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-2 votes

B.C. for time-decaying delta barrier inside an infinite well

To solve the time-dependent Schrödinger equation for the infinite well with the given potential, we need to apply the boundary conditions and the conditions imposed by the delta potential. Let's go ...
William Bell's user avatar
-2 votes

Numerically Solving the Schrödinger Equation for Coulombic Barrier in Fusion

Solving the Time-Dependent Schrödinger Equation for the Tunneling Problem in Fusion Processes Problem Statement The time-dependent Schrödinger equation (TDSE) for a particle of mass ( m ) encountering ...
William Bell's user avatar
1 vote
Accepted

Plane waves, angular momentum, and the 2D Schrödinger equation

I believe the answer to your question is simply that plane waves are not angular momentum eigenfunctions, as a direct calculation shows: \begin{align} (x\partial_y - y\partial_x)e^{ik_x x+ ik_y y} = (...
d_b's user avatar
  • 8,259
2 votes

Normal Base for Hilbert Space of delta Potential Well

A complete set of solutions of the eigenvalue problem $$-\frac{1}{2m} \phi^{\prime \prime} (x) -V \delta(x) \phi(x) =E \phi(x), \quad V\gt0,$$ is given by $$\begin{align} \phi_B(x)&=\sqrt{mV} e^{-...
Hyperon's user avatar
  • 6,506
1 vote

Normal Base for Hilbert Space of delta Potential Well

The usual approach for scattering problems is to construct a basis consisting of a continuous part (scattered waves) and a discrete part (the bound states). You will then of course get combinations of ...
Jos Bergervoet's user avatar
0 votes

Is spontaneous symmetry breaking possible without wave function collapse?

There is no relation with the unfortunate concept of wave function collapse. The double well potential that you give has solutions that are even or odd under inversion, but still symmetric. It is only ...
my2cts's user avatar
  • 24.5k
2 votes

Is spontaneous symmetry breaking possible without wave function collapse?

An initially symmetric wave function will remain symmetric, but for the system in isolation it will not really "fall to the bottom" since its energy cannot escape, it will just spread out (...
Jos Bergervoet's user avatar
0 votes

Is the spherical outgoing wave solution to the Schrodinger equation is not a member of $L^2$?

The simplest form of the problem of this kind is the energy eigenstate for free particles wich is a plane wave and diverges if you take the integral of the square from -infinity to +infinitiy. One ...
atilla gurel's user avatar
0 votes

Numerical solution to Schrödinger equation - eigenvalues

The “phase method” http://gbxafs.iit.edu/phase-method/ easily solves it. It’s new. Take a look.
user403303's user avatar
1 vote
Accepted

Regarding to the asymptotic solution of quantum harmonic oscillator

You need to be careful on defining the asymptotics. From the equation: $$ u''+\left(\epsilon-x^2-\frac{l(l+1)}{x^2}\right)u = 0 $$ you want to know the behaviour of $u$ at infinity. The issue is that ...
LPZ's user avatar
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0 votes

Why must the angular part of the Schrodinger Equation be an eigenfunction of $L^2$?

That the angular part of $\psi$ "must" be an eigenfunction of $L^2$ is only true if you write the wavefunction as the product of two parts: an angular part and a radial part. If you don't ...
Jos Bergervoet's user avatar
1 vote
Accepted

Is the spherical outgoing wave solution to the Schrodinger equation is not a member of $L^2$?

Integrating in spherical coordinates, we have $$ \|f\|_2 = \int |f|^2d^3\mathbf{R} = \int_0^\infty\left(\frac{1}{R^2}\right)\left(R^2dR\,d^2\Omega\right) = 4\pi\int_0^\infty dR = \infty. $$ So $f$ is ...
eyeballfrog's user avatar
1 vote

Experimental constraints on time evolution of quantum states

There are many experiments trying to test various fundamental assumptions and properties of quantum mechanics. I am not aware of any direct exclusion experiments of the extension proposed in the OP, ...
Wolpertinger's user avatar
  • 11.6k
3 votes

Why is the time derivative of the wavefunction proportional to a linear operator on it?

If I can remember well, Landau and Lifschitz first introduce the probabilistic interpretation of the wave function, being $$\rho(\mathbf{r},t) = \psi(\mathbf{r},t)^* \psi(\mathbf{r},t) \ ,$$ so that ...
basics's user avatar
  • 9,138
3 votes
Accepted

Why is the time derivative of the wavefunction proportional to a linear operator on it?

In the text you have pasted, they are not really making arguments, derivations or assumptions, they are making statements: The wave function determines the quantum state at all times. This can be ...
Codename 47's user avatar
  • 2,496
1 vote
Accepted

Help unpacking Schrödinger's Hamiltonian

In Schrödinger's equation, the Hamiltonian is just $$ \hat{H} = \hat{T} + \hat{V}, $$ where $\hat{T}$ and $\hat{V}$ are the kinetic and potential energy of the system, respectively. Usually we work in ...
Gabriel Ybarra Marcaida's user avatar
1 vote

Using QM for a Classical Mechanics problem

You can sort of do that. Fully understanding how classical mechanics emerges from quantum mechanics is very complicated, and I've even heard some physicists speculating that it will require a full ...
Níckolas Alves's user avatar
2 votes

Experimental constraints on time evolution of quantum states

It sounds like you are asking for confirmation/refutation of the time evolution of the system. There's a new manuscript on the arxiv that watches Gaussian wave functions evolve into broader Gaussian ...
Dr. Nate's user avatar
  • 411
1 vote

How to find the wavefunction that solves an infinite square well with a delta function well in the middle?

There is also a solution possible for $E<0$. Define $\kappa^2 = 2m |E|/\hbar^2$ with $\kappa > 0$. Then for $0<x<L/2$ we have $$\psi(x)=Ce^{\kappa(x-L/2)} -Ce^{-\kappa L/2)}$$ and for $L/2&...
Hubert van Luytelaar's user avatar
4 votes

Uncertainity in position in 1D potential box

The standard deviation has an exact mathematical definition. It sounds like your main hang up is that you have internalized it as the hard limits across which the wave function is nonzero, but that's ...
BioPhysicist's user avatar
  • 56.8k
0 votes

Estimate bound state energy for shallow finite well

Sometimes it is useful to rewrite the original equation in a different form. So let's get from equation $$ z\tan z = \sqrt{z_0^2-z^2} $$ to equation $$ z^2 = z_0^2 - z^2\tan^2 z\tag{1} $$ From the ...
Gec's user avatar
  • 5,387
0 votes

On using Python to solve Time Independent Schrodinger Equation, the eigenfunctions have their values "pushed" to one of the boundaries?

You may want to try out a new approach: "The Phase Method" (http://gbxafs.iit.edu/phase-method). It can easily solve your problem, and others of this type.
Grant Bunker's user avatar
1 vote
Accepted

Reflection of quantum particle colliding with a potential barrier

Your last statement is basically correct. For a single particle, the current $\vec{J}$ is measuring a current of probability density $|\psi|^2$. However, to physically interpret what such a ...
Andrew's user avatar
  • 49.8k
0 votes

Are reflection and transmission coefficients in 1D problem are independent of the direction in which we choose as incident?

In general the reflection and transmission coefficients are not independent of direction for arbitrary potentials. For a discussion of the conditions under which the transmission coefficient is ...
alanf's user avatar
  • 8,091
1 vote

Are reflection and transmission coefficients in 1D problem are independent of the direction in which we choose as incident?

You are right. There is some symmetry, but only enough to relate $T$ and $T'$, not to get $R=R'$. You can even find this as a classical result: (for light or EM waves, or electrical signals) https://...
Jos Bergervoet's user avatar
2 votes
Accepted

Am I reading the Madelung equations for a 1D quantum particle right?

Did you verify your TDSE, dimensionally? Your S, here, atypically, is an action over ℏ, hence dimensionless, $\psi\equiv R e^{iS}$; it produces the well-known continuity and quantum Hamilton-Jacobi ...
Cosmas Zachos's user avatar
1 vote
Accepted

Quantum Harmonic Oscillator With a Linear "Perturbation"

Your main idea is right: it is a matter of making a change of variables. But I think you made a change of variables that could be slightly improved. If I didn't mess up any calculations (please double ...
Níckolas Alves's user avatar

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