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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How do I use first-order perturbation theory to compute the first four energy levels related to a potential well?
I came across a problem whose statement is as follows:
An electron moves in the potential well $P (x) = -\delta$ for $- a <x <0$ and $P (x) = \delta$ for $0 <x <a$ (Fig. 13.7). Use first-...
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Entanglement networks and perspective
So, superposition/entanglement is supposed to be fragile because, if any information leaks from the system, the wave function collapses. However, the very definition of 'information leaking' must set ...
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Unanswered Question on Potential Step Function
I have looked at the questions on this stack exchange and did not find a single convincing answer. Please absolutely remember the mathematical definition of only 4 things as you read this. Probability ...
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Measuring momentum
I'm new here, possibly my apologies for misplacing.
After a measurement in quantum mechanics, the wavefunction collapses to an eigenstate corresponding to the outcome of the measurement. Thus if we ...
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Can we handle the wave function as if it was a real valued function? [duplicate]
I am trying to analyze in general simple one dimensional QM problems.
To be more specific let's consider this kind of Hamiltonian:
$$H=\frac{\hat{p}^2}{2m}+V(\hat{x})$$
From this one we can derive the ...
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3d solutions for 1d Schrödinger equation?
The general Schrödinger equation in 3d is
$$i\hbar\frac{\partial\psi}{\partial t}(\mathbf r, t)=-\frac{\hbar^2}{2m}\nabla^2\psi(\mathbf r, t)+V(\mathbf r)\psi(\mathbf r, t).$$
Now consider that $$V(x, ...
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Quantum model of the atom
please note that I am a high school student trying to understand the quantum model of the atom; I have only the most basic understanding of quantum mechanics.
I am trying to comprehend the wave nature ...
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Direct Series Solution Attempt of the Quantum Harmonic Oscillator
The non relativistic Schrodinger equation of the harmonic oscillator in dimensionless variables is
$$\frac{d^2 \Psi}{d \xi^2} = (\xi^2 - k)\Psi$$
where
$$k \equiv \frac{2E}{\hbar \omega}$$
According ...
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Lorentz transformation of wavefunction
Consider two frames $S$ and $S'$ that are related by a Lorentz transformation.
The wavefunction in $S$ is $\psi(x)$ where $x$ is the spacetime coordinates. In $S'$, the transformed wavefunction is $\...
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Doubt in a solved example from Quantum Mechanics: Concepts and Applications by Nouredine Zettili [closed]
Question 3.7 b) from Quantum Mechanics: Concepts and Applications by Nouredine Zettili, on page no. 188 (solved examples) - I understand all the solutions mentioned therein but can't figure out why ...
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Is the standard Fock state representation of squeezed vacuum not normalized?
It is said in arXiv:1401.4118 that Squeezed vacuum can be represented in the Fock state basis as:
$$|\mathrm{SMSV}\rangle=\frac{1}{\sqrt{\cosh r}} \sum_{n=0}^{\infty}\left(-e^{i \phi} \tanh r\right)^{...
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Can Squeezed Vacuum Produce non-zero $\langle E \rangle$?
I believe that squeed vacuum can be represented in the Fock state basis as:
$|\mathrm{SMSV}\rangle=\frac{1}{\sqrt{\cosh r}} \sum_{n=0}^{\infty}\left(-e^{i \phi} \tanh r\right)^{n} \frac{\sqrt{(2 n) !}}...
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Prove that the parity operator is Hermitian
We know that an operator is Hermitian when:
$\langle f|\hat{O}g\rangle$ = $\langle \hat{O} f|g\rangle$
Parity operator in 1D is simply defined as:
$\hat{\Pi} f(x) = f(-x)$
I don't know anything about ...
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Turning a bra made up of a tensor product of two bras into a ket (and vice-versa)
I know that in general the following statement is true: $$\langle\phi|\chi\rangle = \langle\chi|\phi\rangle^* $$
And for the operator $A$ then the following identity also holds:
$$ \langle \psi| A|\...
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Orthogonal eigenfunctions [closed]
I have to show that two eigenfunctions of an electron in a 1 dimensional infinite square well with different parity and different quantum numbers are orthogonal. I am attempting this by integrating ...
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Wavefunction in external electromagnetic fields
If we consider a spin in an external magnetic field, it starts to precess around the vector of the magnetic field. The same should be true for any angular momentum.
But what does this mean for the ...
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Question from Messiah's Quantum Mechanics
The question that I don't even know where to start on is as follows:
Utilizing the fact that any wave can be considered as a superposition of plane waves, show that in the absence of a field, the ...
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Does the time-independent Schrodinger equation in 1D have an exact and general solution?
The (time-independent) Schrödinger equation is for sure the most important equation in quantum mechanics:
$$-\frac{\hbar^2}{2m}\nabla^{2}\psi(\vec{r}\,)+V(\vec{r}\,)\psi(\vec{r}\,)=E\,\psi(\vec{r}\,).$...
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Mirror/Parity symmetry
I am trying to solve a problem of Griffiths' book.
$\hat{\Pi} \psi(\vec{r}) = \psi(-\vec{r})$ where $\vec{r}$= (x,y,z), eq. (1)
$\hat{\Pi}$ is the parity operator.
The problem says to show that ...
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Average over the Bloch sphere
Consider you have a function of a two-level wave quantum state $f(\vert \psi \rangle ) $, with $\vert \psi \rangle = \alpha \vert 0 \rangle + \beta e^{\rm i \phi} \vert 1 \rangle$.
With no loss of ...
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What will happen if I multiply a ket vector by a complex number?
I was reading Zettili’s Quantum Mechanics book. There I have seen
when a ket (or bra) multiplied by complex number, we also get a ket (or bra)
But how do we infer this by mathematics?
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Complex Nature of the Wavefunction
I am doing this problem and I realized the wavefunction is real. But we also already showed that the wavefunction needs to be complex. Why is it that the wavefunction given here is real? I first ...
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Transition between 2 energy levels - wave function picture
Suppose we have a system that has discrete energy levels (e.g. hydrogen atom, potential well) and the stationary solutions for the wave function are $\psi_n$. I would assume that there should be a way ...
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'Deriving' superposition representations for 3 particles
I'm trying to write up all the possible superposition states for 3 spin-1/2 particles (one spin-up, 2 spin-down). Lets denote $|\uparrow \rangle = |0\rangle, |\downarrow \rangle = |1\rangle$. ...
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Do quantum wave functions rotate through imaginary space?
Watching a visualization of Schrödinger’s equation, I noticed that the wave function for a 2-dimensional particle was placed in a 3-dimensional graph consisting of 2 Real axes and an Imaginary axis. ...
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Is the wave packet of a relativistic quantum particle oblate (flattened) or prolate (elongated)?
A free, non-relativistic wave packet of a quantum particle is usually thought of as having spherical shape. What happens for a particle moving relativistically? For an outside observer, is the wave ...
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WKB approximation difficulty - deciding what term to neglect
Consider the following quantum well:
Region 1 is a classically forbidden region, and hence the WKB wave-function will take the form of equation
$$\psi(x) = \frac{C}{\sqrt{q(x)}}e^{+\int_b^a q(x')dx'/\...
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Question on the State Vector of the Quantum Harmonic Oscillator
My book states that the wavefunctions for the quantum harmonic oscillator are
$$\psi_n(x)=(1/2)^{n/2}H_n \left(\sqrt{\frac {m\omega}\hbar}x \right) \exp \left( -\frac{m\omega}{2\hbar}x^2\right)$$ ...
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Treating the delta potential in a Schroedinger equation in 1D
It is a standard problem in quantum mechanics. For the equation
$$ -\psi'' + g \delta(x) \psi = E \psi ,$$
we integrate from $-\epsilon$ to $+\epsilon$ and thus get the boundary condition
$$ g \psi(0) ...
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Question on Probability Amplitudes
The Born rule implies that the probability density $\rho$ is defined as $$\rho(x,y,z)=|\psi(x,y,z,t_0)|^2$$ at time $t_0$. What is the difference in this probability density and the probability of a ...
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How could wavefunction be the same and yet it has degenerate eigenvalues?
In solid state physics, the schrodinger equation
$$H \psi_{\vec{k}} = E_{\vec{k}} \psi_{\vec{k}}$$
has solutions $\psi_{\vec{k}}(x)$. In the near free electron approximation, I was told that $$\psi_{\...
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Similarity of particle in a box and free particle
It can be shown that particle in a box and free particle have the same energy at certain wavenumbers (at an integer multiple of $\pi/L$ , where $L$ is the length of the box)
I am aware that the ...
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Question on the inner product of wavefunctions [closed]
When taking the inner product of a wavefunction $\Psi$ with itself, denoting the inner product as $(\Psi,\Psi)$, since $$\Psi(x)=\int \psi(x)\vec{x}dx$$ letting $$\overline\Psi(x')=\int \overline\psi(...
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Finite double potential barrier transmission coefficient
TL;DR: I want to calculate the transmission coefficient of a particle travelling into a finite double potential barrier system and I think I've got stuck by the fact that I have 9 unknown variables (...
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Question about an “empty ket” and Dirac's notation
This question is related to this other one and it's about Bra-Kets formalism. Hope I'm not bothering you but the truth is I'm very confused.
Reading 1939 Dirac's publication on Bra-kets notation "...
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Question on Probability Densities in Koopman-Von Neumann Mechanics
A book that I used to learn basic Classical Mechanics, called "No-Nonsense Classical Mechanics" by Jakob Schwichtenberg, defines the probability density in Koopman-Von Neumann Mechanics as
$$...
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Is Huygens principle true for any shape of wavefront?
I have read from a few sources that cylindrical waves propagate leaving a wake behind, differently from spherical and planar waves, which would propagate sharply, ‘cleanly’.
One example is this ...
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Phase difference of a standing wave
Before starting the actual question: I do not want any typical answer that anybody might have thought of or criticism and downvote without even reading the question properly.
I have been googling and ...
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How can matter, the things which we can touch and feel on a macro level, exhibit a wave nature?
Louis de Broglie suggested that, if a particle like electron has momentum and wavelength associated with it (due to Planck's constant), then it might be a wave. The region where it exists are those ...
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Variational method: Why do parameters differ for two trial functions (optimization)?
Below the potential and trial functions:
$$V(x)=(x^2-1)^2-x^2$$
Use the variational method with the two trial wave functions:
$$\psi_{\pm}(x)=A\left(e^{-\frac{(x-x_0)^2}{2\sigma^2}}\pm e^{-\frac{(x+...
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Is there a unique way to construct the overall spatial wavefunction for identical particles?
While studying the quantum mechanics of $N$ identical particles, I stumbled upon formulas for generalizing the spatial wavefunction for bosons:
$$\psi(x_1,...,x_N)=\frac{1}{\sqrt{N!\prod_\alpha N_\...
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For a radially symmetric wavefunction is $\Psi$ allowed to blow up at $r=0$ provided that $|\Psi|^2r^2$ doesn't?
For a spherically symmetric wavefunction the probability is proportional to $|\Psi|^2r^2$, and if the wave function blows up at $r=0$ then at $r=0$ $|\Psi|^2=\infty$, and $r^2=0$ meaning that the ...
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Why do different energy levels affect the probability amplitude of the wave function?
When you solve the Schrödinger equation for the classic particle in a box you get that $$\psi=\sqrt{ \frac{2}{l}} \sin{(\frac {n\pi x}l)}$$
where $x$ is the length from the leftmost point of the box, $...
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Solutions of the Harmonic Oscillator are *not* always a Combination of Separable Solutions?
Are there solutions of the Schrödinger equation that are not a linear combination of separable solutions and how do we find them?
In Griffiths, Quantum, Prob. 2.49, there is a solution of the (time-...
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On the 1D Quantum Mechanics Harmonic Oscillator
I was solving the P. 2.41 of Griffiths' Introduction to Quantum Mechanics.
Nothing really new until I read a proposed solution (from Griffiths' himself) for the problem in which it states that I can ...
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What is the Schrödinger equation used for exactly?
The Schrödinger equation is just another way of writing the conservation of energy, right? So how can you use it to find the quantum wavefunction? I mean in every example I've seen the wavefunction is ...
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Normalising a wave-function
So I have a small confusion when normalising an infinite well wave-function. The wave-function for my problem is $$Ψ(x) = Ae^{i(kx-wt)}+Be^{-i(kx-wt)}+Ce^{i(kx-wt)}+De^{-i(kx-wt)}.\tag{1}$$
Applying ...
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Time evolution of the Gaussian packet
I am trying to get the time evolution for the following initial condition:
$$ \Psi(x,0) = \left(\frac{1}{2\pi \sigma^2} \right)^{\frac{1}{4}} e^{- \left(\frac{ x-x_{0}}{2 \sigma}\right)^{2}} e^{i\frac{...
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How do we prepare the electron state $|s=1/2,s_z=+1/2\rangle$ in the laboratory?
If we are interested in preparing the electron state $|+\rangle=|s=1/2,s_z=+1/2\rangle$ in the laboratory, the obvious thing is to apply a magnetic field along the positive $z$-axis (${\vec B}=B\hat{z}...
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How is the nuclear wavefunction norm defined in normal coordinates?
Lets assume we have a diatomic molecule with a total of six Cartesian coordinates. Lets also assume that the BO-Approximation is valid and we can write the ground state wavefunction like this,
$$
\...
