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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QM Propagator Question
On page 577 of Shankar's 'Principles of Quantum Mechanics' the author gives the Schrodinger propagator:
$$U_s(\textbf{r},t;\textbf{r}',t')=\sum_{n}\psi_n(\textbf{r})\psi^*_n(\textbf{r}')\exp[-iE_n(t-t'...
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What does the Wheeler-DeWitt equation imply about the Schrödinger equation concerning the wave function?
The WDW equation is: $\hat{H}(x)|\psi \rangle=0.$ Schrödinger’s time dependent wave function equation says: $$i\hbar \frac{\mathrm d}{\mathrm dt} | \Psi(t)\rangle=\hat{H}|\Psi(t)\rangle.$$
Does it ...
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Single vs. Double Dirac Potential
Both single and double Dirac potentials have an even ground state solution.
Supposing the 'strengths' $\alpha$ of the single and double Dirac potentials to be the same, i.e., supposing
$V_{single} = -\...
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Schrödinger Equation Energy Requirement $E \geq V_{\min}$
Problem 2.2 of Griffiths' Intro to Quantum Mechanics states that
"$E$ must exceed the minimum value of $V(x)$ for every normalizable solution to the time-independent Schrödinger equation."
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Potential Energy of the Helium Atom
So I just learned how to solve the Schroedinger Equation for the Hydrogen atom. The set up looked like this: $$\left[\dfrac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\dfrac{\partial}{\partial r}\...
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First order state correction for time independent perturbation theory
Following the derivation on the wikipedia page, https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)_ , $|n_{0}\rangle$ is an eigenstate of the unperturbed Hamiltonian $\hat{H}_{0}$, ...
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Assigning initial conditions for Schrodinger's equation
I am self-teaching myself quantum mechanics, and my understanding so far is as follows. In the most general case, we would like to find a wave function $\varphi(x,t) \in \mathcal{H}$, where $\mathcal{...
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Doubt regarding the completeness of ${\psi_n}$ in infinite potential well [duplicate]
The wavefunctions (without the time factor) for an infinite potential well (width: $0$ to $a$):
$$\psi_n=\sqrt{\frac{2}{a}}\sin\left(\frac{n\pi x}{a} \right).$$
The set of $\psi_n$ is complete as any ...
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Why are exact solutions limited to hydrogen-like atoms? [duplicate]
Why can we only find exact solutions to the Schrödinger equation for Hydrogen atoms without estimating. What is the problem with the mathematics of extending the Schrödinger equation to more ...
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Why is the probability current density $J$ zero when $Ψ$ is purely real or imaginary?
My intuition says that there is both a real and imaginary part to Ψ, so having a purely real/imaginary Ψ cannot work.
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Justifying separation of variables in solving the Schrödinger equation in 3D
Consider a 3D infinite square "box", satisfying the time-independent S.E.
$$
\begin{align*}
-\frac{\hbar^2}{2m}\nabla^2 \psi &= E\psi, \;\;\; x,y,z\in[0,a], \\
\psi &= 0, \;\;\;\;\;\...
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Is the form of Schrödinger equation without fields/potentials neglecting self-interaction?
I understand that a charged particle (like electron or proton) in some potential $V(x,y,z)$ would be described by the following form of the Schrödinger equation:
\begin{equation}
i \hbar \frac{\...
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Boundary condition like $\psi(0)= \psi(L) = 0$ [duplicate]
This might be very elementary. But I have been baffled for a while.
For the infinite square well potential, the boundary condition is that $\psi(0) =\psi(L) = 0 $. However, from real analysis, we know ...
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What is the variational energy of two spinless bosons with given interaction potential?
There was a question on my exam quantum mechanics that I wasn't able to solve and I am curious how it is done, I cannot find any reference in the section of pertubation theory that describes systems ...
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Why isn't the delta-function-potential solution the same as the infinite square well solution?
In Griffiths, there is a worked-through derivation for the solutions to the wave function for a delta-square potential case and for the infinite square well case. The infinite square well solution ...
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Using De Broglie matter wave hypothesis for expressing the wave funtion of a macroscopic body
Consider a body formed for $N$ identical particles, that moves freely at constant velocity through a region of space without interacting significantly with anything else. According to De Broglie's ...
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Separatibility of wave-function in Schrodinger equation describing motion of particle in spherical shell [closed]
Question.
Why wave-function describing rotation of particle around spherical surface is separable?
Why solution obtained assuming function is separable is general solution?
I am interested in proof of ...
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Doubt in interaction picture of time dependent perturbation theory
Suppose we have a time dependent Hamiltonian $H(t)$ such that $H(t)=H^{(0)}+\delta H(t)$.
$H^{(0)}$ is a known Hamiltonian and is time independent.
Now define $|\tilde\psi(t)\rangle$ as
$|\tilde\psi(t)...
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Kronig-Penney Model Boundary Value Assertion
I've been doing some work pertaining to Condensed Matter Physics and have been trying to fully understand the derivativion of solutions to the Kronig-Penney Model. I've looked at tutorials from ...
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How to take the derivative of an eigenvalue equation in quantum mechanics?
Equation 2.1 is $$H_0(t)\vert n \rangle = E_n\vert n \rangle$$
Equation 2.8 is $$\langle m \vert \partial_tn \rangle = \frac{\langle m \vert \partial_tH_0 \vert n \rangle}{E_n-E_m}$$
In the following ...
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Eikonal Approximation in Weinberg's Lectures on Quantum Mechanics
In Weinberg's Lectures on Quanutum Mechanics, it is said that
$$H(-i \hbar \nabla, \mathbf{x}) \psi(\mathbf{x})=E \psi(\mathbf{x})\tag{7.10.1}$$
and an ansatz solution motivated by the WKB ...
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Integral of time dependent Hamiltonian
Computing the time evolution of a quantum system described by a time-dependent Hamiltonian, $H(t)$, amounts to constructing the time evolution operator
$$U = \mathcal{T} \exp \Biggl( -i \int_{0}^{t} \...
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Omitting the negative exponential in the plane-wave solution of the Schrodinger equation
The time-independent Schrödinger equation in one dimension for a free particle,
$$\frac{-\hbar^{2}}{2 m} \frac{\partial^{2} \Psi(x)}{\partial x^{2}}=\varepsilon\Psi(x)$$
can be solved as a homogeneous ...
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Help with expanding to second order in perturbation theory
In McIntyre's Quantum Mechanics: a Paradigms approach (pg 318), we solve for the energies of a perturbed 2 level system to get that
$$E_1= E^{(0)}_1+\lambda H'_{11}+\frac{\lambda^2|H'_{12}|^2}{(E^{(0)}...
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Band gap perturbation theory
Say we are given the Hamiltonian
$$H=-\partial_x^2+c\cos{qx}$$
on for example $C^\infty([-L,L])$ and try to approximate its eigenvalues perturbatively.
Without the perturbation the eigenvalues are ...
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QFT-style vs. (NR)QM-style [closed]
This question refers to the differences between usual formalism of ordinary quantum mechanics (QM) and usual formulation of QFT. Speciffically, there are three questions I would like to know:
The ...
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Kronecker delta and Potential at an infinitesimal boundary x0
Help with a model please! This is intuitive I am sorry.
Two orthogonal wavefunctions, kinetic with V=0, both ground state, periodically inphase at the intersection, which is a boundary of ...
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Step potential bound states not bound
According to Griffiths, if the energy is less than the potential at −∞ and +∞ then the state is bound. For the step potential this would be if the energy is less than the step height. But there are no ...
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About disregarding the $Fe^{lx}$ term in the step potential wavefunction
The step potential is of the form
$$V= \begin{cases}
0 & x< 0 \\
V_0 & x>0 \\
\end{cases}$$
For simplicity, let $V_0>0$. If we consider $0<E<V_0$ first: Define $...
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Connection between Laplace's equation and hydrogenic electron Schrodinger equation
Consider Laplace's equation:
$\nabla ^2 V = 0$
This holds for an electric potential $V$ in a region of space where no charges are present. This includes a Coulomb potential of a hydrogen nucleus (...
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How to show that $\psi(x)=-A\exp(-\alpha^2x^2)$ satisfies TISE for $V(x)=\frac 1 2 m\omega_0^2x^2$? [closed]
I'm struggling to approach this 'show that' question:
Write down the time-independent Schrödinger differential equation for $\psi(x)$ in a one-dimensional and time-independent potential $V(x)$. In ...
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What would the wavefunction represent in an observer-less universe?
Suppose the universe has no observers, and the universe's dynamics is governed by the Schrodinger equation. What does the wavefunction represent now? Is it that parts of the universe keep measuring ...
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How does Feynman justify using this propagator expansion on the Dirac equation?
I'm slowly wading through Feynman's series of three papers from 1948-49 (Space-Time Approach to Non-Rel QM, Theory of Positrons, Space-Time Approach to QED). I think they're brilliant and they are ...
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How to solve for the scattering solution of following Schrodinger equation?
Suppose you have non-relativistic fermions scattering off a delta function potential.
It is an easy job to solve $H=-\partial_x^2+\epsilon \delta(x)$ by starting with an eigenfunction of the form $\...
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Why numerical solution of hydrogen Schrodinger equation diverged in higher energy levels?
I am studying Computational Physics by Thijssen. I also read this notebook https://github.com/tamuhey/python_1d_dft/blob/master/numpy_1ddft.ipynb. Then I changed this notebook to solve this ...
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Why are the energies of bound states quantized? [duplicate]
What mathematical condition can be invoked to justify quantization?
I would say that it is the boundary conditions by which the quantization is justified. Because thereby, only certain wave functions ...
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Why isn't the time dependent Schrödinger equation in the momentum basis simply the time independent one? [duplicate]
If $-i \frac{ \partial}{\partial x}$ becomes $p$ in the momentum basis, I would expect the energy to be the same:
$$i\frac{\partial}{\partial t} \to E$$
So the time dependent Schrödinger equation ...
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The composition property of the time-evolution operators
For time-dependent Hamiltonians in Quantum Mechanics and QFT, we define the time-evolution operator as a unitary operator $U (t_2, t_1)$ such that
$$ \tag{1}
|\psi (t_2) \rangle = U (t_2, t_1) |\psi (...
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In what way are eigenfunctions of an observable operator complete? [duplicate]
I am a physics undergraduate reading through Griffiths's 2ed Quantum book.
In section 3.4 (Generalized Statistical Interpretation), Griffiths states:
The eigenfunctions of an observable operator are ...
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Why does the integral of $E\psi(x)dx$ go to zero around the the delta function? [closed]
My lecturer writes:
Firstly, I assume the term with a second derivative is, well, exactly that - a second derivative and therefore intended to be $\frac{d^2\psi(x)}{dx^2}$ and not $\frac{d^2\psi(x)}{...
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Question about quantum mechanics and diatomic molecule
"The double well is a very primitive one-dimensional model for the potential experienced by an electron in a diatomic molecule (the two wells represent the attractive force of the nuclei)"
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Combine the completeness and linearity of Schrodinger equation's solution, can we say that any wavefunction can solve any Schrodinger equation?
From quantum mechanics, I learnt:
Completeness of energy eigenfunction: The energy eigenfunctions of the Schrodinger equation span the space, i.e. any state can be expanded as linear combination of ...
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How can Schrodinger equation be derived based on the argument of Lorentz transformation?
I came across this note which talk about obtaining the Schrodinger equation based on the argument of Lorentz transformation. However, I am not able to follow how it exactly works. Any help would be ...
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If arbitrary wavefunctions can be expanded as energy eigenfunctions of a schodinger equation, is it mean that it can solve schodinger equation anyway? [closed]
We know that the superpostion of solutions also sloves the schodinger eq, and any wavefunction can be expended as superposition of energy eigenstates. Is it means that any wavefunction can solve the ...
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Solving Schrodinger Equation for scattering off a periodic potential
I am interesting in solving the time-independent Schrodinger equation (TISE) for the scenario where we have an electron plane wave of fixed energy incident upon a potential that is infinite and ...
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Isn't it inaccurate to use the Schrödinger equation to find the probability that a macroscopic object will undergo quantum tunneling?
Since Schrödinger's equation doesn't show wavefunction decay or quantum decoherence, isn't it inaccurate to calculate the probability that a person or macroscopic object will quantum tunneling? I ...
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Is the Schrödinger equation the heat equation with imaginary constants?
Playing around with the Schrödinger equation, I separated the time partial derivative this way:
$$\frac{\partial \Psi}{\partial t}=\frac{i\hbar}{2m}\frac{\partial^2\Psi}{\partial x^2}-\frac{i}{\hbar}V\...
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Are there any nonlinear Schrödinger equations?
The 1D Schrödinger equation reads:
$$\frac{\partial \Psi}{\partial t}=\frac{i\hbar}{2m}\frac{\partial^2 \Psi}{\partial x^2}-\frac{i}{\hbar}V\Psi.$$
Now, generally we have $V=V(x)$ (or it dependending ...
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Depression in central peak of real part of diffusing Gaussian
I'm attempting to use Crank-Nicolson scheme to solve the TDSE in 1D. The initial wave is a real Gaussian that has been normalised. As the solution evolves, a depression grows in the central peak of ...
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How can quantum tunneling happen conceptually?
I have read in Griffiths' Quantum Mechanics that there is a phenomenon called tunneling, where a particle has some nonzero probability of passing through a potential even if $E < V(x)_{max}$.
What ...