# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Wave function for the free electron gas

In the free electron gas model, we suppose that the electrons are non interacting and that they occupy a 3 dimensional infinite potential well. Solving the time independent Schrodinger equation for ...
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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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### Comparing the quantum version of the infinite well to classical version

In the quantum version of the infinite well, the energy eigenvalues can be precisely determined. The energy is all in the form of kinetic energy, $E=\frac{p^2}{2m}$, and so, classically, the magnitude ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...