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 does a sine wave evolve under the Schrodinger equation?

On pg 72 of "Something Deeply Hidden," Sean Carroll discusses how the uncertainty principle is just a consequence of the Schrodinger equation. He writes: Consider a simple sine wave, ...
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Question regarding gravity and quantum entanglement

We know that the Earth is revolving in an elliptical orbit around the Sun. However, if the Sun suddenly disappeared, the information would then travel to the earth, at the speed of light, and the ...
2 votes
2 answers
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Particle density vs. Probability Density in Quantum Mechanics

I am currently reading trough "Bose-Einstein Condensation and Superfluidity" by Pitaevksii and Stringari and noticed some inconsistencies in my reasoning. In Chapter 5 (Non-uniform Bose ...
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2 answers
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In Quantum Mechanics is it possible to apply time evolution operator to wavefunction?

If I consider a wavefunction that is the superposition of Hamiltonian eigenfunctions, for example like: $$\psi(x)=\frac{1}{\sqrt{2}}\psi_1(x)+\frac{1}{\sqrt{2}}\psi_2(x)$$ with $\hat{H}\psi_1(x)=E_1\...
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About solutions of Schrodinger equation [closed]

If $\psi_1$ and $\psi_2$ are two independent solutions of the time independent Schrodinger equation, then is the product $\psi_1\psi_2$ also a solution of the same Schrodinger equation? If it's not ...
1 vote
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Reference request for a mathematical motivation for the Born rule

I was reading the popular science book The Hidden Reality by Brian Greene. My question is about a part in the notes at the end of the book. It is chapter 8, note 9. Brian Greene describes a ...
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2 answers
215 views

Understanding the solution of the infinite spherical well

I have been reading Griffith's Introduction to Quantum Mechanics, and I just went over the solution of the infinite spherical well. He gives it as $$\psi _{nlm}(r,\theta, \phi) = A_{nl}j_l\left(\beta_{...
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1 answer
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Probabilities of eigenfunctions

I am struggling to understand how to get the probabilities of each eigenstate occurring from a wavefunction that is a linear combination of eigenfunctions. If we have a wavefunction $$\Psi = A ( e^{...
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1 answer
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Pure state vs mixed state in this example

Consider, I have a quantum state $|\Psi\rangle$, such that : $$|\Psi\rangle=c_1|\psi_1\rangle+c_2|\psi_2\rangle$$ This is defined as a pure state, since I have complete information about the system. ...
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Delta function: Intuitive way for boundary conditions

Giving the Schrödinger equation $$-\dfrac{\hbar^2}{2\,m}\,{\partial_x}^2\psi(x)+ V(x)\,\psi(x) = E\,\psi(x)$$ with potential $V(x) = V_0\,\delta(x)$. Solving this equation using an ordinary Ansatz ...
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4 answers
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Is continuity of the wavefunction "put in by hand" for the Dirac delta potentials?

In 1d, for $V(x) = g\delta(x)$, integrating the TISE yields (assuming that $\psi$ is bounded$^\dagger$, so as to suppress the term containing $E$) $$ -\frac{\hbar^2}{2m} \left( \psi'(\varepsilon) - \...
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Why is this a negative frequency?

In general, i have noticed that i have simple accepted the fact that $$\psi = e^{i(kx-wt)}$$ represents a positive frequency, and $$\psi = e^{i(kx+wt)}$$ represents a negative frequency. After a time ...
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2 answers
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Single-particle wavefunction in Slater determinant

The ground state of $N$ non-interacting fermions can be written using a Slater determinant as: $$ \Phi_{GS}(\textbf{r}_{1}, ..., \textbf{r}_{N}) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \phi_{\mu_{1}}(\...
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1 answer
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Variational Method

A particle is moving in one dimension under a potential $V(x)$ such that, for large positive values of $x$, $V(x) \approx kx ^\beta$, where $k>0$ and $\beta$ $\geq$ 1. If the wave function in this ...
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1 answer
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Why is it always $\psi= A \sin kx + B \cos kx$ to solve a wave function?

Why is it always $\psi= A \sin kx + B \cos kx$ to solve a wave function instead of the one with $e^{ikx}$? Both are the solutions but the one with $e^{ikx}$ is seldom used.
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Relation $\Delta x \Delta p \approx \hbar$. Show that the condition $\Delta p \ll p$ guarantees that the packet does not spread

Consider a wave packet that satisfies the relation $\Delta x \Delta p \approx \hbar$. Show that the condition $\Delta p \ll p$ guarantees that the packet does not spread appreciably in the time it ...
2 votes
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Does this relativistic generalization of the Schrodinger equation make sense? [duplicate]

So I'm aware that the correct relativistic approach to quantum mechanics is through quantum fields, but I'm still interested in the question that follows. We know the Schrodinger equation in free ...
2 votes
2 answers
147 views

Traversing between nodes (Zero Probability) in Quantum Mechanics

To elucidate the question I have in mind, let us consider a very simple QM system that has exact solution, say Particle in a 1D Infinite Well. Since this is a well studied problem I will simply state ...
1 vote
1 answer
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1D bound state for a real potential

The prof says: "for 1Dimensional bound states with a real potential, the wave function is real, up to a phase". The proof goes like this: 1D bound states are never degenerated. So $\Psi_{...
5 votes
1 answer
461 views

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 ...
0 votes
1 answer
305 views

Black hole wave function

Does a black hole have a wave function? Matter falling into a black hole has a wave function. Is the wave function of this matter destroyed or converted to information on the surface of the black hole?...
3 votes
2 answers
188 views

How does $s$ subshell not have a node in the center despite the nucleus being there?

In most images of $1s$ subshell I see that there's no node shown at the center, and even the formula $n-\ell-1$ gives 0 as the answer. But, isn't the nucleus experimentally proven to be at the center? ...
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1 answer
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Regarding Griffith quantum mechanics problem 2.47: Square double well

I have a query regarding part b) of the question. I do not understand in particular why $E_1$ and $E_2$ will vary as a function of $b$. With my understanding of the double rectangular potential ...
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1 answer
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Why does applying the kinetic energy operator to a free particle result in a divergent integral?

The wavefunction of a free particle is just $$\psi = Ae^{i(kx-\omega t)}$$ and when you plug this into the Schrodinger equation you get the dispersion relation $$E = \frac{\hbar^2 k^2}{2m}$$ However, ...
2 votes
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How to derive Landau level with semiclassical approach?

I'm trying to derive the Landau level by applying semiclassical dynamics and the time-dependent Schrodinger equation. From that, I success to derive $E = \hbar\omega_c n$, but I fail to derive the ...
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1 answer
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Find the expectation value of angular momentum $L_z$ of a wave function in energy eigenstates

$\psi_{nlm}(r,\theta,\varphi,t)$ as expansion in energy eigenstates: $$\psi(r,\theta,\varphi,t) =\sum_{n=1}^{\infty}\sum_{l=0}^{n-1}\sum_{m=-1}^{l}c_{nlm}ψ_{nlm} (r,θ,ϕ) \exp \left(-\frac{iE_n t}{\...
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1 answer
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Reality, locality, and universality in the EPR paradox

Apologies if this has been asked before. I did some searching but didn't see it anywhere asked quite like this. Thanks in advance for any insights. Caveat: I am an organic chemist and thus ...
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3 answers
702 views

Electrons are 3 dimensional quantized waves (wave functions)?

I thought that electron wave functions were only mathematical of were to find the electron. Why don't atoms collapse if they are mostly empty space? What is the shape of an electron? This physics ...
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2 answers
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Why does it mean to say a quantum state's "penetration" is comparable to the distance over which it fluctuates?

Consider the following diagram of an energy eigenstate in a harmonic potential. The textbook from which I got this image says Also, in the classical limit the quantum mechanical probability ...
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Phase shift in square potential barrier when $E>V$

I'm trying to understand what happens to the phase of the wave reflected by a potential barrier when the energy $E$ is greater than the height of the barrier (i.e. $E>V0$) in the region $0<x<...
1 vote
2 answers
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Confusion about probability of finding a particle

The wave representation of a particle is said to be $\psi(x,t)=A\exp\left[i(kx−\omega t)\right]$. The probability of the particle to be found at position x at time t is calculated to be $\left|\psi\...
2 votes
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Could one, in principle, make any predictions using the wavefunction of the universe? [closed]

Do physicists talk about the wavefunction of the universe? What does that wavefunction even mean? Usually, wavefunctions describe probabilities of measurements of a system. But in this case, every ...
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1 answer
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Why sum of squares of the magnitudes of Fourier coefficients in Infinite Square Well equals one but it is not so in regular Fourier analysis?

My question is basically this.. In regular math, Fourier Coefficients give the "amount" a particular frequency is available in any periodic signal. The squares of sum of coefficients is not ...
2 votes
2 answers
67 views

Why not all Berry phase just vanished?

I just learned that for any real wavefuntions, berry phase equals zero. But in Griffiths' Problem 2.1(b), he proved that any complex wavefuntion can be written as linear combination of REAL ...
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1 answer
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Wave functions for two electrons in an infinite 2D potential well

Consider two electrons in a square 2D infinite potential well i.e $V=0\ for \ 0<x<a, 0<y<a, \ \ V=\infty$ everywhere else. Determine the energy and wavefunction(s) for the first excited ...
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1 answer
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Ordinarily continuous function of the wave function

I just started studying quantum mechanics using the textbook Introduction to Quantum Mechanics by Griffith. Under the section of solving the Shrodinger equation for a Dirac delta potential, he ...
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Boundary Condition for Free Wave Function

In Secion 3.5 of Quantum Field Theory An Integrated Approach, Fredkin, the author talks about Aharanov-Bohm Effect, where it says Define the wave function $$\Psi(\boldsymbol{r})=e^{i \theta(\...
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How to evaluate a non-banal derivate?

I need to evaluate the following derivate: $$\frac{dF}{d\Psi} = \frac{d}{d\Psi}\left[\beta\Delta\Psi+\alpha\left|\Psi\right|^2\Psi+\mu\Psi-i\vec{v}\cdot\bar{\nabla}\Psi\right]$$ where $\Psi$ is a ...
1 vote
3 answers
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General solution to the time-dependent 1d Particle in a box

In Griffith's text, he shows that $$\psi_n(x) = \sqrt{\frac{2}{a} } \sin \left(\frac{n \pi }{a}x \right)$$ is the solution to the time-independent Shrodinger equation for the 1d "infinite well&...
2 votes
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A wave function normalized for a given time $t=0$ is normalized for every time $t \gt 0$ [closed]

Given $\Psi(x,t)$ a wave function such that $$1=\int_{-\infty}^{\infty}\Psi^{*}(x,0)\Psi(x,0)dx$$ Prove that $\Psi(x,t)$ is normalized for every $t \gt 0$. My approach on this has been the following: ...
2 votes
1 answer
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What is the meaning of this wave function?

In these notes here the tight binding model for graphene is worked out. The tight Binding Hamiltonian is the usual: $$H=-t\sum_{\langle i,j\rangle}(a_{i}^{\dagger}b_{j}+h.c.)$$ where two different ...
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1 answer
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How to get the weight of an eigenstate inside the state of the system without knowing the state?

Let us suppose we have a system in a state $\Psi$, with: $\Psi = \sum_m c_m \psi_m$ Let us further suppose that we don't know what $\Psi$ or the $c_m$ are, but that we know what the $\psi_m$ are since ...
2 votes
2 answers
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The probability of finding the electron in the $\rm H$-atom

In the book Arthur Beiser - Concepts of modern physics [page 213] author separates the variables in the polar Schrödinger equation assuming: $$\psi_{nlm}=R(r)\Phi(\phi)\Theta(\theta)$$ then there a ...
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1 answer
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What is the physical meaning of the eigenstates of an operator in quantum mechanics?

Let us suppose that we have an Hamiltonian that describes a quantum system. If one would like to know all of the possible values that the energy of the system described by that hamiltonian, one has to ...
1 vote
1 answer
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How does the wavefunction of an antiparticle differ from that of the particle?

In this question I was answered that the invertion of wave function does not give antiparticles. Then how does the wavefunction of an antiparticle look, given the wavefunction of the corresponding ...
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Degeneracy of wavefunction in 1 dimension

Suppose we have a one-dimensional bound state, with the degenerate eigenstates given by $\psi(x)$ and $\phi(x)$. Using the Wronskian, we can show that there is no degeneracy, as the two functions are ...
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1 answer
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Energy (Hamiltonian) of Trial Wavefunction

Here I give a part of derivation of Hartree-Fock equations in case where basis functions (wavefunctions) are orthonormal and real: $$ \langle \psi_i | \psi_j \rangle = \langle \psi_j | \psi_i \rangle =...
1 vote
1 answer
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Definition of a wave packet

In Shankar's QM book page 168, the author stated a wave packet is any wave function with reasonably well-defined position and momentum. What does he mean by resonably well-defined position and ...
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Uncertainty of waves

Here in the pictures I have written about some question I have been thinking about a long time, what do you think? Link to the chapter I am talking about: http://www.its.caltech.edu/~matilde/...
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Finding coefficients for wave function when Fourier transform is not possible

I am looking at a wave function moving towards a potential step with potential $V_0$ for $x>0$ while having a total energy that is smaller than $V_0$. I already know how you can find the unbound ...

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