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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Heisenberg Picture unclarities
In our lecture about this topic, the two following statements were made:
An Operator $O_S$ (Schroedinger Picture) can be time dependent. But only when the operator $O_S$ is time independent, the ...
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Expressing a state in Schröndinger picture and time evolution operator
I have two question, regarding the derivation of two expression that I will post below:
If we have $\{ |\phi_{n,\tau}\rangle\}$ the set of eigenvectors of the Hamiltonian of a system. This set, is at ...
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Connecting asymptotic scattering solutions to short-distance numerical potentials
Problem
I'm dealing with a one dimensional quantum mechanical scattering problem in a finite region, say $x\in\left[0,L\right]$. At first, this problem is defined as a finite-difference problem, i.e., ...
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WKB approximation derivation for $E<V$
I understand that we can write any complex wavefunction on polar form $A\exp(iθ)$ with both $A,θ$ real. Following the logic of Griffiths on WKB (here, page 291):
We write the energy wavefunction in ...
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Galilean covariance of the Schrödinger equation without choosing a representation
The most general form of Schrödinger equation is
$$i \hbar \frac{d}{d t}\Psi(t) = H\Psi(t) \tag 1,$$
where $\psi(t)$ is an element of a Hilbert space $\mathcal H$ (not necessarily $L^2$), and $H$ is a ...
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A classical localization of a wave function?
We know that the state of a quantum particle defined on the real line is represented by its wave function $\psi(x)$ that is the position probability amplitude. We also know that the momentum ...
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Why do most introductory texts on QM use the Schrödinger formulation rather than Heisenberg's matrix mechanics? [closed]
I know they're mathematically equivalent, and that makes intuitive sense, seeing as linear differential equations can in general be solved using matrices and other linear algebra approaches. In fact, ...
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Free quantum particles with space-dependent mass
I'm interested in solving the 1D quantum problem for a free particle with the Hamiltonian
$$H = -\partial_{x}\frac{\hbar^2}{2m(x)}\partial_{x}$$
in the finite interval $x\in\left[0,N\right]$. This ...
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Potentials with continuous spectra except for discrete set of values
Section 18 of Landau & Lifschitz's Quantum Mechanics discusses how the Schrödinger equation with a potential that vanishes at spatial infinity can have a continuous spectrum, a discrete spectrum, ...
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Interpretation of time evolution in Quantum Mechanics
Let a quantum mechanical system, at time $t=0$, be described by:
$$
|\psi(0)\rangle = c_1(0) |E_1\rangle + c_2(0) |E_2\rangle \;,
$$
here $|E_1\rangle$, $|E_2\rangle$ are energy eigenstates.
Now, for ...
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Momentum Space Solutions of the 1D Schrodinger equation in the Potential $f(t)x$ [closed]
Yesterday, I asked a question about solving the 1D Schrodinger equation in a time varying potential $f(t)x$ using a method solely in configuration space. Although this approach does not directly ...
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Basis representation for isotropic 2D quantum harmonic oscillator
The basis functions of the 2D isotropic quantum harmonic oscillator are of the form
$$ \psi_{n,\ell} (r,\varphi) = A_{n\ell}(r)e^{i\ell\varphi}$$
where $A_{n\ell}(r) = \frac{\sqrt{2 \times p!}}{\sqrt{(...
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Problem on deep potential well [closed]
The problem is to find out the Schrödinger equation for a system of two particles $m_1$ and $m_2$ confined in a one dimensional well.
I have previously solved the problem of a single particle in a one ...
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Solving the 1D Schrödinger equation under the potential $V(x) = f(t)x$ [closed]
Following the sketch given in this answer, I hoped to solve the 1+1 dimensional Schrodinger equation under a potential $f(t)x$ by using a time dependent boost.
$$\left(\frac{-\hbar^2}{2m}\frac{\...
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Schroedinger equation for wave functional (QFT)
As far as I'm aware you can solve for the wave functional $\Psi[\phi]$ of a field using the Schrodinger equation $$i\hbar\frac{\partial \Psi}{\partial t}=H\Psi.$$
Should $H$ here be the Hamiltonian, ...
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Ground state of hydrogen atom, $e^{-\kappa r }$ or $e^{-\kappa r }/ r$
The ground state must be an $s$-wave state, so it depends only on the radius $r$. I cannot remember the exact form, but I know it must be one of the two.
Is there any simple way to determine which one ...
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Do solutions of the Schrödinger equation for multiple particles automatically obey spin-statistics?
Consider the Hamilitonian for a general two-electron system subject to an external potential $V_\mathrm{ext}$ and an interaction potential $V_\mathrm{ee}$. In this case
$$H\psi(x, y) = -\frac{1}{2} \...
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Does the wave function of system plus detector satisfy the Schrödinger equation?
Let $S$ be a quantum system and let $D$ be a detector. Suppose that $D+S$ does not interact with the environment. Now when $D$ makes a measurement of $S$, the wave function of $S$ collapses. Therefore,...
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How does a state change in time for a free particle?
I was reading my textbook and there was this question: For the fundamental state of an infinite square well:
$$ \left| \psi \right \rangle = \sqrt {\frac{2}{L}} \sin \left( \frac{\pi x}{L} \right)$$
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Is a continuum of bound states possible in a finite system Hamiltonian with a Coulomb potential?
I am wondering if it is possible to have a continuum of bound states in a finite system, for example a molecular system with a fixed number of nuclei and electrons. As a chemist, I'm used to ...
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Examples when wavefunction is changing but probability density is constant?
What are examples of wavefunction that changes with time but the square of wavefunction is constant?
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In a finite quantum well with a constant electric field, is it possible to find a bound state? [closed]
You have this equation
$-\frac{\hbar}{2m}\frac{\partial^2 \Psi}{\partial x^2}+(V(x)-eFx)\Psi=E \Psi$ where $V(x)$ is the potential for a finite quantum well.
Is there solution which is a real bound ...
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Schrödinger Equation as a limit of von Neumann equation
How would I derive the Schrödinger Equation as a limit of the von Neumann equation?
The quantum Liouville equation (von Neumann equation) is given by
$$i \hbar \: \partial_t \rho = [ H, \rho ] \quad .$...
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Force required to remove an electron from an atom [closed]
For a certain experiment, it would be useful to me to calculate the force required to remove an electron from an atom of varying atomic number. As I understand it, calculating the energy required is ...
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What happens step-by-step (and why) when a particle tries to escape an infinite potential well?
I am aware that the following question might be quite elementary. My background is mainly in mathematics and my physics education is limited to high-school level material (discounting analogues made ...
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Liouville-von Neumann $\frac{dI}{dt}=\frac{\partial I}{\partial t}+[I,H]$ invariants for the $3+1$-dimensional Schrodinger equation
Before I pose my question, let us assume that we are trying to solve the Schrodinger equation with a time-dependent Hamiltonian in one spatial dimension. We may take the Hamiltonian to be of the form
$...
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Calculating location of a complex pole of a Scattering Matrix
I am asked to calculate a pole in the lower complex momentum plane of an element from the Scattering Matrix for the potential $ V(x) = V_{0}(δ(x-1)+δ(x+1)$ . The potential parity even so it easier to ...
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The von Neumann series for time-dependent perturbation theory
i'm currently trying to understand the derivation of Fermi's Golden Rule as outlined in Schwabel's book "Quantum mechanics". When integrating the Schrödinger Equation, the "von Neumann&...
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How to apply Schrödinger equation to superposition with time dependent factors?
The question comes from reading through either of these two papers:
https://doi.org/10.1103/PhysRevB.35.3629
https://arxiv.org/abs/1811.05886
The question is on the time dependence of a state like:
$$|...
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Why is Griffiths using ordinary power series to solve Hydrogen atom problem?
The hydrogen atom problem leads to a differential equation of the form
$$\rho\frac{d^2v}{d\rho^2}+2(\ell+1-\rho)\frac{dv}{d\rho}+[\rho_0-2(\ell+1)]v=0\tag{4.61}$$ where $\rho_0$ is a constant. In ...
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Different "Pictures" in Quantum Mechanics [duplicate]
What are some examples of 'pictures' in quantum mechanics besides the famous Schrodinger and Heisenberg pictures?
In the Schrodinger picture, one takes the time evolution operator to be acting on ...
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Schrodinger Equation: Time Dependent, Periodic Potential $$V(x,t)=\begin{cases}V_0x&:t\in[0,\frac{T}{2})\\-V_0x&: t \in [\frac{T}{2},T) \end{cases}$$
Imagine that we have a particle in a cylinder of finite length and neglible radius. We can then assume that the system is axisymmetric and can be solved in one dimension.
Let us consider a time ...
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Can We Add a Dimension To The TISE To Get All Steady States In Time?
Suppose I have the Time Independent Schrodinger Equation (TISE) in 1D, if I add a second dimension, does the state in 2D now represent all the 1D states in time?
Similarly, if I have the TISE in 3D ...
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Are the probability amplitudes in the continuous case, always the coefficients of the eigenstates of the wave-fun. expansion?
Assume that we have a Hamiltonian eigenvalue problem with continuous energy eigenvalues $E$. Griffiths says that the inner product of an eigenstate $ψ$ with the total wavefunction $Ψ$ gives the ...
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Is this a valid reformulation of QM?
So let us consider the Schrodinger equation but written differently:
$$ - i \hbar \lim_{b \to 0}(\frac{\hat U(b t) - I}{b}) \psi = - \hbar^2\lim_{a \to 0}(\frac{\hat T(a \vec s) + \hat T(- a \vec s) - ...
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How to prove the bound and scattering states theorem? [duplicate]
In Griffiths it is mentioned that if the energy eigenvalue is less than the value of the potential at + and - infinity, then we have bound states. If however the energy is bigger than the potential at ...
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How is the non-locality of a theory apparent from its mathematical form?
I am reading Relativistic Quantum Mechanics by Bjorken and Drell and on page 5 they present the following attempt at a relativistic Hamiltonian for a free particle
\begin{equation}
i\hbar\frac{\...
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When solving the Schrodinger Equation, where do we add the condition that $E$ is real?
I'm reading through Griffiths, and I noticed two seemingly contradictory facts:
In Chapter 1, it is proved that for any square-integrable function solving the Schrodinger Equation, $$\frac{d}{dt}\...
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Schrödinger equation and $\rm U(1)$ group
I watched two youtube videos: https://youtu.be/paQLJKtiAEE and https://youtu.be/V5kgruUjVBs. Now, I compared those two videos. One talk about the breaking of the Schrödinger equation after applying ...
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When spectrum of eigenvalues of Schrodinger equation is continuous or discrete? [duplicate]
There is a point which I do not understand about the Schrödinger equation. I will try to explain the issue.
Consider the Schrödinger equation:
$\hat H = \frac{ \hat p^2}{2m} + U(\hat x)$
and we are ...
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What happens in the square potential barrier during Klein Scattering in 1D graphene?
I have a question about Klein Scattering in a square potential barrier, $U$, incident on 1D graphene. I have managed to get these equations from continuity equations at $x=0$ and $x=L$ of the ...
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Wavelength of wavefunction graph [closed]
So my professor loves to make like 10 exam questions where we interpret some graphs of quantum wave functions. Now I really do not understand how he reads off the wavelength for instance in these ...
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Delta function in discontinuous derivative of a wavefunction
In variational principle problem; we need to find $\frac{d^2\psi}{dx^2}$ to find $\langle T \rangle$; if
\begin{align}
\psi = \begin{cases}
A \cos ( \pi x / a ) , & -a/2<x<a/2\\
0 ...
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What is the hydrogen atom wavefunction in cartesian coordinate? [closed]
The ground state wavefunction of the hydrogen atom is given in spherical coordinates in the form of this:
$$\Psi(r)= exp(-r)$$
where r is the radius from the center of the atom.
Suppose we would like ...
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Where no observer exists, does this mean the wavefunction never collapses?
In most places across the universe, there is no conceivably sentient candidate to act as an "observer" to this system.
Are we to believe that, in the emptiness of intergalactic space, or ...
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Why are the coefficients of vectors (that are not eigenstates) the probability amplitudes?
In the adiabatic approximation, we assume that:
are the solutions of the eigenvalue problem:
We know however that the ψn(t) 's here may not be actual solutions to the time dependent Schrodinger ...
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Understanding (non-degenerate and degenerate) Perturbation theory
I was learning the basics of perturbation theory (PT) in QM, and I had quite a bit of trouble with it, especially (time independent) degenerate PT. My reference books are primarily Sakurai (became too ...
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Particle creation by inducing a set of values on the wavefunction? [closed]
The following 10 numbers:
completely specifies a self-propagating Schrodinger wavefunction:
If one is able to produce/induce these 10 excitation numbers directly onto a pre-existing wavefunction, is ...
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On the infinite potential well: an apparent paradox [duplicate]
$$
V(x) = \left\{
\begin{split}
0&\quad \operatorname{in}\ [0, a] \\
+\infty&\quad \operatorname{elsewhere}
\end{split}
\qquad a>0,
\right.
$$
The Schrödinger equation for stationary ...
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What is the wavefunction of a traveling particle? [closed]
The following animation (from here) depicts the wavefunction of a free particle traveling in space:
Aside from the (beautiful) picture, the actual wavefunction is not given for this traveling ...
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