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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Is gravity caused by the length contraction of a wave in an inelastic but tensioned medium? [closed]
As far as I know, all waves require medium through which they are propagated. A medium that couples two adjecent points in space together, so that the change in amplitude of one point causes a change ...
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Does rotational symmetry imply reflection symmetry for electrostatic interactions?
Consider two charge distributions $\rho_A(\mathbf{x})$ and $\rho_B(\mathbf{x})$. Suppose that the ground state energy of a system of $n$ electrons in a potential generated by the sum of these two ...
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Can a wave function discontinuous in the time variable be a solution of the Schrödinger equation?
It is well known that wave functions that are discontinuous in the space variable cannot be solutions of the Schrödinger equation, because the Schrödinger equation is a second-order differential ...
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Differential equation for the operator relating Heisenberg and interaction-picture fields
I'm following Schwartz's QFT textbook and he finds a differential equation for the operator
$$U(t,t_0) \equiv e^{iH_0(t-t_0)}S(t,t_0)\tag{p.85}$$
via the following steps:
$$ i\partial_tU(t,t_0) = i(\...
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Scaled Hamiltonian and sped up evolution
Suppose there is a time-dependent Hamiltonian and the Schrodinger equation is solved.
$$
i\hbar \partial_t U(t) = H(t) U(t)
$$
Now, how easy is it to solve a scaled version of the Hamiltonian (e.g., $...
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Understanding periodic boundary conditions [closed]
For a free particle, the solution of the (time independent) Schroedinger equation is
$$\psi = \frac{1}{\sqrt{V}} e^{ip\cdot r /\hbar}$$
Since over infinite space this is undefined (not normalizable) ...
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Derivation of SECOND-order time-dependent perturbation theory (TDPT)
Are there any detailed derivation of the second oder term of TDPT? I found a pdf note on google, but an important equation in this pdf maybe wrong.
How can I get (17) from (15) and (16)? It seems ...
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Why in QM the solution to Laguerre equations are ONLY Laguerre polynomials?
I am studying eigenfunction methods to solve Fokker-Planck equations and I got stuck with a calculation that is related to some typical problems in QM. In particular, the radial part of an hydrogen ...
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What is the difference between the superposition principle and completeness relation in quantum mechanics? [duplicate]
As far as I know, we say that any wavefunction which is a superposition of the solutions of the Schrödinger equation are also valid solutions.
On the other hand, according to completeness relation we ...
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Hellmann-Feynman theorem and the derivation of the Lippmann-Schwinger equation
When deriving the Lippmann-Schwinger equation, one denotes
$$H_\text{free}|\phi\rangle = E|\phi\rangle \tag{1}$$
with $H_\text{free}$ as the free Hamiltonian and
$$H|\psi\rangle = E|\psi\rangle \tag{2}...
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A confusion on the $T$-matrix in scattering theory
In scattering theory,
the Lippmann-Schwinger equation gives
$$|\psi^{+}\rangle=|\phi\rangle+G_{0}V |\psi^{+}\rangle,$$ where $(E-H)|\psi^{+}\rangle=0$ and $(E-H_{0})|\phi\rangle=0$, $$G_{0}=(E-H_{0}+\...
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How do I "force" a hamiltonian to describe a particle moving with a velocity in a specific direction?
I want to describe a system of two particles, each in a local harmonic potential, with an intermolecular potential between them. One of the particles is moving toward the other one. I wish to be able ...
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How does one use empirical pseudopotentials to model disordered random alloys?
How does one use empirical pseudopotentials to model disordered random alloys?
I am working on modeling random alloys and came across the following paper: https://link.aps.org/pdf/10.1103/PhysRevB.85....
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Double and single delta-function potential well energy similarity
I am working through Griffiths QM at present, and I came across a question that asks me to find
"the bound state energies in the limiting cases (i) $a \to 0$ and (ii) $a \to \infty$ (holding $\...
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Time dependent perturbation theory validity and initial condition
This question concerns the validity of the perturbation theory formula that is so commonly found. The section to which I explicitly refer is section 18 of Shankar, 2nd edition (pg 473 on). Per usual ...
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What are the linear combination of slater determinant for three electrons on three different spatial orbitals? [closed]
I read the following link : https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/08%3A_Multielectron_Atoms/8.06%...
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Is Sakurai's Modern Quantum Mechanics explanation that the oscillation period between $|S\rangle$ and $|A\rangle$ is infinite wrong?
[Errors that persist in the 3rd edition of Sakurai's textbook?]
The content dealing with the symmetry double-well potential contains an error in the coefficient of $|A\rangle$ in $|R,\,t\rangle$, ...
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Complex probability density of wave function in the first region for potential well
this is my first question in this forum. My original language is Spanish, so I'm sorry if you don't understand me completely.
So, I go with the problem.
I'm working the square well potential with ...
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Schrödinger equation with gravitational potential
In "On the Gravitization of Quantum Mechanics 1: Quantum State Reduction" by Roger Penrose, illustrates two different approaches to incorporating gravitational field into a quantum system.
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Reduced dynamics for the pure state of the system
First consider a closed system.
If it is known a priori that the initial state of the system is a pure state $| \phi \rangle$, then the von Neumann equation for the density matrix is
$$
\frac{d \rho}{...
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Quantum mechanics with a classical Chern-Simons term
In this post, quantum mechanics falls under what is traditionally called "first quantization". This is in contrast to quantum field theory which traditionally falls under "second ...
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Why can non-differentiable solutions to the Schrödinger equation be ignored?
To clarify the question, let's consider the particle in a box (infinite potential $V$ outside [0,1], potential 0 inside [0,1]). (But the problems illustrated here also apply to particles in a non-...
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How to Derive the Time Evolution Equation for Quantum Phase?
In quantum mechanics, the wavefunction $\psi(x,t)$ outputs a complex number that describes the probability amplitude of finding a particle in a particular place and time. The complex number can be ...
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Time dependency of Wave function and its probability density function (PDF) [closed]
When we study the Schrodinger wave equation, we have a time dependent wave function $\Psi(x,t)$, and when we deduce its Probability Density function we come to know there is no time dependence in the ...
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Uniform dynamics in quantum mechanics
I've found in the book "Quantum Processes System & Information" of Benjamin Schumacher the following definition of "uniform dynamic":
it often happens that the basic dynamical ...
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Correct way to take complex conjugate of the Schrodinger equation
There are one or two questions on this site regarding the complex conjugate of Schrodinger equation but they do not clear my doubt.
Question : The Schrodinger equation is $$iħ \frac{∂Ψ}{∂t} = HΨ $$ ...
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What, exactly, does Schrödinger's wave equation describe (just in plain English, without any of the math please) [closed]
I've been studying the philosophical foundations of quantum mechanics and would love to find out if my understanding is correct. Could anyone please let me know if the below description is accurate? ...
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Equation of motion of quantum fluctuations/ quasi particles
I have a question in my mind, that is haunting me for some time now. I am looking for an equation of motion for a quasiparticle. My actual problem is originated in the Gross-Pitaevski model and the ...
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Field theories where the potential solves a linear Schrodinger equation
Are there physical situations/applications where the potential solves a linear time-dependent Schrodinger equation, or where the gradient of a solution to the Schrodinger equation (after somehow ...
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Calculating the expectation value of the angular momentum operator
I'm not looking for the exact answer to the question, but rather why a certain way of solving it is chosen. We agree on the answer, but why is the approach different. I'm afraid it's a sign of me not ...
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Quantum Mechanical Current Normalisation
Consider an electron leaving a metal. The quantum-mechanical current operator, is given (Landau and Lifshitz, 1974) by
$$
j_x\left[\psi_{\mathrm{f}}\right]=\frac{\hbar i}{2 m}\left(\psi_{\mathrm{f}} \...
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Hydrogen radial equation solution's boundary condition for $r \to 0$ [duplicate]
I am studying the hydrogen atom and I am analysing the radial equation: $$\left[\frac{-\hbar^2}{2m} \frac{\partial^2}{\partial r^2} + \frac{\hbar^2l(l+1)}{2m}+ V(r)\right]u=Eu$$ with $V(r)$ equal to ...
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Negative kinetic energy on a step potential
I'm doing an introductory course on quantum mechanics. I'm having trouble with the explanation of the kinetic energy on the classically forbbiden region on a step potential ($V=0$ for $x<0$, $V=V_0$...
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Relationship between unitaries generated by a Hamiltonian and its negative sign
Consider two unitary operations $U_1$ and $U_2$ defined by:
$\partial_t U_1 = -iH_1U_1$ and $\partial_t U_2 = iH_1U_2$
Here, $U_1$ is generated by $H_1$ and $U_2$ is generated by $-H_1$, with the ...
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Question on 1D Scattering Resonances
I'm reading Henley and Garcia's Subatomic Physics. To introduce the concept of resonances they use a 1D square well scattering example. Resonances are where the transmission coefficient goes to one.
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Time derivative of complex conjugate wave function [duplicate]
We have
$$\frac{\partial \Psi}{\partial t} = \frac{i\hbar}{2m} \frac{\partial^2 \Psi}{\partial x^2} - \frac{i}{\hbar}V\Psi$$$$\frac{\partial \Psi^*}{\partial t} = -\frac{i\hbar}{2m} \frac{\partial^2 \...
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Derivation of Schrödinger equation in Feynman-Hibbs
I am going through the derivation in chapter 4-1 of "Quantum Mechanics and Path Integrals. Emended Edition" by Feynman and Hibbs. The chapter starts with a proof of the equivalence of the ...
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Unperturbed eigenvector combination for degenerate case in perturbation theory
My question has arised from the previously asked question. In short: I have Hamiltonian with a perturbation such that $\hat{H} = \hat{H_0} + \lambda \hat{V}$. I know eigenvectors for the unperturbed ...
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How can I convert a non-local potential $V (r, r') $ into a local but energy-dependent potential, i.e., $V (r, E) $?
I need to localize a nuclear non-local potential. Using both these non-local and local potential in the Schrödinger equation we should have approximately close eigenvalues.
I tried the way followed by ...
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Is Schrodinger's cat a problem of how we define identity?
I apologize if the question is somehow silly or useless. I was reading about the infamous Schrödinger's cat paradox and I thought that if we consider that a cat is composed of numerous atomic ...
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Does this double well potential contradict the fact that there is no degeneracy for one-dimensional bound states?
This example is taken from Modern Quantum Mechanics by Sakurai.
Consider a symmetric double well potential in one-dimension with a barrier of height $V_0$ and width $a$ at the middle. The eigenstates ...
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What's the meaning of the momentum operator?
I understand that a wave function $\psi(x, t)$ tells me that the probability to find the particle at position $x$ is $|\psi(x, t)|^2$. In the Schrodinger equation, we use the momentum operator $\hat{p}...
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Time-evolution operator in QFT
I am self studying QFT on the book "A modern introduction to quantum field theory" by Maggiore and I am reading the chapter about the Dyson series (chapter 5.3).
It states the following ...
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Relativistic Schrödinger Equation: How is it relativistic and can it be useful? [duplicate]
As is well known, the usual Schrödinger equation,
$$\mathrm{i}\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar^2}{2m}\Delta\psi+V\psi,$$
is not relativistic. It can be derived formally by applying ...
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I need to find the state of the system at a general time, knowing the Hamiltonian and the state at $t=0$ [closed]
The Hamiltonian for a certain three-level system is represented by the matrix
$$H = \begin{pmatrix}a & 0 & b \\ 0 & c & 0 \\ b & 0 & a\end{pmatrix},$$
where $a$, $b$, and $c$ ...
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How to get a lower bound of the ground state energy?
The variational principle gives an upper bound of the ground state energy. Thus it is quite easy to get an upper bound for the ground state energy. Every variational wave function will provide one.
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Bound states between neutrinos using Schrödinger's equation?
I would like to see if it's possible that neutrinos (with sufficiently slow velocities) could form bound states in a universe with matter (such as ours)
There is a cosmic neutrino background in the ...
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Is it possible to derive Schrödinger's equation from Hamilton's equations?
Accepting the postulates of quantum mechanics, so promoting the classical dynamical variables to operators with appropriate commutation relations, is it possible to "derive" Schrödinger's ...
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Determining the Sign of $E$ When Solving the Time-Independent Schrödinger Equation
I am having trouble understanding how to choose the sign of $E$ when solving the time-independent Schrödinger equation. I understand that for potentials where $V(\pm\infty) < E$, we want ...
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Finite-time effects in Landau Zener
Consider a two level system with a Landau-Zener Hamiltonian of the form
$$\hat{H}=\begin{pmatrix}v t&\beta\\\beta&-v t\end{pmatrix}.$$
The Landau-Zener formula provides a closed form for the ...
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