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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Estimate time for a simple quantum evolution process
Consider the Hamiltonian given by the sum of two projectors $$H=-\gamma N P_s-P_w,$$ where $|s\rangle=\sum_{j=1}^N |j\rangle/\sqrt{N}$ is the uniform state on the $N$ orthonormal nodes $|j\rangle$, ...
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How to get Airy differential equation?
A particle with mass m is under the following potential
$$V(x) = \begin{align}
0 |x|\leq L ;
c(|x|-L) |x|>L
\end{align}$$
and the schrodinger equation:
$$ \psi (x)'' + \frac {2m}{\hbar^2}[...
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Localized wave packet in a large box
If I understand it correctly, the solutions to the Schrodinger equation for a free particle in a box are standing waves. What would happen if a localized wave packet (free particle), which is ...
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TISE for hydrogen atom [closed]
I am given the radial component of the time independent Schrödinger equation for spherically symmetric electron wavefunctions: $$- \frac{\hbar^2}{2mr^2} \frac d {dr} \left(r^2 \frac {d \psi}{dr} \...
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Schrodinger equation for free EM field
The question come from the fact that I've seen for the first time in my life the quantization of a field, in particular of the free em field.
I've study how it is possible to write the energy of the ...
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2answers
37 views
Schroedinger equation and an infinite universe
Given that the Schroedinger equation states that a particle can be found an infinite distance away from its "center" and the universe is infinite, why don't we find infinite particles at any given ...
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1answer
46 views
Reasoning about the quantum mechanical characterization of energy
I have the Schrödinger equation:
$$\dfrac{-\hbar^2}{2m} \nabla^2 \Psi + V \Psi = i \hbar \dfrac{\partial{\Psi}}{\partial{t}},$$
where $m$ is the particle's mass, $V$ is the potential energy operator,...
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Calculation of a 2D scattering length with different masses along x and y - contact interaction
I want to calculate the 2D scattering length of two particles with equal (but anisotropic) masses interacting via a pseudo contact potential. In a relative coordinate system, the Schrödinger equation ...
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Energy spectrum for Linear potential [closed]
A particle in one dimension $(\infty<x<\infty)$ is subjected to a linear potential $\lambda x$ where $\lambda$ is a positive constant.
(a) Find the approximate expressions for energy ...
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3answers
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Deriving the time-independent form of Schrödinger's equation
The motion of particles is governed by Schrödinger's equation,
$$\dfrac{-\hbar^2}{2m} \nabla^2 \Psi + V \Psi = i \hbar \dfrac{\partial{\Psi}}{\partial{t}},$$
where $m$ is the particle's mass, $V$ ...
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1answer
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What is the meaning of Schrodinger equation solution for bound state of delta potential well?
Let's assume that we have delta potential well with $V = -\lambda\delta(x)$, where $\lambda >0$. Now if we solve Schrodinger equation, we get one eigenvalue $E_b=-\frac{m\lambda^2}{\hbar^2}$ with ...
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Why are wavefunctions in Quantum Mechanics shown as complex Circular waves instead of real Planar waves?
I'm currently learning Quantum Mechanics from online video lectures and resources. In most of the web articles and videos, the wave functions are shown as circular waves $e^{i\omega t}$ instead of ...
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Is “Particle in a box” actually a misnomer?
In the usual statement of the Particle in a Box problem, we assume two infinite potential barriers, to hold its wavefunction constrained, so it goes to zero on both ends:
But instead of invoking some ...
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Is orbital and wave function are same thing?
As we know that wave functions are the solution of schrodinger wave equation which contains all the information about an electron. We also tought that these wave functions are the atomic orbitals of ...
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How to obtain large order perturbation series for cubic anharmonic oscillator?
Consider the potential
$$V(x)= \frac{x^2}{2} + gx^3.\tag{1}$$
Then the time-independent Schrödinger equation becomes
$$\left(-\frac{1}{2}\frac{d^2}{dx^2} + \frac{x^2}{2} + gx^3 \right)\psi = E(g) \...
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1answer
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Potential energy of particle in delta function potential
What is the potential energy of a particle in the single bound state $\psi_b(x)=\frac{\sqrt{m\alpha}}{\hbar}e^{-\frac{m\alpha}{\hbar^2}|x|}$ of the Dirac-delta potential well
$$V(x) = -\alpha \delta(x)...
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Quantum mechanics - potential step problem
I've done potential steps where V > E0 and V < E0, but not where it's equal to 0. How would I go about answering this question? Any help is appreciated.
See below.
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Is the Hamiltonian fully defined by a quantum state (vector)? [duplicate]
From what I have read, the evolution of a quantum state is determined by the Hamiltonian (Schrodinger equation). However, I'm trying to understand if the Hamiltonian itself can be fully derived from ...
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How can a wave function that is both solution of classical wave equation and solution of Schrödinger equation be written?
Do wee need to solve the classical wave equation and Schrödinger equation together?
Schrödinger equation has first time derivative while classical wave equation has the second time derivative. In ...
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1answer
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TISE and uncertainty in energy
We use time independent schrodinger equation to find Stationary state solution for some potentials. My question is that, these Stationary state solutions are physically reliable or not? I am asking ...
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114 views
Number of bound solutions of electronic Schrödinger equation
How can I tell how many solutions I will have for an electronic Schrödinger equation ?
For example, solving it for the hydrogen atom we get infinitely many solutions \begin{equation}
...
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1answer
48 views
Quantum numbers in various situations [closed]
In the following questions I will be only talking about spatial states so we can safely ignore the spin state.
1) First of all I want to ask why is it that 3 quantum numbers are all that is needed to ...
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1answer
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Constructing solution to the time-dependent Schrödinger's equation
Given the initial state: $$\Psi(x,t=0)=c_1 \psi_1(x)+c_2\psi_2(x)+c_yy(x)$$ where $\psi_1$ and $\psi_2$ are eigenstates of $\hat{H}$ and $y(x)$ is a normalizable function but is not eigenstate of $\...
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Solving TISE for particle in the box for Infinite square well
While Solving the TISE for a particle an infinite square well with potential given by:
$$
U(x) = \left\{
\begin{array}{ll}
0 & \quad -L/2 \leq x \leq L/2 \\
\infty &...
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2answers
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Why does quantum tunneling increase de-broglie wavelength?
The picture (taken from a textbook) shows how quantum tunneling occurs with electrons.
Why does the de-Broglie wavelength of the electron change when doing this? It does not make intuitive sense to ...
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1answer
62 views
What happens when you try to derive the GR formula from Schroedinger?
I have seen Newton derived from Schroedinger (Ehrenfest) and I have seen Newton derived from the General Relativity equation. I assume that it has been tried to derive GR from QM. Is this the main ...
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1answer
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Numerical way of finding energy spectrum of $N$-body Schrodinger equation
For a single particle trapped in a potential, one can discretize the Time Independent Schrodinger Equation and hence find the eigenvalues of the corresponding Hamiltonian by diagonalising numerically.
...
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1answer
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Writing down a metric tensor given parametric equation of the surface
Let me begin by saying this question isn't related to GR.
I'm reading a paper (see https://arxiv.org/abs/0903.0798v1) that talks about deriving a Schrodinger equation for an electron confined on a ...
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1answer
36 views
When does the wave function spread over the volume of a box?
I have heard colloquially that for any initial state, a particle enclosed in some volume $V$ will spread itself relatively evenly over that volume after large time, so that $|\psi(\vec{x})|^2\approx 1/...
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What is the name and meaning of the integral of potential energy?
The paper, Significance of Electromagnetic Potentials in the Quantum Theory by Aharonov and Bohm makes reference to a term, $S$. I'm curious as to if this term has a name and to its meaning.
$S$ is ...
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How to choose boundary conditions for numerical solution of Schrodinger's equation whose solutions are expected to die out “at infinity”?
I am using the "Shooting method" for solving the TISE with a "reasonably arbitrary" potential in 1D,with boundary conditions such that the eigenfunctions $\psi_n\to0$ as $x\to\infty$(And another ...
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1answer
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Why is the Laplace operator used in the Schrödinger equation? [closed]
Why is the Laplacian necessary in the time-dependent Schrödinger equation in a position basis for a non-relativistic particle?
$$i \hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\...
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2answers
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Discontinuous derivative of wavefunctions in the infinite square well potential problem?
I am intrigued about two points given in an answer to a similar question (https://physics.stackexchange.com/a/38198/262985).
On one hand, it is stated that wavefunctions inside the well (excluding ...
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1answer
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How to find the matrix elements of $ \hat{P}^2 $ in the $X$ basis?
In a resolution of a question in Shankar's book (https://www.physicspages.com/pdf/Shankar/Shankar%20Exercises%2005.01.02.pdf), the derivation of the matrix elements of $ P^2 $ is obtained as follows
$...
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1answer
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Question about Schrodinger equation in atomic units vs in SI units
In SI units, we see that the Schrodinger equation can be written as
\begin{align}
i\hbar\partial_t \Psi= \left(-\frac{\hbar^2}{2m_e}\Delta_x+V(x)\right)\Psi
\end{align}
whereas, in atomic units, we ...
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Solving an NMR Schrodinger equation
Fairly standard differential equations question. I always struggled with this as an undergrad because I skipped the diff. eq. math course. It has obviously come back to bite me.
I am trying to solve ...
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1answer
62 views
Are all bound states normalizeable?
Following Griffiths one may define a bound state to be an energy eigenstate
$$H|E\rangle=E|E\rangle\tag{1}$$
with an energy being smaller than the potential far away from the origin in the sense
$$\...
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1answer
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Confusion in Barrier Tunelling formula for physics
is the formula T = e^-2(alpha)a
where alpha = ((2m(V-E))^1/2)/h
or is it T = 16(E/V)[ 1-E/V]e^-2[(alpha)(a)]?
my book seems to have the second formula but Ive seen online solutions using the first ...
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1answer
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Quantum Tunneling in Dirac Delta potential
In quantum mechanics phenomenon of tunneling is well understood ; we know that there is some finite probability to find the particle in classically forbidden region but potential of this forbidden ...
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3answers
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Why do we take the wave function to be zero at the edges of the box when solving the Schrödinger Equation for particle in a box?
The evanescent wave would be penetrating the box so why don't we account for that even if it decays, there might be some part protruding the box with walls of infinite potential.
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1answer
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What is the physical reason the angular quantum number appears in the relativistic correction to the Schrödinger equation with a Coulomb potential?
The energy levels derived for the Schrödinger equation for the Coulomb potential are
\begin{equation}
E_{n}=-m c^{2}\frac{(Z \alpha)^{2}}{2 n^{2}}.
\end{equation}
If you add the relativistic ...
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1answer
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Under what assumptions does a state following the TDSE converge to its ground state?
Until $t=0$ a system is in an eigenstate $\psi_0(x)$ of the Hamiltonian $\hat{H}_0$. The time-evolution is the trivial phase factor. Now at $t=0$ the system changes to $\hat{H}$ (you can assume it is ...
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Decomposing the Schrödinger Equation
I heard that the Schrödinger Equation is (naturally) to two coupled first-order real differential equations, where one is a continuity equation for the probability amplitude and the other is somehow ...
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1answer
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Infinite well particle probability where it will most likely be found
Suppose a particle is in the first excited state of an infinite square well of width $L$, where would the particle most likely be found if its position were measured?
So by looking at the probability ...
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2answers
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Why do we put an electron in a potential well to derive the Schrödinger equation?
I do understand the math behind and the "derivation". And I do understand why it is smart for our equations to put it in a potential well with infinite thick walls and a length.
What I don't get: why ...
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3answers
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When Is It Appropriate To Use The Ladder Operator Method in Quantum Mechanics?
I'm trying to understand when it is intuitively obvious that the ladder method would be best used to tackle a problem in quantum mechanics.
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Finding the number of nodes in bound states of an attractive Yukawa Potential
Consider a particle of mass $m$ moving under the influence of an attractive Yukawa potential $V(r)=-ge^{-\alpha r}/r$.
(a) Write the Schrodinger equation for the radial wavefunction, $u(r)$. ...
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What happens when an electron has a positive value of energy and it is in a square well finite potential?
We know that in a finite square well potential like this one , when an electron has less energy than 0 (its energy is between 0 and -V0), it will be bound in the well. If I have an electron in the ...
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1answer
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Evaluation of an integral in the variational treatment of the ground state of $\rm He$ atom
The hamiltonian of the electrons in an He atom, in CGS units, is
$$
\hat{H} = -\frac{\hslash^2}{2m}\left(\nabla_1^2 + \nabla_2^2\right) - Ze^2\left(\frac{1}{r_1} + \frac{1}{r_2}\right) + \frac{e^2}{r_{...
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Perturbation expansion in bound scattering states for double Dirac barriers [closed]
I was working my way through scattering theory notes by David Tong.In there,he discusses the analytical property of the $S$-matrix and uses it for the resonance states for the Double - Dirac potential ...
