Questions tagged [wavefunction]

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. DO NOT USE THIS TAG for classical waves.

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Which position and momentum distributions arise from some wave function?

Consider a particle in one dimension with wave function $\psi$. The probability density function describing how likely it is to find it in a given position is given by $f(x)=\left|\psi(x)\right|^2$. ...
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Does the Schrodinger Equation yield a unique wave function and density?

I am learning DFT and the Hohenberg Kohn Theorem of Existence. And it says that there is a one-to-one correspondence between the external potential and the density. However the proofs that I have seen ...
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Projection operator (relative angular momentum) in FQHE Toy hamiltonian

I am working on Fractional Quantum Hall Effect and reading these lecture notes http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf. As all others sources I have found, none of them precisely define the ...
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Connection between quantum field and the wavefunction

The general question "What is a quantum field?" has been asked here before, but I'm looking for specific help in trying to iron out the details of my own personal interpretation and understanding. In ...
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Spherical symmetry of Cooper pair wave function

Can someone please explain to me how the wave function of a Cooper pair is spherically symmetric?
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The ground state of arbitrary Potential Function

How can one say that the number of nodes in the ground state must be nodeless . And how one can ensure that, when one gets up in the energy spectrum, for consecutive States the difference of number of ...
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Irreducible representation of derivatives of diabatic electronic wave functions

My question regards the symmetry properties of the derivatives of diabatic electronic wavefunctions. I think the question goes somewhat beyond the standard group theoretical machinery taught in ...
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Intuitive understanding in QFT

I recently read a bit about the Schrodinger picture in QFT and wavefunctionals, see e.g. Polchinski's String Theory lectures, and I wanted to ask if the intuitive understanding of QFT I got is "right"...
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What is the interpretation of a wave function of the Universe in Hawking's no boundary proposal?

In the path integral formalism we have an in state $\Psi_{in}[\phi]$ and and out state and we find the amplitude for going from one to the other: $$\Delta[\Psi_{in},\Psi_{out}] = \int \Psi_{in}[\phi]...
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Functionals of quantum states in QFT

Almost every book and article I can think of represents states of QFT using the Heisenberg picture of Hilbert space vectors, but Visser in "Lorentzian wormholes" does mention that you can also ...
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Why do electrons in an atom only occupy stationary states, without superposition?

In simple quantum mechanical problems such as the infinite square well, we solve the Time Independent Schrodinger's equation by separation of variable, effectively getting the energy eigenstates of ...
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Harmonic oscillator with delta function

For a one-dimensional Quantum Harmonic oscillator with delta function, the Hamiltonian $$\hat{H}=-\frac{1}{2m}\frac{d^2}{dx^2}+\frac{m\omega^2}{2}x^2+\frac{g}{2m}\delta({x})$$ The potntial is even so ...
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How to marginalize a Lagrangian density?

I'm trying to replicate a result from this paper: Physical Review A 76, 063614 (2007). It's for a class in classical mechanics, so we're only concerned with Lagrangian densities and such. I must ...
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What is the Schrödinger equation in position velocity space?

One way I've seen the Schrödinger equation expressed for the position wave function is $$\frac{i\hbar\partial\Psi\left(\vec{r},t\right)}{\partial{t}}=-\frac{\hbar^2}{2m}\nabla^2\Psi\left(\vec{r},t\...
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Large-scale rotational invariance in lattice space

It is often claimed among physicists that rotational invariance can emerge at large scales in lattice space. Let's focus on quantum mechanics for now. I interpret this claim as follows (I am a ...
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Continuum solutions for the Dirac equation in Coulomb potential - numerical codes

Following the representation used in [1, pag. 11] the solution of the Dirac equation in polar coordinates for energy $E$ is of the type: $$ \psi_{E\kappa m}(\bf{r})= \dfrac{1}{r} \Bigg( \begin{matrix} ...
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Why there aren't classical field associated with fermions?

In A Introduction to Nuclear Physics by GreenWood, It's written Bosons are particles that obey Bose-Einstein statistics and are characterized by the property that any number of particles may be ...
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How do you normalize this wave function?

I have a basic question in elementary quantum mechanics: Consider the Hamiltonian $$H = -\frac{\hbar^2}{2m}\partial^2_x - V_0 \delta(x),$$ where $\delta(x)$ is the Dirac function. The eigen wave ...
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Radial wave function matrix elements

For hydrogen atom radial wave function is the analytic form of the matrix elements, $$\langle n'\ell'|r^k|n\ell\rangle,$$ known? I am especially interested in $k=-2$ and $k=-3$. Notation: $$|n\ell\...
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Is there any utility in conceptualizing a 1D wavefunction as a space-curve?

I was watching animations of different wavefunctions that occur in physics, and it annoyed me that the real and imaginary parts are often graphed on the same axis separately. I wanted to know whether ...
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How does $\psi(x) =\exp(\frac{iq}{\hbar}\int^xA(x')\cdot dx')\phi(x)$ remove the gauge field for a free particle?

In what sense does writing $$\psi(x) =\exp(\frac{iq}{\hbar}\int^xA(x')\cdot dx')\phi(x)$$ "formally remove the gauge field" for a free particle in the Hamiltonian $$H\psi=\frac{1}{2m}(-i\hbar\nabla -...
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Wave function as a section of a complex line bundle to do QM in polar coordinates

If you want to change the coordinates of a Wave function $\Psi$ in 2D QM from cartesian to polar coordinates in a naive way one encounters a problem, namely the (naively defined) radial momentum ...
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Orthogonalizing a Gaussian Basis

Given a discrete Gaussian basis $$G = \{\lvert n\rangle, n \in \mathbb{Z}\},$$ where $$\langle x\rvert n \rangle = \exp\left(\dfrac{-(x-nL)^2}{2}\right),$$ with $L$ fixed. Does there exist a set of ...
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Can one add a discrete set of functions to complete the bound states of the hydrogen atom?

Though being an infinite orthonormal set of functions, the bound states $\Psi_{nlm}$ of the hydrogen atom do not form a basis of the Hilbert space $L^2(\mathbb{R}^3)$ due to the continuous spectrum, i....
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Finding total flux of probability current through a sphere

For a wavefunction: $$\Psi(\textbf{x}) = e^{ikz} + \dfrac{f(\theta)}{r}e^{ikr}$$ Where $z = r\cos(\theta)$. The probability current $J$ is then given by: $$J(\textbf{x}) = J_1(\textbf{x}) + J_2(\...
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Who knows experiments on wave packets spread?

I'm looking for some experimental evidence of quantum wave packets spread to propose to my students. I notice that a huge attention is devoted in books to the theoretical frame, while almost nothing ...
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Why does a system being in an $s$-wave mean that the spatial wavefunction is symmetric? Does it?

Probably being silly here, but me and my fellow undergrads can't seem to come up with an exact answer to why, if a system is in an $s$-wave, edit: or any orbital with an even value of $\ell$, the ...
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Is there an intuitive interpretation of the shape of the angular momentum eigenstate?

I was watching a MIT lecture video on angular momentum eigenstate. Toward the end of the lecture, the professor had shown some plots of the first few spherical harmonics, in an attempt to explain ...
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Overall symmetry of pion wave function

Mesons are bosons, therefore their wavefunction must be symmetric under particle exchange. Overall, the meson wave function ($\text{WF}$) has the following contributions: $$\text{WF} = \lvert \text{...
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Envelope of wavepacket and group velocity

In this answer a possible derivation of the group velocity is provided. It is, anyway, based on the assumption that there will always be a point where all the cosines will sum with the same phase: ...
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Wave packets in Dirac equation

Gaussian wave packets remain Gaussians after evolution in case of the Schrodinger equation. It is a very useful property of these wave packets. I don't think the same is true for a Gaussian wave ...
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Normalizability of the Hartle-Hawking state in Liouville theory

I'm confused about how to normalize the Hartle-Hawking state in 2D quantum gravity. We can compute the HH state for two circles of length $\ell_1$ and $\ell_2$ in the matrix model as $\langle W(\ell_1)...
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Analytic form of the normalization constant for Laughlin wavefunction

Is there any analytic form of the normalization constant for Laughlin wavefunction $$\prod_{i < j} (z_i-z_j)^{1/\nu} e^{-\sum_i |z_i|^2/4}$$ where $\nu$ is the filling factor?
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Where in experiment do you encounter Lorentizan wavefunction?

Is there an experimental system, or such that can be observed in nature where a particle's wave function assumes a form - $\psi(x)\propto \frac{1}{\sqrt{x^2+1}}$ such that $|\psi(x)|^2$ is Lorentzian? ...
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Can we ever "measure" a quantum field at a given point?

In quantum field theory, all particles are "excitations" of their corresponding fields. Is it possible to somehow "measure" the "value" of such quantum fields at any point in the space (like what is ...
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How to derive Landau level with semiclassical approach?

I'm trying to derive the Landau level by applying semiclassical dynamics and the time-dependent Schrodinger equation. From that, I success to derive $E = \hbar\omega_c n$, but I fail to derive the ...
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Why not all Berry phase just vanished?

I just learned that for any real wavefuntions, berry phase equals zero. But in Griffiths' Problem 2.1(b), he proved that any complex wavefuntion can be written as linear combination of REAL ...
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Interpretation and units of propagators

Quantum field theory is usually expressed in natural units in which $\hbar=c=1$. This simplifies equations and one can always get back to other units by inserting $\hbar$ and $c$ in appropriate places....
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Marginalisation of a joint probability distribution in bra-ket notation

Given a wave function $\Psi(\vec r_1, \vec r_2)$, where $\vec r_1$ and $\vec r_2$ are the positions of particle 1 and 2, respectively, the probability of finding particle 1 at position $\vec r$ (...
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Normalization in vector field in QFT after non-relativistic expansion

I encountered this equation when I was reading the article "Black Hole Superradiance Signatures of Ultralight Vectors" $$A_\mu=\frac{1}{\sqrt{2m}}\Big(\psi_\mu (\vec{r})\exp(-i\omega t)+\...
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Nuclear cusp condition

Does the nuclear cusp condition $$ \frac{\partial}{\partial r_\alpha} \bar{\rho}(r_\alpha) \vert_{r_\alpha = 0} = -2 Z_\alpha \bar{\rho}(0) $$ hold only for the ground state of a quantum mechanical ...
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Probability Waves vs. Amplitude Waves

It is often asserted, and it is common knowledge, that the waves associated with a particle are probability waves. This seems reasonable. But what about $E=hf$? This does not seem to be about ...
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Probability distribution function of the photon's scattering angle

What is the normalized probability distribution function of the photon's scattering angle, $\theta$, in Compton scattering effect when a photon hits an electron?
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Wave-function in geometric algebra, rotors and electromagnetism

I am confused by the appearance of the electromagnetism bivector in the formulation of the wave-function in space-time algebra. David Hestenes suggests that the wave-function can be written as $$ \psi ...
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Overlap of state subspaces in QM

In many-body QM we are rarely able to solve exactly for (some or all of) the eigenstates of the Hamiltonian. In some fields of condensed matter physics there has been a successful "business" ...
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Probability of two electrons of different energy levels contained in a single infinite potential well being found in same region

What is the probability of two electrons in a single infinite potential well centered at 0, one in the ground state, the other in the first excited state, being in the same region? I know by the Pauli ...
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How to imagine wavefunction branching?

This is a question particularly geared toward the Many Worlds interpretation, but I think it could be translated to other approaches as well. I am not sure I understand exactly what sort of events ...
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Hologram: At the "crux" of the hologram, is there any potential for wave-wave interaction?

Starting with Gabor's original 1948 paper: https://www.nature.com/articles/Art56 And continuing where that paper left off with Gabor's Nobel Prize paper, https://www.nobelprize.org/uploads/2018/06/...
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Entanglement entropy inequalities for a sum of wavefunctions

I am learning about entropy in quantum mechanics, and I am trying to develop some tools and intuition. One tool that I have found very helpful has been $$\sum_{i=1}^k \lambda_i \mathbb{S}[\rho_i] \...
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Is there any dependence between the shape and dynamics of the electric field produced by an electron and 'shape' of its wave function?

In classical electromagnetism the electron is described by a point charge that generates an electric field with spherical symmetry when the electron is static but at the quantum level the electron is ...
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