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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373 views

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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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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429 views

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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373 views

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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236 views

1D Quantum scattering from $V(x) = e^x$

Define $V(x) = e^{x}$, $x \in \mathbb{R}$ and consider the Hamiltonian $H = - \frac{d^2}{dx^2} + V(x)$. The eigenvalue problem is $$ -\psi''(x) + e^{x} \psi(x) = E \psi(x)\,, \quad x \in \mathbb{R}\,. ...
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514 views

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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319 views

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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27 views

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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154 views

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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63 views

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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Which one of them is the time-reversed wave-function, $\psi^{\ast }\left( x,t\right) $ or $\psi^{\ast}\left( x,-t\right) $?

If the wave function $\psi\left( x,t\right) $ is a solution of the spinless time-independent Schr$\ddot{\mathrm{o}}$dinger equation, $$ i\hbar\frac{\partial}{\partial t}\psi\left( x,t\right) =\...
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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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1answer
79 views

How to evaluate the probability when a particle is detected?

Everyone knows the standard probability interpretation of the quantum mechanics. For example, the wave function of some particle at some time $t$ is $\psi (x,t)$. Therefore, if the particle is ...
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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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1answer
227 views

Normalized probability distribution from the Coulomb/Rutherford scattering amplitude?

My question appears elementary, but I have been pretty vexed trying to answer it precisely. Can one use the Rutherford/Coulomb scattering amplitude to get a finite, normalized momentum-space ...
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144 views

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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213 views

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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473 views

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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3answers
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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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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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1answer
20 views

Intuiting qualitatively the shapes of the eigenfunctions of a finite well-like potential, using the infinite well eigenfunctions as an inspiration

Consider, for example, the third excited state of an infinite square well: Now consider the following potential: If we wanted to sketch the rough shape of the third excited eigenfunction of this ...
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1answer
106 views

Transformation of Wave function under gauge transformation

I am trying to obtain the transformed wavefunction $\psi$ under gauge transformation in the presence of the E-M field. So Schrödinger's equation is (in the units $c=1$ and $\hbar$ = 1) $$i\frac{d\psi}{...
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83 views

Wavefunction and a central potential $V(r)$ that is singular at origin

I read the following line from Weinberg's Lectures in Quantum Mechanics (pg 34): As long as $V(r)$ is not extremely singular at $r=0$, the wave function $\psi$ must be a smooth function of the ...
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46 views

Quantum tunneling for bound states

In QM, take a particle in a bound state in $\mathbb{R}^n$ subject to a potential (need not be smooth and not necessarily bounded above, but is bounded from below, say, something that might roughly ...
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2answers
60 views

Number of nodes in Hartree-Fock solution

The Hartree-Fock equation for atoms is of the form $\left[\frac{d}{dr^2}+f(r)-\epsilon\right]P(r)=g(r) \tag1$ Usually algorithms to solve this equation assumes that the number of nodes of $P(r)$, ...
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1answer
71 views

Understanding propagator nature of QM Green's function

I'm trying to understand the many-body Green's functions, but first I want to understand Greens functions in QM. I'm reading this article, but I'm having trouble with eq. 17. The equation states: $$ \...
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218 views

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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1answer
255 views

Symmetric potential well different solutions

I have solved $H|\psi\rangle=E_{n}|\psi\rangle$ with $V(x)=0$ from $-a<x<a$ and $\infty$ otherwise. If I propose a solution of the form $\psi(x)=A_{n}e^{ikx}+B_{n}e^{-ikx}$ I arrive to the ...
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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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859 views

Expressing position wave function in momentum space for a single state

I am doing an introductory quantum mechanics course, I have been told that the momentum space wave function is essentially the Fourier transform of the position-space wave function. I.e. $$ \Phi(p,t)=\...
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1answer
67 views

Scaling Problem with Variational Method

$\def\braket#1{\langle#1\rangle}$ I am attempting to solve a particular Hamiltonian by variational method. The wavefunction that I have selected is as follows: $$ \Psi = Ne^{\frac{-kr}{2}}\sum_{i=0}...
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1answer
104 views

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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Proof that 1D bound states are non-degenerate without Wave Mechanics

Most proofs I see that all bound states of a quantum system are non-degenerate use the wave-function representation. How can we prove it using only Dirac notation, without resorting to the wave ...
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81 views

Does a pilot-wave theory need to be stochastic in a discrete space-time?

I have found an article about Bohmian mechanics on a lattice with discrete space and time, the link of which is given below: https://arxiv.org/abs/1606.02883 Here the motion of quantum particles is ...
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1answer
81 views

Gravitationally-induced slowing-down of the spreading of a wave packet

The spreading of a wave packet is very fast in quantum mechanics: for an electron, a gaussian wave packet spreads from one angström to 600km in one second! In his famous QM book, Sakurai mentions that ...
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Two ways of thinking about the Dirac equation

My impression is that there are two ways of thinking about the Dirac equation: Quantum Mechanically: Here we think of the spinor $\phi$ as a generalization of the Schrodinger wave function which ...
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1answer
209 views

The Schrodinger equation with strange potential

The particle of mass $m$ moves in potential $$V(x) = \dfrac{\alpha \left( \left( 2 \alpha +1 \right)x^2-a^2 \right)}{m \left( a^2 + x^2\right)^2},$$ and $\alpha > 1/4$. Find the energy and the ...
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Nearly free electron model, periodicyt and first Brillouin zone

I am taking an introductory course in condensed matter physics, and am absolutely stumped by the concept of the Brillouin zone and backfolding of the dispersion curve in the nearly free electron model....
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Reconstruct quantum state from probabilities

Given a quantum state, the Born rule lets us compute probabilities. Conversely, given probabilities, can we reconstruct the quantum state? I think the answer is almost trivially positive but how ...
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1answer
58 views

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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1answer
210 views

Energy spectrum for a step potential

Most of the books tend to give this explanation that for a bound physical system, the energy and momentum eigen values have discrete spectrum and otherwise, they have a continuous spectrum, which I ...
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67 views

Why does a system being in an s-wave mean that the spacial 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 l, the spacial wf ...
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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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How do I calculate the overlap integral of H2?

I'm given a problem where I need to "evaluate the overlap integral for two 1s orbitals as a function of interatomic spacing, R". This is what I think I need to do: $$S = \int \psi^*_a (\vec{r})\...
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201 views

Must a photon ever be thought of as 3-dimensional

I am still (ref. my previous questions) trying to understand the nature of photons, so here goes: It seems most physicists imagine the photon as a wave packet, which can be mathematically constructed ...
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538 views

Dirac Delta potential and perturbation

I have a Dirac Delta potential as follows : $$V(x)= - \alpha \delta (x)$$ I know how to solve that problem. There is exactly one bound state. Now let's say I have an initial wave function in this ...

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