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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When is a Schrodinger equation equivalent to a Fokker-Planck equation

In these notes on kinetic theory, Tong shows that the Fokker-Planck operator for a particle undergoing overdamped Langevin dynamics in a potential $V$ is equivalent to a Schrodinger operator with ...
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Relation between Schrödinger's equation and commutator of position and momentum [closed]

Schrödinger's equation is $$\hat H\vert\psi\rangle=i\hbar\ \partial_t \vert\psi\rangle$$ I was trying to motivate this equation in a very hand-wavy way and tried to get to $$\hat H\vert\psi\rangle=K\...
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Interpretation of a stochastic Schrodinger equation

Suppose we have a linear stochastic Schrodinger equation (SDE) describing the evolution of a system in a finite-dimensional Hilbert space: $$ d\psi(t) = \left(-iH(t) - \frac{1}{2}\sum_{j=1}^dR_j^{\...
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Why is proper guess of wave function essential in variation method? [closed]

Proper Trial wave function in variation method with equations where necessary
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Schrödinger equation, 2D delta function potential, and confusion

Apropos of nothing in particular, I thought I would play around with the Schrödinger equation in 2D with a delta function potential. To keep things simple I thought I would concentrate on the bound ...
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Understanding equation for eigenvalues of a Hamiltonian

I'm reading the paper Hamiltonian Truncation Study of Supersymmetric Quantum Mechanics. I'm not understanding a claim they make about the eigenvalues of a certain Hamiltonian. In particular, how eqn 3....
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Is Schrödingers "quantization" as a symptom of the Infinite Potential Well the same as in the linear algebra equivalent? [closed]

Is the quantization observed when "simulating" the Schrödinger equation in an Infinite Potential Well equivalent to the certain stationary eigenvalues you can obtain from the Hamilton ...
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How is the Schrodinger kernel also a propagator? [duplicate]

I asked a very similar question earlier, but I think the question was unclear and so I'm hoping this post is more focused. If this question is still unfocused please let me know and I will revise it. ...
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Schrodinger kernel as a propagator [duplicate]

Let $e^{-it\hat{H}/\hbar}$ be the time evolution operator for a Hamiltonian $\hat{H}$ and $K(x,t)$ its associated integral kernel, i.e. $$\varphi(x,t) = e^{-it\hat{H}/\hbar}\varphi_0(x) = \int_{\...
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Help with solution for Schrödinger equation [closed]

solve one dimensional Schrödinger equation $-(ћ/2m)*(d²ψ/dx²)+(c_1x+c_2/x²)*ψ=Eψ$ for : $x≥0 c1>0 c2>0$ and $E<0$
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Resource for WKB approximation formula

Is there any source that explicitly writes down the WKB "function" (to be defined soon) in orders of time derivative of the frequency over the frequency? Of course only to some finite order. ...
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A simple example when the single-valuedness of the state function implies quantizedness

We know that in the case of the Aharonov-Bohm effect, the condition of the single-valuedness of the state function does not imply the quantizedness of the magnetic flux (see eg. here). However, in ...
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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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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 ...
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Estimate (time dependent) Hamiltonian from given time evolution of a density matrix

Basically the question is, whether you can give some estimation for the Hamiltonian of a system, given the time evolution of a density matrix $\rho$ under the assumption that it obeys the von-Neumann ...
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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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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}{\...
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Using the Bloch Ansatz in the Stationary Schrödinger Equation to Derive Dynamical Diffraction Intensity Equations

I am working on deriving the intensity equations for the dynamical diffraction of neutrons following along with a paper by Hartmut Lemmel (Hartmut Lemmel. Dynamical diffraction of neutrons and ...
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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 ...
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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 } ...
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Landau levels in symmetric gauge, what is the constraint on the quantum numbers?

After solving the Schrödinger equation for the charged particle in a constant and homogeneous magnetic field, using the symmetric gauge $\vec{A} = \frac{B}{2} (-y, x, 0)$, we could find the Landau ...
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Uniqueness of many-electron ground state in rotational invariant external potential

Does there exist a proof/conjecture/counterexample, to the statement that the fermionic ground state, of the many-electron Schrödinger operator (including spin-orbit interaction) with spherically ...
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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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On the validity of energy eigenvalues obtained when solving the Schrödinger equation for a particle in a 1D box

I'm having trouble understanding the legitimacy of solving the Schrödinger equation for a particle confined in an infinite square well. Aren't we supposed to solve it for the whole space and not just ...
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Connection between superconductivity and breaking of $U(1)$ symmetry in superconductors

$\newcommand{\Ket}[1]{\left|#1\right>}$Suppose I have a total Hamiltonian $H = H_0 + V$ given by the usual kinetic term $$H_0 = \frac{\hbar^2}{2m} \sum_{\mathbf{k}, \sigma = \uparrow, \downarrow} \;...
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Normalizing the wave-function [closed]

I want to show that the wave-function of $$\Psi(x,0) = \frac{1}{\sqrt{ 5 }}(2\psi_{2}(x)-\psi_{3}(x)) $$ for an infinite potential well of length $a$ is normalized. With the time-independent function ...
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Why Schrodinger equation does not contains internal energy?

well i was confused about something when we are comparing Schrodinger equation by classical mechanics, we can see it contains the total energy (or the Hamiltonian) equals the kinetic energy plus the ...
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How does the Green's function related the wavefunctions at different space-time points in Schrödinger's equation?

I have been trying to study Quantum Field Theory and have come across Green's Functions for the first time. While referring to Tom Lancaster's book Quantum Field Theory for the Gifted Amateur, the ...
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Normalizable solutions to the time-independent Schrödinger equation are real [duplicate]

In Griffiths Introduction to Quantum Mechanics (3ed.) problem 2.1, we are asked to prove that the normalizable solutions to the time-independent Schrödinger equation can always be chosen to be real, ...
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Analogy between the Electromagnetic Field and the Schrodinger Equation

In this answer my2cts says "The electromagnetic field is to photons what the Schrödinger or Klein-Gordon wave function is to electrons." Could someone expand on this further? Is this just a ...
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Generator of time shift when the Hamiltonian is time dependent

Let's consider the unitary group $\hat{S_{\tau}^†}$ such that :$$\hat{S^†_{\tau}}|\psi(t)\rangle=|\psi(t-\tau)\rangle$$ Since we know that: $$\hat{U}(t,t_0)|\psi(t_0)\rangle=|\psi(t)\rangle$$ Where ${...
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Parity of a bound state determined by potential

Some time ago in my QM class, we were working with an infinite well potential, and my professor told us we could know beforehand the bound states we were going to obtain for said potential would have ...
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Is the initial state the eigenstate of a Hamiltonian?

Solutions to the Schrödinger equation can take the form $ \psi(r,t)=\psi(r)f(t) $, where $f(t) = e^{\frac{-iEt}{\hbar}}$, $$ H \psi(r) = E \psi(r) ,$$ where $\psi(r)$ is the eigenstate of a ...
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Why do the Schrödinger and Dirac equations contain the mass?

I know the Schrödinger equation is bascially the "quantized" Hamiltonian formalism from classical mechanics, and the Dirac equation is the special-relativistic version. But these equations ...
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Global phase of wave function in quantum mechanics and Fubini-Study metric

I have a basic question about projective representations in quantum mechanics. In projective representation we identify the class of normalized states in Hilbert space as the same physical state as ...
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Schrodinger equation with $\hbar =1$

The Schrodinger equation is given by: $$i \hbar \frac{d}{dt}|\psi(t)\rangle = H(t)|\psi(t)\rangle.$$ Sometimes, physicists set $\hbar=1$. I suppose that they achieve this by changing the scaling and ...
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Numerical resolution of Schrödinger 1D Time Independent Equation, why do Energies not following the expected pattern? [closed]

I want to solve numerically the 1D time independent Schrödinger: $$-\dfrac{\hbar^2}{2m} \dfrac{d^2 \psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)$$ For starter, lets say we solve the Particle In a Box ...
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Separation of variables: When can we say that a function of $x$ and $t$ (for example) is a function of $x$ times a function of $t$?

I've seen this a lot in physics so far. For example in the stationary state solution to the Schrodinger equation for hydrogenic atoms, it's commonly approached using $\psi(r,\theta,\phi)=R(r)\Theta(\...
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Difference between stationary states, collision states, scattering states, and bound states

A few weeks ago, I was presented one-dimensional systems in my QM class, and of course one-dimensional potentials too. Nonetheless, I'm still a bit unclear about the terminology my professor uses. ...
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Creating Schrödinger cat states with trapped ions

We will consider an ion is in an harmonic trap. The ion has two internal states \lvert g\rangle and \lvert s\rangle and it interacts with a laser that induces a state-dependent force. The quantum ...
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Time Evolution of Eigenkets in the Heisenberg picture

I'm reading Modern Quantum Mechanics by Jun John Sakurai and in section 2.2 he talks about Base Kets and Transition Amplitudes. He goes to show, that $|a',t\rangle=\mathcal{U}^\dagger|a'\rangle$, (...
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Derivative of $c(t)$ in Adiabatic Approximation

In Sakurai's Modern Quantum Mechanics, second edition, $5.6.10$ is $$\begin{aligned} \dot{c}_m(t)=-\sum_nc_n(t)e^{i[\theta_n(t)-\theta_m(t)]}\langle m;t|\left[\frac\partial{\partial t}|n;t\rangle\...
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How to Choose which space to work in for Schrodingers equation?

I'm working out of Shankar's principles of quantum mechanics book. And overall, I think I get the gist of how to solve problems with Schrodinger's Equation. I recall in my Modern Physics course, we ...
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How do I find the wavefunction equation for a translated potential? [closed]

Let me explain myself. In my case, I know the wavefunction equation of a infinite-U potential which has the form: $$V(x)=\begin{cases} V_0,\ \ |x|\leq \frac a2\\ \infty, \ \ |x|>\frac a2\end{cases}$...
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Infinite potential well suddenly expanding

Problem statement: an electron is in its fundamental state in an infinite (1-dimensional) potential well, its walls being located at $x=0$ and $x=a$. Suddenly, the right wall moves from $x=a$ to $x=2a$...
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How is energy defined in QM?

I have been introduced to QM this spring semester. One thing that I couldn't understand is that how do they define the energy of an electron. For example while solving "Particle in a box" ...
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Simulating the helium atom using the Schrödinger equation numerically?

I recently set up a numeric solver of the Schrödinger equation and can receive solutions for single-particle quantum mechanical problems. I became interested in simulating atoms, since there is a ...
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Taking the non-relativistic limit of the Dirac Lagrangian

I know the usual derivation (as well as the Foldy–Wouthuysen derivation) to obtain Schrödinger equation from the Dirac equation. See for example U Alberta Phys 512. But it it possible to go from $$i\...
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Expectation Value Involving $s$-Wave Solutions to Central Potential

I previously posted a question regarding the expectation value described below, but it was closed because the question was not developed enough. Since I was given the option to delete it, I deleted it;...
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Deducing the ground state from a known first excited state

I am studying Schrodinger equation with a potential of hyberpolic functions. $$ H \psi = - \psi''(x) + \Big[1-\frac{12}{1+b \cosh{(2x)}} + \frac{15\,(1-b^2)}{[1+b \cosh{(2x)}]^2}\Big]\psi(x) $$ The ...
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