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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Asymmetric delta potential well modelling wave function [closed]

I'm trying to model the wave function for an asymmetric delta function potential well, in which the left side of the well is at a potential of $0$, however the right side of the well has been moved ...
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How to get Airy differential equation?

A particle with mass m is under the following potential: $$V(x) = \begin{align} 0 |x|\leq L ; c(|x|-L) |x|>L \end{align}$$ and the schrodinger equation: $$ \psi (x)'' + \frac {2m}{\hbar^2} ...
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Time-dependent state probability harmonic oscillator [closed]

For my homework i am considering a harmonic oscillator which´s wavefunction at $t=0$ is the superposition of the eigenstates $\psi_n$. $$ \psi(x,t=0) = \sum\nolimits_{n} c_n \cdot \psi_n(x) $$ Now i ...
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TISE for hydrogen atom [closed]

I am given the radial component of the time independent Schrödinger equation for spherically symmetric electron wavefunctions: $$- \frac{\hbar^2}{2mr^2} \frac d {dr} \left(r^2 \frac {d \psi}{dr} \...
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Is it possible to reverse the spreading in space of a particle's wavefunction and if not, are all processes asymmetric in time?

Already more than a year ago, scientists claimed to have reversed the direction of time. What they actually did (see this article) to have reversed the direction of time. Which was hugely exaggerated. ...
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Energy spectrum for Linear potential [closed]

A particle in one dimension $(\infty<x<\infty)$ is subjected to a linear potential $\lambda x$ where $\lambda$ is a positive constant. (a) Find the approximate expressions for energy ...
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Deriving the time-independent form of Schrödinger's equation

The motion of particles is governed by Schrödinger's equation, $$\dfrac{-\hbar^2}{2m} \nabla^2 \Psi + V \Psi = i \hbar \dfrac{\partial{\Psi}}{\partial{t}},$$ where $m$ is the particle's mass, $V$ ...
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What is the meaning of Schrodinger equation solution for bound state of delta potential well?

Let's assume that we have delta potential well with $V = -\lambda\delta(x)$, where $\lambda >0$. Now if we solve Schrodinger equation, we get one eigenvalue $E_b=-\frac{m\lambda^2}{\hbar^2}$ with ...
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3answers
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Confused about definition of three dimensional position operator in QM

My QM text defines the position operator as follows: The position operator $X= (X_1,X_2,X_3)$ is such that for $j=1,2,3: \ X_j \psi(x,y,z)= x_j \psi(x,y,z)$. To me this can mean two things. 1) $...
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Why are wavefunctions in Quantum Mechanics shown as complex Circular waves instead of real Planar waves?

I'm currently learning Quantum Mechanics from online video lectures and resources. In most of the web articles and videos, the wave functions are shown as circular waves $e^{i\omega t}$ instead of ...
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Time dependent perturbation, particle on a segment

Consider a particle of mass $M$ moving alongside a segment of length $a$. Wavefunction: $$\psi_n(x)=\sqrt\frac{2}{a} \sin(\frac{n\pi}{a}x)$$ Energy: $$E_n=\frac{\hbar^2\pi^2 n^2}{2Ma^2}$$ Time-...
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What really are eigenstates? [closed]

As far I've understood/misunderstood, there exists a State Vector in an infinite Hilbert Space. When an operator acts on an eigenstate it yields an eigenvalue times the eigenstate. What is the ...
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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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On particle diffraction and its relation to the statistical interpretation of the wave function

Particles can be diffracted due to their quantum nature and that is understood by their wave-like behavior. Clearly seen in e.g. plane wave solutions of the Schrodinger equation or a superposition of ...
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1answer
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Is orbital and wave function are same thing?

As we know that wave functions are the solution of schrodinger wave equation which contains all the information about an electron. We also tought that these wave functions are the atomic orbitals of ...
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Do we get different Normalization constants when we solve for the Simple Harmonic Oscillator using the Analytic Approach and the Operator Approach? [closed]

I was trying to solve for $n=4$ of the Simple Harmonic Oscillator using the Analytic Approach and the Algebraic Approach (As Griffiths classifies it). After following the standard procedure of ...
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Potential energy of particle in delta function potential

What is the potential energy of a particle in the single bound state $\psi_b(x)=\frac{\sqrt{m\alpha}}{\hbar}e^{-\frac{m\alpha}{\hbar^2}|x|}$ of the Dirac-delta potential well $$V(x) = -\alpha \delta(x)...
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1answer
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Inverse of Wave Reduction

Let's consider a system of three apparatuses. Sequentially these act as A device that measures momentum of an electron Parallel plates where electric field between them changes randomly. Same device ...
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1answer
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Quantum mechanically, is superposition a sum or a product?

This question may sound like a no-brainer, but I'm getting confused after watching this lecture (cf. the slide at minute 5:07). The context is to motivate the quantization of a field which, for the ...
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2answers
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About the physical definition of the Planck's constant

In Quantum Mechanics (QM), the Planck's constant has the S.I. units of (energy) x (time). So, how should one interpret that?! How should I understand the physical meaning of the following relations? ...
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Hellmann-Feynman theorem for nearly degenerate states

If we have: $$\tag{1} E(P)=\int \psi_a(P)^{*} H(P) \psi_b(P)\ \mathrm{d} \tau $$ Then taking the derivative wrt. the parameter P yields: $$ \begin{aligned} \frac{\mathrm{d} E}{\mathrm{d} P} &=\int\...
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“Probabilities are the ghosts of quantum mechanical amplitudes”

I came across this quote today; [Quantum computers] process information using quantum mechanical amplitudes. And probabilities are sort of the ghosts of amplitudes after they have been degraded to ...
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1answer
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Quantum mechanics - potential step problem

I've done potential steps where V > E0 and V < E0, but not where it's equal to 0. How would I go about answering this question? Any help is appreciated. See below.
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Is the Hamiltonian fully defined by a quantum state (vector)? [duplicate]

From what I have read, the evolution of a quantum state is determined by the Hamiltonian (Schrodinger equation). However, I'm trying to understand if the Hamiltonian itself can be fully derived from ...
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Representation of wavefunction as superposition of eigenstates

In Quantum Physics it is postulated that, any general state $\psi$ can be represented as superposition of eigenstates with constant coefficients corresponding to any observable. Say, I have all the ...
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The superposition principle in quantum mechanics

This is a question parallel to this question The importance of the phase in quantum mechanics. In introductionary quantum mechanics I have always heared the mantra The superposition principle ...
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The importance of the phase in quantum mechanics

In introductory quantum mechanics I have always heard the mantra The phase of a wave function doesn't have physical meaning. So the states $| \psi \rangle$ and $\lambda|\psi \rangle$ with $|\lambda|...
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5answers
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Zero Solution in infinite square well?

Consider the well: $$V(x) = \begin{cases} \infty&\text{if }x<0 \\ 0&\text{if }x\in\left(0,L\right) \\ \infty&\text{if }x>L. \end{cases}$$ Solving the time independent ...
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Wavelength as an observable in quantum mechanics?

Recently I was discussing a problem with one of my students in which she found that two states of the particle in a box were orthogonal and was then asked to give an example of an observable that ...
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Symmetrisation of (fermionic) two-particle system without vs. with spin in wave function

I'm using D. J. Griffiths's textbook Introduction to Quantum Mechanics (3rd ed.) for my introductory university course on the subject. In chapter 5 (starting at section 5.1.1), he discusses the ...
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Collapse of wavefunction to its eigenfunction upon measurement

In quantum mechanics, it is postulated that to every observable, we have an associated operator. It is further postulated that when we do a measurement on a system, the measured value is one of the ...
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Do the hydrogen atom's electron orbitals have Gaussian probability density functions?

In this article they show the following diagram: Are all the diagrams in the little boxes really just Gaussian probability density functions with mean and variance (or covariance)? If not, what kind ...
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Perturbation theory - integrals over different wavefunctions

I am reading a research paper on how to calculate g-tensors* computationally (Correlated four-component EPR g-tensors for doublet molecules). In the theory section, they write "[...]from (quasi-)...
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1answer
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TISE and uncertainty in energy

We use time independent schrodinger equation to find Stationary state solution for some potentials. My question is that, these Stationary state solutions are physically reliable or not? I am asking ...
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1answer
50 views

Annihilation operator acting on density operator in the position representation

I have a silly question. I have a state $\hat{\rho}$ and I make the transformation $\hat{\rho}'=\hat{a}\hat{\rho}\hat{a}^\dagger$ (I want to subtract a photon). I expand in the position basis the ...
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1answer
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Normalisation of the following wavefunction: $\psi(\theta,\phi)=\cos(\theta)$ [closed]

Normalisation of the following wavefunction: $\psi(\theta,\phi)=\cos(\theta)$ So I thought about setting the following $N\int \cos(\theta)\cos^*(\theta) d\theta=1$ But then maybe I thought I was ...
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1answer
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Constructing solution to the time-dependent Schrödinger's equation

Given the initial state: $$\Psi(x,t=0)=c_1 \psi_1(x)+c_2\psi_2(x)+c_yy(x)$$ where $\psi_1$ and $\psi_2$ are eigenstates of $\hat{H}$ and $y(x)$ is a normalizable function but is not eigenstate of $\...
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Solving TISE for particle in the box for Infinite square well

While Solving the TISE for a particle an infinite square well with potential given by: $$ U(x) = \left\{ \begin{array}{ll} 0 & \quad -L/2 \leq x \leq L/2 \\ \infty &...
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2answers
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How to get effective quantum numbers of a linear combination of $\rm H$-atom wavefunctions?

The convention for the Hydrogen atom's interpretation subject to the laws of quantum mechanics is that you can prove the quantization of $|L|$, $L_z$, and Energy through quantum numbers $\ell$, $m_\...
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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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Why does quantum tunneling increase de-broglie wavelength?

The picture (taken from a textbook) shows how quantum tunneling occurs with electrons. Why does the de-Broglie wavelength of the electron change when doing this? It does not make intuitive sense to ...
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1answer
294 views

How to get the operator in Dirac bra-ket notation?

Suppose I have an operator given as $$T_a \psi(x) = \psi(x+a)$$ Is there a way I can get the operator in Dirac bra-ket notation. I am a newbie to QM. Please do give me hints.
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Foliation of wavefunction

Is it meaningful to take different time slices (and other hypersurfaces) through a quantum wavefunction $\Psi$? Below is a spacetime diagram, with axes $x$ and $t$. The wide band represents the ...
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Expectation value of $p^2$ for an arbitrary real state function

I'm trying to obtain the expectation value of momentum squared for a real state function $(\Psi^*=\Psi)$. I already know that \begin{equation} \left <p \right >=0 \end{equation} for a real ...
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1answer
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When does the wave function spread over the volume of a box?

I have heard colloquially that for any initial state, a particle enclosed in some volume $V$ will spread itself relatively evenly over that volume after large time, so that $|\psi(\vec{x})|^2\approx 1/...
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1answer
114 views

Integral kernel of the adjoint of operators

In the general case, the linear operator $\widehat{L}$ can be associated to a linear integral, $$\Phi(\xi)\equiv\widehat{L}\Psi(\xi)=\int L(\xi,\xi')\Psi(\xi')d\xi',$$ where $L(\xi,\xi')$ is the ...
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1answer
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Trouble understanding method used in the proof of Kinetic Energy operator is Hermitian

I am trying to understand why do we go from line 1 to line 2 in the way its shown? Why can't we directly just take the momentum operator squared outside and jump to line 3?
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How to choose boundary conditions for numerical solution of Schrodinger's equation whose solutions are expected to die out “at infinity”?

I am using the "Shooting method" for solving the TISE with a "reasonably arbitrary" potential in 1D,with boundary conditions such that the eigenfunctions $\psi_n\to0$ as $x\to\infty$(And another ...
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1answer
56 views

Why is the Laplace operator used in the Schrödinger equation? [closed]

Why is the Laplacian necessary in the time-dependent Schrödinger equation in a position basis for a non-relativistic particle? $$i \hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\...
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2answers
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Discontinuous derivative of wavefunctions in the infinite square well potential problem?

I am intrigued about two points given in an answer to a similar question (https://physics.stackexchange.com/a/38198/262985). On one hand, it is stated that wavefunctions inside the well (excluding ...

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