe. (Greater or lesser, depending on which church you belong to.)
Or maybe ...
3
votes
2answers
318 views
Is the electron wave function defined during photon emission
I have heard the term quantum leap to describe the (instantaneous?) transition from a higher energy orbital to a lower energy orbital. Yet, I understand that this transition time has now been ...
3
votes
1answer
194 views
Projection of states after measurement
Continuing from the my previous 2-state system problem, I am told that the observable corresponding to the linear operator $\hat{L}$ is measured and we get the +1 state. Then it asks for the ...
3
votes
1answer
333 views
Finding $\psi(x,t)$ for a free particle starting from a Gaussian wave profile $\psi(x)$
Consider a free-particle with a Gaussian wavefunction,
$$\psi(x)~=~\left(\frac{a}{\pi}\right)^{1/4}e^{-\frac12a x^2},$$
find $\psi(x,t)$.
The wavefunction is already normalized, so the next thing to ...
3
votes
2answers
462 views
How do I figure out the probability of finding a particle between two barriers?
Given a delta function $\alpha\delta(x+a)$ and an infinite energy potential barrier at $[0,\infty)$, calculate the scattered state, calculate the probability of reflection as a function of ...
2
votes
3answers
220 views
Is this interpretation of $\psi=\frac{1}{\sqrt{\pi a^{3}}}e^{-r/a}$ correct?
Apologies if this is stating the obvious, but I'm a non-physicist trying to understand Griffiths' discussion of the hydrogen atom in chapter 4 of Introduction to Quantum Mechanics. The wave equation ...
2
votes
3answers
208 views
How do I integrate $\frac{1}{\Psi}\frac{\partial \Psi}{\partial x} = Cx$
How do I integrate the following?
$$\frac{1}{\Psi}\frac{\partial \Psi}{\partial x} = Cx$$
where $C$ is a constant.
I'm supposed to get a Gaussian function out of the above by integrating but don't ...
2
votes
5answers
218 views
wave superposition of electrons and quarks
Is quantum wave superposition of electrons and quarks possible?
If not, can different types of elementary particles be mixed in wave superposition?
2
votes
3answers
398 views
Is the wave function objective or subjective?
Here is a question I am curious about.
Is the wave function objective or subjective, or is such a question meaningless?
Conventionally, subjectivity is as follows: if a quantity is subjective then ...
2
votes
2answers
68 views
Interpretation of de Broglie wave
Until what point can the de Broglie wave be thought as a real wave?
I mean, is it made of something?
What amplitude does it have? Is it a sine wave?
How can it be related to the wavefunction of the ...
2
votes
3answers
274 views
What is the rationale behind representing a state function by a complex valued function in QM?
What is the rationale behind representing a state function of an electron with a complex valued function $\Psi$. If only the probabilistic argument was required then why not represent it with just a ...
2
votes
2answers
791 views
Speed of a particle in quantum mechanics: phase velocity vs. group velocity
Given that one usually defines two different velocities for a wave, these being the phase velocity and the group velocity, I was asking their meaning for the associated particle in quantum mechanics.
...
2
votes
1answer
755 views
Probability current
Conservation of probability: Suppose a wavefunction has ${\partial \mathbb P \over \partial t} = -t f(x,t)$ and ${\partial j \over \partial x} = i f(x,t)$. How does it follow that ${\partial \mathbb P ...
2
votes
2answers
122 views
Vector representation of wavefunction in quantum mechanics?
I am new to quantum mechanics, and I just studied some parts of "wave mechanics" version of quantum mechanics. But I heard that wavefunction can be represented as vector in Hilbert space. In my eye, ...
2
votes
3answers
302 views
Confused over complex representation of the wave
My quantum mechanics textbook says that the following is a representation of a wave traveling in the +$x$ direction:$$\Psi(x,t)=Ae^{i\left(kx-\omega t\right)}\tag1$$
I'm having trouble visualizing ...
2
votes
1answer
90 views
Confused over the presence of 2 expressions for $\Psi(x,t)$
I'm following Griffiths' Introduction to Quantum Mechanics, and I see that he's got 2 different expressions for $\Psi(x,t)$. One of them is ...
2
votes
1answer
135 views
What does the notation $|x_1,x_2\rangle$ mean?
I would like clarification on an equation in the paper "Free matter wave packet teleportation via cold-molecule dynamics", L. Fisch and G. Kurizki, Europhysics Letters 75 (2006), pp. 847-853, DOI: ...
