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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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Need verification if this simple derivation of the Schrödinger equation is valid

By 1924 it was well observed that matter (as well as light) has wave-particle duality (later named quantum), and the wavelength-momentum-energy relation of quanta $$\lambda=\frac{h}{p}\;\;\...
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Effects of measurement on particle energy

According to quantum mechanics, once you measure a particle's energy, its wave function collapse into some state, an eigenfunction with some eigenvalue (which is the particle energy). But if a ...
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Does Fuzzy time interpretation is able to solve the problem of Quantum mechanics unification and General Relativity? [on hold]

Does Fuzzy time interpretation (more exactly fuzzy time-space interpretation) is able to solve the problem of Quantum mechanics unification and General Relativity? Fuzzy time-space is an ...
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Symmetric potential well different solutions

I have solved $H|\psi\rangle=E_{n}|\psi\rangle$ with $V(x)=0$ from $-a<x<a$ and $\infty$ otherwise. If I propose a solution of the form $\psi(x)=A_{n}e^{ikx}+B_{n}e^{-ikx}$ I arrive to the ...
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Do large objects follow Schrödinger equation?

I have read here that double slit experiment can be performed on molecules. Matter-wave interference with particles selected from a molecular library with masses exceeding 10000 amu https://...
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Why can $|\Psi (t=0)\rangle $ be written as a coherent superposition of some eigenkets?

Why can $|\Psi (t=0)\rangle $ be written as a coherent superposition of some eigenkets? One of the approaches to solve time dependent Schrodinger equation $i\hbar \frac{\partial |\Psi(t)\rangle}{\...
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Derivation of 3D simple harmonic oscillator energies in spherical coordinates

I'm trying to show the permitted energies of the 3D simple harmonic oscillator (which is spherically symmetrical) are: $E_n = \hbar \omega(N + \dfrac{3}{2})$ In particular, $V(x) = \dfrac{1}{2} m \...
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Does gravity affect the time evolution of a QM wave function?

We know that the Schrödinger equation describes the time evolution of a wave function, but how does gravity affect that evolution? For example, does the wave spread slower in a strong gravitational ...
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Why do we assume the potential is independent of time in the Schrödinger equation?

In just about every text I read (online or in paper), when they handle the time-dependent Schrödinger Equation, I see something along the lines of "we always assume the potential is independent of ...
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Do atomic orbitals “pulse” in time?

I understand that atomic orbitals are solutions to the time-independent Schrödinger equation, and that they are are analogous to standing waves ("stationary states"). However, even a standing ...
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49 views

Schrödinger equation in unknown potential well [closed]

Have a question from my class which I'm struggling with. We have a particle $m$ with wavefunction $φ=Ax \exp(-ax)$ when $x≥0$, and $φ=0$ otherwise. We are asked to show that the double derivative $...
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Why wavefunction becomes exponentially smaller during quantum tunneling?

I am interested in quantum tunneling and I am wondering why the wavefunction of a particle would becomes smaller so that there is a slight possibility of finding it at the other side of a big energy ...
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In Bohmian mechanics, it the uncertainty due to non-locality?

In the pilot-wave interpretation of quantum mechanics, each particle is driven by the pilot wave on the universal configuration space, and therefore its trajectory is determined nonlocally, and ...
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Adiabatic Approximation, Solving the Schrödinger equation

In the adiabatic approximation one looks at the Hamiltonian $$ H_0 = \sum_{i = 1}^{N_e} \frac{\vec{p}_i^2}{2m_e} + \sum_{i < j} \frac{e^2}{|\vec{r}_i - \vec{r}_j|} + \sum_{k < l} \frac{Z_k ...
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Can this kind of TDSE be solved by series?

I have a particular kind of time dependent Schrodinger equation: $$i \hbar \frac{\partial}{\partial t} \Psi (t) =(\hat{H_0}+i \frac{t}{\tau} \hat{H_1}) \Psi (t) \\ \Psi (0) = \Psi_0 $$ Both $\...
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Should physical solution to Schrödinger eq. always be real in one dimencional space? [duplicate]

one dimensional Schrödinger equation: $$ \left[-\frac{\hbar^{2}}{2 m}\frac{\partial^2\psi(x)}{\partial{x}^2} +V(x)\right] \psi(x)=E \psi(x)$$ I know that to calculate the eigenfunctions $ \psi(x) $ ...
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Discrete energy levels of electrons in isolated atom

My question is a duplicate of this. Please consider the equation $\nabla^2\psi + (2m/\hbar^2)[E-V]\psi=0$ (1) Potential of electron revolving hydrogen atom is given as $V=\frac{-e}{4\pi\...
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In the Schrodinger Equation for the Hydrogen Atom does $\frac{{\partial}f(x)}{{\partial}x}$ equal $f(x)\frac{\partial}{{\partial}x}$? [closed]

I was looking at the Schrodinger Equation for the Hydrogen Atom, and saw it in the form $$\left(-\frac{\hbar^2}{2{\mu}r^2}\left(\frac{\partial}{{\partial}r}\left(r^2\frac{\partial}{{\partial}r}\right)+...
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Changing widths of potential wells

I have the following question presented to me: A particle of mass M is moving in a one-dimensional infinite potential well of length $a$, with walls at $x=0$ and $x=a$, described by the potential: $...
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Feynman's derivation of Schrödinger equation. Potential spatial dependence

I am working on the book "Quantum Mechanics and Path Integrals" from Feynman and Hibbs. When finding the correspondence with Schrödinger equation he takes $$\eqalign{&\psi(x,t+\epsilon) = {}\...
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Stability condition of a finite difference algorithm

While studying finite difference methods of TDSE I found myself stuck on a reasoning step: The largest possible spatial curvature for the wave function, $\frac{\partial^{2}}{\partial x^{2}} \Psi(x, ...
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Wave-packet backwards emission

I am trying to reproduce the results of a TDSE two wave-packet algorithm described in "Computational Physics: Problem Solving with Python" by Landau, Paez and Bordeianu, in section 22.4 (Wave Packet–...
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Solving the problem using Many Body Perturbation Theory

I am trying to solve the following Hamiltonian using Many body perturbation Theory. $$H=\sum_{i=1}^{N}\Bigg[\frac{P_{i}^{2}}{2m} -\sum_{i,j}\frac{1}{|\vec{r}_{i}-\vec{R}_{j}|}\Bigg]$$. I split this ...
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Domain of the infinite square well hamiltonian

I am reading the book by Gitman et al. 'self-adjoint extensions in quantum mechanics'. In the book, they give a precise definition of the domain of the hamiltonian of an infinite square well. For ...
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Schrödinger equation variable substitution

Here is the equation sets: $$\frac{-h^2}{2m}\frac{d^2\psi}{dx^2}+\frac{h^2}{2m}\left(\frac{\rho(\rho-1)}{\sinh^2x}-\frac{\lambda(\lambda-1)}{\cosh^2x}\right)\psi=E\psi\tag{1}$$ there is a variable ...
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Why is the $k$-space in multiples of $2\pi/L$?

So when you find the solution to the Schrödinger equation you get that the wave function can have $k=n\pi/L$, $n=1, 2,3 \dots $ The problem I have is that when calculating the density of states of a ...
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Expressing the Schrödinger equation in 2nd quantised language

For times sake, I will only write about the non-interacting part of the Hamiltonian, $$H_0=\sum_{j=1}\left(-\frac{\hbar^2}{2m}\frac{\partial}{\partial x_j^2}+U(x_j)\right)$$ where $U(x_j)$ is some ...
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Feynman's derivation of the Schroedinger equation by expanding path integrals to first order in $\epsilon$

As discussed in the answer to How can one derive Schrödinger equation?, one should be able to "derive" the Schrodinger equation from the path integral formulation of quantum mechanics. However, ...
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Schrödinger equation from Dirac–von Neumann axioms

Using Dirac–von Neumann axioms, it isn't too difficult to derive $$\frac{d}{dt}\left|\psi\right>=ikH\left|\psi\right>,$$ where $H$ is some hermitian operator. However, how would one show that $k=...
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Direction of propagating wave in quantum barrier

Consider a step potential $V(x)$ where \begin{align} V(x) & = 0; \quad x\leq 0 \\ V(x) & = V_0; \quad x> 0 \end{align} Now consider the case where $E_0<V_0$. The solutions of the ...
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Wave function of a particle under $V(x)$ (QM)

Suppose I have a particle with mass $m$ and it's under potential of a certain $V(x)$. (NOT an infinite or finite potential well) Also given is the wave function at time $t=0$, $\psi(x,0)$. What is ...
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Spectrum of sinusoidal potential

In 1D quantum mechanics problems, the energy spectrum is often determined by the limits of the potential at $\pm$ infinity. Generally, the spectrum is continuous non-degenerate when energy is above ...
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WKB near turning point by means of complex integration (Landau & Lifshitz, Quantum Mechanics) [duplicate]

The question is basically about section 47. in Landau's Quantum Mechanics (non-relativistic theory) everything is fine until the sentence (at the beginning of the second page) where he says: it is ...
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Why does it matter that the propagator is related to the Green's function for the Schrodinger equation?

If $L = i \hbar \hat{H} - \dfrac{d}{dt}$, then $ L \psi(x,t) = 0$ is the Schrodinger equation. It is well known that we can solve the Schrodinger equation with initial condition $\psi(x,0) = f(x)$ ...
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The order of excited states

The energy level of electron in an infinite square well in three dimensions is given by $E_{n_1 n_2 n_3} =\frac{ \hbar^2 \pi^2}{2mL^2}(n_1^2 + n_2^2 +n_3^2)$. It is understood that $E_{111}$ ...
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Discrete time evolution in a non-Euclidean space?

The time independent schrödinger equation can be written as $$i\frac{\partial \psi}{ \partial t}=H\psi$$ if we consider the case of a 1D particle we can evolve it in time by discretising the ...
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Relation between the propagator and probability for the infinite well

This may be an easy question, but I am really confused about it. For the infinite square well, the (time-dependent) energy eigenfunctions are (inside the well):$$\psi_n(x,t) = \sqrt{2/L}\:e^{-iE_nt/\...
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A particle on a ring: orthogonality of eigenstates

Let us consider the quantum-mechanical problem---a particle on a ring of a circumference as $2\pi$ with a magnetic flux $A$ inserted through it: \begin{eqnarray} H=(-i\partial_\phi-A)^2, \end{...
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How can theories about 1D or 2D systems be generalized for 3D systems?

I was watching a lecture video from MITx $^\dagger$ by professor Barton Zwiebach. He proved a pretty cool theorem "every attractive 1-dimensional potential has a bound state"; however, that only holds ...
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64 views

Delta function eigenstate for non-zero potential

Consider the potential $V(x)=\frac{2}{x^2}$ and let $\frac{\hbar^2}{2m}=1$ for convenience. Now consider the function $\psi(x)=\delta(x)$. According to Griffiths (electrodynamics book) problem 1.45(a),...
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Qualitative understanding of negative step potential problem

In QM textbooks the single step potential problem is explained in great detail. However, it is hard to understand what happens when $V_{0}<0$. Could anyone please explain qualitatively, how the ...
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Is $\langle \phi_m|\dot{\phi}_n\rangle$ assumed real in electronic excitation theory?

I'm studying a topic of the Nikitin's book (see pages 101 and 105) which deals with nonadiabatic electronic transitions, considering the two-state approximation. I think that the author make ...
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How can I use the Wronskian to show the following relation? [closed]

I cannot solve the part(a) and (b) mathematically. Have no idea how to start solving the problem by using the property of wronskian.
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Is time-evolving wave function verified by direct measurement?

In quantum mechanics, if we know the initial wave function and the external potential, we can predict its wave function in any future point in time with Schrodinger's equation. Is there direct ...
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Wave function evolution of an electron [closed]

In many basic quantum mechanics books the wave packet of an electron is described. It will say that the wave packet will broaden as time evolves because of dispersion. But suppose the electron just ...
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Deriving the Pauli-Schrödinger equation from the Dirac equation

Since the Schrödinger Pauli equation describes a non-relativistic spin ½ particle. This equation must be an approximation of the Dirac equation in an electromagnetic field. I was trying to derive this ...
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Solving the free particle problem in momentum space

$\newcommand{\ket}[1]{|#1\rangle}$$\newcommand{\bra}[1]{\langle#1|}$(Note: this question was asked before here but I didn't follow the answer.) For the free particle, Schrödinger's equation is given ...
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What is the role of $\hbar$ in quantum mechanics? [duplicate]

Planck's constant $\hbar$ appears in the Schrodinger equation: $$i\hbar \frac{d|\psi\rangle}{dt}\ = \hat{H}|\psi\rangle$$ which implies for stationary states, $$|\psi(x,t)\rangle=e^{-iE_n/\hbar}|\...
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513 views

Picking the different solutions to the time independent Schrodinger eqaution

The time independent Schrodinger equation $$-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2}+V\psi = E\psi$$ can have many different solutions of $\psi$ for a particular value of $E$. For example, if we ...
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1D Time independent Schrodinger eq. with limit

The one dimensional Schrodinger equation for an exponential potential $$-{\hbar^2\over 2m} \frac{d^2\psi}{dx^2}+-\alpha\frac{e^{-\vert x\vert\over\epsilon}}{2\epsilon}\psi=E\psi$$ I am interested ...