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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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Free particle in spherical coordinates

I'm trying to solve the very simple equation: $$-\frac{\hbar^2}{2m}\nabla^2 \psi = E\psi$$ but in polar coordinates. I used separation of variables to find out that my wave function is of the form: $$\...
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Show that in the case of stationary states with a discrete spectrum, the value moment average is zero [closed]

I have a excersise that say: Show that in the case of stationary states with a discrete spectrum, the value moment average is zero. I was searching in the internet and all say me that I should use: $\...
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Finding wavefunction using given probability distribution function [closed]

The probability distribution of the position of a quantum particle in one dimension is $$P(x) = \alpha e^{-\beta x^2}.$$ The expectation value of its momentum has a magnitude (p) and points in the ...
Amrit Sagar Kar's user avatar
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Transformation of wavefunction

While learning QM, I was wondering how would the wavefunction of a particle, suppose charged particle, look for different observers moving with respect to each other. To begin with, let the electric ...
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The $L^2$ operator not returning the expected value (2) when applied on the (2,1,0) Hydrogen wave function [closed]

Let the Hydrogen wave function for the state $n=2, l=1, m=0$ be described as: $$\psi_{210} =cos(\sigma )*f(r)$$ Since the squared angular momentum eigenvalue is $L^2=l(l+1)$, I would expect it to be 2 ...
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Is there a meaning to the expectation of the conjugate product of an observable? [closed]

The expectation of an observable $A$ is taken to be $\int \psi^* A\psi dx$. We can also define a function $f(x) = (A\psi)(x)$ and $g(x)= f^*(x)f(x)$. Does $g$ or its expectation $$\int g \cdot \psi^* \...
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How to calculate the inner product $ \langle T|x \rangle = \langle\frac{p^2}{2\mu} |x\rangle$? [closed]

If I have a wave function $ \psi (x) = \langle x|\psi \rangle$, now I want to use kinetic energy representation $$ T = \frac{p^2}{2\mu} ,$$ where $ \mu $ is the mass of particle. I try to \begin{align*...
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Regarding to the asymptotic solution of quantum harmonic oscillator

In quantum mechanics, the radial equation of the SHO takes the form \begin{align} \frac{d^2 u}{dx^2}+\left(\epsilon-x^2-\frac{l(l+1)}{x^2}\right)u=0, \end{align} where $x=\sqrt{\frac{m\omega}{\hbar}}r$...
Mr. Anomaly's user avatar
1 vote
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Is the spherical outgoing wave solution to the Schrodinger equation is not a member of $L^2$?

I was reading a discussion about the Mott problem, where the authors discuss the outgoing spherical wave solutions to the Helmholtz equations $\nabla^2 f = - k^2 f$. This equation can also be ...
Jo Carlo's user avatar
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Wavefunction with determinate momentum

In page 100 Griffiths' Introduction to Quantum Mechanics, Griffiths states that the eigenvector of $\hat{p}$ in the position basis is $\frac{1}{\sqrt {2\pi\hbar}}e^{\frac{ipx}{\hbar}}$ and states that ...
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Why is the time derivative of the wavefunction proportional to a linear operator on it? [closed]

I am currently trying to self-study quantum mechanics. From what I have read, it is said that knowing the wave function at some instant determines its behavior at all feature instants, I came across ...
Gauss_fan's user avatar
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Proof that separation of variables leads to a complete basis of wave function in spherical coordinates [duplicate]

In griffith's introduction to quantum mechanics (chapter 4), there is an analysis of the stationary states of a particle given a potential function $V(r)$ that only depends on the radial distance $r$, ...
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When solving a paraxial Helmholtz equation ... ? Amplitude vs Wave function

Given the paraxial Helmholtz-equation (PHHE). Why are the solutions formed by the amplitude function of a wave function, but not the wave function itself? Example, a form of a wave function is defined ...
plantpot's user avatar
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Uncertainity in position in 1D potential box

In a question of a usual 1D box for a particle between $-L/2$ to $L/2$ i had to compute $\Delta x$ and $\Delta p$ for the particle. The solution used the formulas- $$\Delta x = \sqrt{\langle x^2 \...
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How do you apply a transformation, or a superposition in David Hetens' geometric formulation of the wavefunction?

In David Hestenes' formulation of the wavefunction in geometric algebra, we have: $$ \psi(x) = \sqrt{\rho(x)} R(x) e^{-ib(x)/2} $$ where R(x) is a rotor. For simplicity, let us now consider a two-...
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How to Perform Fourier Transform on a Quantum State of Spin-1/2 Particle?

I am currently studying quantum mechanics and need help understanding how to perform the Fourier transform of a particular state. I have a spin-1/2 particle whose momentum and spin state at time $t=0$ ...
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Incorrect factor in radial wavefunction $R_{2,1}$ of hydrogen atom

First of all, let me state that this isn’t a homework question but rather my personal annoyance. I can provide a proof in the comments if you think otherwise. The general equation for radial wave ...
Mr. Science's user avatar
3 votes
2 answers
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Density Matrix vs Wavefunction Formalism

What is the main motivation and advantages of the density matrix formalism compared to the wavefunction formalism? From what I understand, the density matrix is more commonly used when you want to ...
Rich Hard Fine Man's user avatar
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A missing link in the logical chain about the Aharonov-Bohm effect

The usual treatment of the Aharonov-Bohm effect (which appeared already in Aharonov and Bohm's original paper) takes two particular local solutions of the Schrödinger equation, $\psi_1$ and $\psi_2$. ...
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Reflection of quantum particle colliding with a potential barrier

Let a quantum particle be subject to a one dimensional step potential barrier $V$ such that: $$V(x)=\begin{cases}0, \ x<0 \\ V_0, \ x>0\end{cases}$$ where the particle's energy is $E>V_0>0$...
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Amplitude superposition for different kinds of particles

We have seen that the probability of finding a particle at a particular point is the square of its wave function. In the double slit experiment, we notice that wave functions add up and the resultant ...
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How to decide for correct value of phase difference?

A sinusoidal wave is propagating along a stretched string that lies along x-axis. The wave is moving in +x-direction. Figure shows the graph of transverse displacement of particles of the string at x =...
Garv Chaudha's user avatar
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Are reflection and transmission coefficients in 1D problem are independent of the direction in which we choose as incident?

I was watching a lecture series of Quantum mechanics of Professor V. Balakrishnan, There was a problem session, “For an arbitrary potential barrier (any potential function of position and it need not ...
Vivek Panchal 's user avatar
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Why do delta wells and delta barriers behave equivalently, but normal wells and normal barriers don't?

In Griffith's Intro to QM, it is shown that for $V(x)=-\alpha \delta(x)$ regardless of whether $\alpha$ is positive (a "delta well") or negative (a "delta barrier"), the ...
user56834's user avatar
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Operators algebra for quantum mechanics [closed]

I am taking my first quantum mechanics course and I am a bit lost in operators algebra. These are the main questions I have: Why can we write this kind of equations? $$ Ô \psi = o\psi $$ What I mean ...
Tymothée Waldner's user avatar
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1 answer
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Confusion in de Broglie wave and wave function

Recently I saw some questions related to the same topic and got to know that matter-wave functions(Ψ) are superpositions of multiple de Broglie waves corresponding to multiple momenta, which is ...
Nihal Popat's user avatar
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Difference between the expectation value of an operator and operator applied to wave function?

Expectation value of any operator $\hat{Q}$ is defined as, $$ \left\langle\psi_n\mid\hat{Q}\mid \psi_n\right\rangle $$ and action of the operator $\hat{Q}$ on wavefunction is defined as $$ \hat{Q} \...
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Schroedinger equation with value of wave function in polar coordinates?

I'm trying to get a better sense of what causes an increase in the magnitude and phase of the wave function at a given point. Is there a way to rewrite the schroedinger equation such that it ...
user56834's user avatar
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2 answers
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Basic confusion about evolution of wave function of a free particle

I am going through Griffith's introduction to quantum mechanics. An example for a free particle is given where $$\Psi(x,0) = \begin {cases}A \quad \text{if } x\in [-a,a]\\ 0\quad \text{otherwise}\end{...
user56834's user avatar
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2 votes
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Are projective representiations of a Lie group a representation of the semi-direct product of the group with $U(1)$ if the norm is preserved?

Let's say we have a function $f(x_{\mu},t)$ that transforms under the action of an $N$-parameter group $G(a_{\nu})$. Then a projective representation of $G(a_\nu)$ in the $f(x_\mu,t)$ basis would ...
Ilya Iakoub's user avatar
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Momentum Eigenvalues for Particle in a Box

A question from my college exams is as follows: Find out the eigenfunctions and eigenvalues of the momentum of a particle of mass $m$ moving inside an infinite one-dimensional potential well of width ...
L lawliet's user avatar
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1 answer
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Is a fermionic boson possible?

We know that bosons need an overall symmetric wavefunction. So is it possible for a boson to have an anti-symmetric spatial wavefunction and an anti-symmetric spin wavefunction? Such that upon ...
Despaxir's user avatar
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3 answers
204 views

Basic doubt in quantum mechanics

Do entities like electrons, which are considered point particles in Classical Mechanics, actually have a definite position at a particular time (irrespective of it can be measured or not)?
Users's user avatar
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Functional analysis question about operator on quantum wave functions

If I have two time-independent wave functions $\psi_{t_{1}}$ and $\psi_{t_{2}}$ and define an operator $\hat{U}$ such that $$\psi_{t_{2}} = \hat{U}_{t_{1},t_{2}}(\psi_{t_{1}})$$ and $$\psi_{t_{2}}(x) =...
Adam Kabbeke's user avatar
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0 answers
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Is the overall (distinguishble-particle) ground state for a many-body identical particle Hamiltonian also immediately the bosonic ground state?

Consider the following many-body Hamiltonian of $N$ particles in an external trapping potential with inter-particle interactions: \begin{align} \hat{H}= \sum_{i=1}^{N} \left[-\frac{\hbar^2}{2m} \...
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Self-interference of nuclear decay

Consider a stationary atom undergoing radioactive decay. The probability density function for decaying at time $t$ is given by an exponential distribution: $$p(t)=\lambda e^{-\lambda x}$$ When ...
Riemann's user avatar
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Can any meaning be given to a path integral with no fixed end point?

A path integral has the interpreted as the probability a particle goes from $A$ to $B$ in time $t$. Such a path integral is given by $$\langle x_B, t|x_A, 0\rangle = \frac{1}{Z} \int_{\textrm{paths } ...
CBBAM's user avatar
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1 answer
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A simple question in quanum mechanics on position and momenum eigenstates

The eigenfunctions (eigenstates) for the momentum of a particle are given by the plane waves $$\phi(x,t) = \sin(kx - \omega t)$$ If we sum a large number of these waves in a range from $0$ to $k_m$, ...
Anky Physics's user avatar
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Closed expression of eigenfunctions of a two dimensional isotropic harmonic oscillator

Where can one find the closed expression of the eigenfunctions of the 2d isotropic harmonic oscillator? I saw something like this: $$ \psi_{n_r m }(r, \theta) \propto e^{im\theta} r^{|m|} e^{-r^2/2} F(...
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3 votes
4 answers
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Does QM recognise empty waves?

If a particle (photon) goes through a Mach-Zehnder interferometer it is accepted in quantum mechanics texts that in passes in both channels after first beam splitter BS1 and propagates there until BS2....
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Homogeneity of Schroedinger equation implies norm conservation

I am trying to understand how homogeneity of Schroedinger equation implies norm conservation. I know that we are considering the non-relativistic case, where particle number is conserved, so we do not ...
imbAF's user avatar
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Group velocity, phase velocity and signal velocity for axion like particles

In dark matter models of axion-like particles (ALPs), sometimes we get the field $$\phi=2\phi_0\sin(m_\phi c^2 t/\hbar)\cos(k_\phi x)$$ This is like an stationary field with amplitude $\phi_0$ (in m/s ...
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Scattering Matrix and the Lippmann-Schwinger equation in QM

I am currently studying scattering theory from the Sakurai's quantum mechanics. I have previously studied this subject from Griffith's quantum mechanics. In the latter textbook, scattering matrices ...
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3 votes
1 answer
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Question about Griffiths' proof that $\Psi$ stays normalized

In "Introducion to Quantum Mechanixs", at p. 16, Griffiths writes what follows: Now, if $\Psi$ is just assumed to be in $L^2(\mathbb{R})$, this does not imply that $|\frac{\partial\Psi}{\...
Uagi's user avatar
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Doubts on Particle in a box model [closed]

I have following three doubts. For a particle in a box problem, a particle is moving within a box of length a. The normalization constant is $\sqrt{\frac{2}{a}}$. My question is if we take a negative ...
str's user avatar
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4 votes
1 answer
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When do two state functions represent the same quantum state?

According to the standard quantum mechanics, quantum states are one-dimensional subspaces of a separable Hilbert space. In practice, this Hilbert space is $L^2(M)$ where $M$ is the classical ...
mma's user avatar
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What is the difference between a vector and a representation of a vector in QM?

What does the phrase The wave function is a representation of the abstract quantum state. Or more generally, $A$ is a representation of a vector $\vec V $ mean? What is the difference between a ...
GedankenExperimentalist's user avatar
1 vote
1 answer
157 views

Why is probability outside the infinite square well zero? [duplicate]

In an infinite square well, potential energy is given below, why is the probability of finding a particle in the position of infinite potential energy zero? $$V(x)=\begin{cases} 0,& \text{if } ...
GedankenExperimentalist's user avatar
1 vote
1 answer
57 views

Gaussian wave packet with complex coefficients [closed]

I am trying to obtain a representation of the momentum-space wavefunction $<p'|\alpha>$ Its position space wavefunction is given as $$ <x'|\alpha> = N \exp [-(a+ib)x'^2 +(c+id)x'] $$ where ...
raccoon's user avatar
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6 votes
1 answer
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Minimization over a function is equivalent to the problem of finding the minimum energy eigenstate in an infinite potential well?

I'm reading this paper [Eqs.(10,11)] and met the following problem. The author states that the following minimization problem $$ \underset{\tilde{g}\left( \mu \right)}{\min}\,\,\int_a^b{\left| \frac{\...
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