Questions tagged [schroedinger-equation]

Partial differential equation which describes the time evolution of the wavefunction of a quantum system. It is one of the first and most fundamental equations of quantum mechanics.

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Schrodinger equation of linear combination of quantum states

We know that the solution for $i\hbar \frac{\partial}{\partial t}|\psi (t) \rangle = H|\psi (t)\rangle $ where $H$ is time-independent Hamiltonian, is $|\psi(t)\rangle = e^{-iHt/\hbar}|\psi(t=0)\...
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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 ...
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About the solutions of Paraxial Equation and Schrödinger Equation

There is an analogy between the Schrödinger Equation for a free particle: $$ -\frac{\hslash^2}{2m} \nabla^2 \psi (x,y,t) = i\hslash \frac{\partial \psi (x,y,t)}{\partial t} $$ and the Paraxial ...
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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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Deriving non-relativistic potentials from QFT

Some systems, like atoms, are described well by quantum mechanics, where one just gives the Hamiltonian in the form $H=T+V$ and computes the eigenvalues and eigenvectors of this operator to figure out ...
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ODE solver for Hamiltonian evolution of a finitely-dimensional pure quantum state

I want to simulate evolution of a finitely-dimensional pure quantum state: $$\frac{d\psi}{dt} = - i H(t) \psi$$ The wavefunction $\psi$ is a $n$ dimensional complex vector, and the Hamiltonian $H$ is ...
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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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Discrete Spectrum vs Continuous Spectrum and Bounded, Scattering States

Apolgies in advance if this is a confusing ramble and multitude of questions, I'm not quite sure how to articulate myself. I am currently reading up on quantum mechanics and seem to have confused ...
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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 ...
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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_{...
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Solving Schrodinger equation with a harmonic oscillator potential

This is referenced from the textbook Introduction to Quantum Mechanics by Griffith. I am learning about the application of ladder operators to solve algebraically the Shrodinger equation for harmonic ...
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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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How was kinetic energy for a particle measured?

To my understanding, the original objective of the Schrödinger equation was to find the total amount of energy within the system. So, for the time independent particle in a box, which has 0 potential ...
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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<...
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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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The Hamiltonian of Schrödinger field: symmetrized and anti-symmetrized form

I have troubles to prove the eq.(1.1) in the article of S.Kamefuchi & Y.Takahashi: "A generalization of field quantization and statistics" Nucl.phys 36. (1962) 177-206: $$ H = \frac{1}{...
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$S$-Matrix and Relation to Phase Shifts

Given some potential $V(x)$, we can describe the amplitude of incoming and outgoing waves through the scattering matrix $S$ whereby $$\begin{pmatrix} B \\ F \end{pmatrix}= \begin{pmatrix} S_{11} & ...
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Eigenstates meet the condition but resulting Linear Combination state doesn't

There is something I don't understand. When we solve the Time Independent S.E. for example for a particle on a circle $[0,L]$ we have our state $\Psi$ that writes as a Linear Combination of $\Psi_n$ (...
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Stationary Schrödinger Equation in Momentum space

Given the time dependent equation: $$\partial_t\,\hat{\psi}(p,t) = \dfrac{p^2}{2\,m}\,\hat{\psi}(p,t) + \hat{\psi}(p,t)\star{\hat{V}(p)}$$ and forcing through some kind of separation: $\hat{\psi}(p,t) ...
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Product rule for bras and kets

For the time evolution of expectation value of an operator $\Omega$, we can write $$\frac{d}{dt}\langle\psi|\Omega |\psi\rangle=\langle\dot\psi|\Omega|\psi\rangle+\langle\psi|\dot\Omega|\psi\rangle+\...
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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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Why Does Exchange Symmetry Stabilize Quantum Systems?

When trying to solve Schrödinger equation approximately, we usually represent wavefunction of the system as the Slater determinant of some set of spin orbitals to take into account that fermions (...
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Energy-Momentum tensor in the non-relativistic limit of Klein-Gordon Field

Assume we have a real Klein Gordon field $\phi(x,y,z,t)$, and we do the non-relativistic expansion of it in terms of a complex field $\psi(x,y,z,t)$ $$\phi=\frac{1}{\sqrt{2m}}(\psi e^{-imt}+\psi^* e^{...
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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: ...
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QM: How do electrons affect each other's potential energy?

I'm trying to learn quantum mechanics. And I'm confused. The question is, do electrons affect each other's wave functions, i.e. potential energies? How do they affect each other orbital shapes, how do ...
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Why is there a non-zero probability density of finding an $l=0$ electron at the origin of a Hydrogen-like atom?

A well known result for the $l=0$ hydrogenic functions is that $$\psi_{nlm_l}=R_{nl}(r)Y_{lm_l}$$ $$|\psi_{n00}|^2=\frac{Z^3}{\pi a_0^3n^3}$$ where $R_{nl}$ and $Y_{lm_l}$ are the radial function and ...
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Why doesn't the Hydrogen atom, as described by the Dirac equation, collapse?

In Griffiths quantum mechanics, it's noted that the exact energies for the Dirac equation, involving fine structure, are $$E_{nj} = mc^2 \left\{ \left[1 + \left(\frac{\alpha}{n-(j+1/2)+\sqrt{(j+1/2)^2 ...
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Fourier-Transforming the Schrödinger Equation in order to solve it?

In Quantum all time favourites equation is given by: $$-\dfrac{\hbar^2}{2\,m}\,{\partial_{x}}^2\psi(x,t) = i\,\hbar\,\partial_t\,\psi(x,t)$$ What happens if you were to apply a Fourier-Transform on ...
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How can we ignore the diverging term $e^\infty$ in the integral?

In Question (2.20) of Griffiths' Quantum Mechanics book, they have given this Solution. In the Solution of question 2.20(b), they omitted $e^{(ik-a) \infty}$ (or may have considered $e^{(ik-a) \infty}=...
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Schrödinger equation obtain $ψ(x,t)$ from $ψ(x,0)$

In this answer of the post "Wave packet expression and Fourier transforms" it is said that for the S.E. we have this property: If we start with an initial profile $ψ(x,0)=e^{ikx}$, then the ...
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First derivative boundary condition for scattering phase shift

My understanding is that in the partial-wave approximation we can solve for the phase shifts $\delta_{\ell}$ by (in my case, numerically) integrating the radial equation $$\frac{d^2 u_\ell}{dr^2} + \...
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Visualising the time dependent 2d schrodinger equation

So I managed to visualise the 1d schrodinger equation using the following algorithm: First solving the time independent schrodinger equation (1d) for the particle in a box potential, $$\psi_n(x) = \...
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Density of states of a finite potential well

Considering a finite square potential well. The solution of it gives the isolated bound states (below zero) and continuous scattering states (above zero). Here the isolated and continuous are the ...
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Deriving Schrödinger equation from Ehrenfest Theorem

Im reading the following article https://arxiv.org/abs/1105.4014. There they derive the Schrödinger equation from Ehrenfest‘s Theorem. By starting with the following equations: \begin{equation} \...
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Quantum mechanics of a moving bound state in the lab frame

Consider a non-relativistic quantum-mechanical problem of two bodies interacting via a confining potential $V(\vec{\mathbf{r}})$ (say, a hydrogen atom) moving with a constant speed $\vec{v}$ relative ...
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Are the shapes of atomic orbitals direct consequence of the Schrödinger equation?

I am trying to understand whether the shapes of the orbitals are inevitable given the standard model. They would probably change if we change the fine tuning of the fundamental physical constants, ...
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Negative time derivative of state function

I have been working on a relatively simple problem. Just take a quantum wave function for which a physical requirement is that an arbitrary displacement of x or an arbitrary shift of t should not ...
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Where does this time-dependent wavefunction of $\Psi(x,t)= \sum_{n=1}^\infty \psi_n(x)\exp(\frac{-in^2\pi^2\hbar t}{2mL^2})$ come from?

I was reading this blog post on simulating the probability desnity of the Schrodinger equation but there was one equation that I could not quite understand. Firstly, defining $$V(x) = \begin{cases} 0 ...
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What does Haag's theorem say about the Schrodinger picture?

Suppose there are two interacting fields $\phi _1 $ and $\phi_2 $. Let $\psi [\phi_1, \phi_2]$ be a functional with the two fields as the input functions and complex numbers as the output, such that ...
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Time-independent amplitude to go from one point to another in Feynman lectures (free particle)

In the third chapter of Feynman Lectures Volume III, I found this expression Suppose a particle with a definite energy is going in empty space from a location $\boldsymbol{r_1}$ to a location $\...
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Radial Schrödinger equation: from $R_l(r)$ to $u_l(r)$

I am in the 3-dimensional radial Schrödinger equation, in the spherical coordinates, where we try to find the separable solutions $$\psi(r) = R_l(r) Y_l^m(\theta, \varphi) \equiv \frac{u_l(r)}{r}Y_l^m(...
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Feynman path integral in an EM field

I'm studying Feynman and Hibbs, Quantum Mechanics and Path Integrals In problem 4-2, the book says for a particle of charge $e$ in an EM field the Lagrangian is $$L=\frac{m}{2}\dot{\boldsymbol{x}}^2+\...
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What happens if the width $a$ of the potential well gets large or goes to infinity?

What happens to the wavefunction if for a 1D infinite potential well of certain width $a$, we let $a$ go-to infinity? I think then is just a free particle and therefore it can be described with a ...
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The Schrödinger-Equation. A linear differential equation of what order?

Am I right in saying, that one cannot generally assign an order to the Schrödinger-Equation (SE)? E.g. if one considers a particle in a potential, the hamiltonian contains the second derivative of a ...
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Is it possible to approximate the shapes of molecules based on the solutions to the Schrodinger Equation of the Hydrogen Atom?

I understand that the Schrodinger equation of the Hydrogen Atom can be used to figure out the number of electrons that can fit in each shell electron shell, and the number of electrons that can fit in ...
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In non-relativistic QM are coordinate systems $(\vec{r_1},t)$, and $(\vec{r_2},t)$ indistinguishable if $\vec{r_2}=\vec{r_1}+\vec{u}t$?

As I understand it non QM reduces to classical physics when planks constant is negligible compared to the relevant action, and in non relativistic classical physics coordinate systems $(\vec{r_1},t)$, ...
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Can the cases of multiple fermions in any spherically potential be approximated by the Schrodinger Equation for a single fermion?

In this video https://www.youtube.com/watch?v=tq_y1qOmUBE&t=783s it's mentioned that the structor of atoms of multiple particles can be approximated using the Schrodinger Equation of the Hydrogen ...
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Time dependent Schrodinger equation through variation principle - questions about derivation

I'm reading a text which discusses time dependent variation principle (Geometry of the Time-Dependent Variational Principle in Quantum Mechanics by Kramer and Saraceno), and there is some part of a ...
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Definition of transmission and reflection coefficients for a particle

Quick intro: A 1D quantum particle is subject to the potential $$ V(x) = \begin{cases} 0 \;\;\;\;\; x\leq 0\\ V_0 \;\;\; x > 0 \end{cases} $$ I am trying to understand the definition of ...
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Why does the energy of an electron depend on the size of the well?

In the square well the energy states of the electron depend on the width of the square well. That means that by changing the physical shape of the confinement region (making it a parabolic well) or ...
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