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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What are the experiments performed to determine the position of an electron inside an atom to verify the probability wave function data?
What are the experiments performed to determine the position of an electron inside the atom to verify the probability wave function data? Is it possible to do those experiments in real life?
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Time-independent Klein-Gordon PDE
Given the KG PDE:
$$\psi_{tt} - \psi_{xx} + m^2 \psi = 0.$$
Wikipedia describes the time-independent variant of this as just setting $\psi_{tt}=0$.
My question is this:
For the Schrödinger ...
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2answers
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Conceptual understanding of Schrödinger equation
So I followed this lecture:
https://www.youtube.com/watch?v=qu-jyrwW6hw
which starts of with the statement:
If you have a Schrödinger equation for an energy eigenstate you have
$$-\frac{\hbar}{2m}...
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0answers
37 views
E greater than V [closed]
Show that E must exceed the minimum value of V(x) for every normalisable solution to the time-independent Schrödinger equation?
This question was given to be proved in David Griffith's ' ...
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1answer
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Infinite square well: wall with infinitesimal thickness
Given an infinite square well, it doesn't matter how thick the wall is, the particle is trapped inside the two walls. If we make the wall of arbitrarily small but finite thickness, the particle is ...
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1answer
46 views
Scaling Problem with Variational Method
$\def\braket#1{\langle#1\rangle}$
I am attempting to solve a particular Hamiltonian by variational method. The wavefunction that I have selected is as follows:
$$
\Psi = Ne^{\frac{-kr}{2}}\sum_{i=0}...
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1answer
55 views
Numerical approximation of the wavefunction in a delta-potential [closed]
I am trying to approximate the wavefunction of a particle in a delta potential $U(x) = -U_0 \delta(x)$ with $V_0 \gt 0$. I am using the following formula to calculate the wavefunction:
$\psi(x+\Delta ...
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4answers
148 views
Does the Schrodinger wave function associated with a non-moving free particle change in time?
I'm a bit confused by an answer given on this question. In the answer with the animation of a moving free (chargeless) particle and a non-moving free particle (or a free particle with a non-zero ...
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0answers
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Possible error in this paper on the eigenfunctions of the Schrödinger equation for a regular tetrahedron [on hold]
I am working on trying to construct the spectrum for a particle trapped inside a impenetrable regular Tetrahedron. This problem is different in nature then solving for right and equilateral triangles ...
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1answer
34 views
Why do you need to add variables when multiplying through by a constant? [closed]
I was reading the following paper: http://vixra.org/abs/1206.0055 when I came across equation 9 on page two. The author claims that when multiplying equation 8 through by the constant A, you need to ...
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1answer
26 views
Double Gaussian well bound states parameters
I'm currently simulating a one-dimensional double Gaussian well potential numerically, and have been asked to find parameters corresponding to the overlap in potential and wave functions from my ...
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1answer
60 views
Help me make sense of the spectrum for the quantum wave function of an infinitely hard equilateral triangle
I'm trying to solve the spectrum for a equilateral Tetrahedron with infinitely hard walls. My first guess is to sum up a infinite amount of separable solutions to match the boundary conditions on the ...
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2answers
91 views
Non-Relativistic Limit of Klein-Gordon Probability Density
In the lecture notes accompanying an introductory course in relativistic quantum mechanics, the Klein-Gordon probability density and current are defined as:
$$
\begin{eqnarray}
P & = & \dfrac{...
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2answers
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E.L. Equations in QFT
In QFT, we use the Lagrangian to construct the Hamiltonian, and in the Interaction Picture (with regards to the Free Field Hamiltonian) use the full Hamiltonian to calculate the changes in the field (...
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4answers
87 views
Does a particle with infinite energy escape an infinite well?
Currently, my modern physics class is going over particles in finite and infinite wells, general quantum formalism, and tunneling.
What happens to a particle as it gains an infinite amount of energy? ...
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2answers
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The allowed energies of 3D harmonic oscillator [closed]
I'm trying to calculate the allowed energies of each state for 3D harmonic oscillator.
$$
E_n = (n_x+\textstyle\frac{1}{2})\hbar \omega_x+ (n_y+\textstyle\frac{1}{2})\hbar\omega_y+ (n_z+\textstyle\...
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0answers
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Physical meaning that an energy functional has no minimizer
It is well known that the Hamiltonian of a system might not have a minimizer, even the Hamiltonian is bounded below. For example, let us consider the cubic time independent Schrödinger equation
\begin{...
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2answers
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Meaning of $E$ in time-independent Schrödinger's Equation (high school)
I've just learned the time-independent Schrödinger's equation as
$$-\frac{\hbar^2}{2m}\cfrac{d^2\psi(x)}{dx^2}+V(x)\psi(x)=E\psi(x).$$
Does $E\psi(x)$ mean that $E$ is a constant (that the kinetic ...
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1answer
33 views
Algebraic trouble in gauge invariance of Schrodinger equation
I've been trying to prove a component of a proof the gauge invariance of the schrodinger equation. Specifically the part in the first answer here where this is stated:
$$\big(\frac{\nabla}{i}-q(\vec{...
0
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1answer
67 views
Finding an energy of a particle in an infinite potential well
This question arises from a discussion I recently had with my friend. We were talking about a particle in an infinite potential well. The particle is in an arbitrary wavefunction $\Psi$. When one ...
0
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0answers
36 views
Taylor expansion for a double well/perturbed infinite square well
I'm trying to estimate the ground state energy for a perturbed infinite square well directly. The potential is piecewise constant
$$V(x)=\begin{cases}&\infty, \qquad x<-a/2\\ &0 \qquad -a/2&...
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1answer
37 views
Gram-Schmidt process and degenerate subspace of the solutions to the Schrodinger's equation
So I know that in QM each linear combination of a degenerate set of wavefunctions is also a solution to the Schrodinger's equation (SE). The degenerate wavefunctions must be orthogonal to the non-...
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0answers
72 views
Why are energy levels in a quantum well discrete?
Assuming the potential is zero in $x$- and $y$-direction and zero between $z=0$ and $z=L_z$ and infinite for $z<0$ or $z>L_z$. This means the particle is confined in a layer which is infinitely ...
0
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1answer
82 views
Schrödinger wave equation - mass component
In the Schrödinger wave equation, where does the $8\pi^2 m/h$ come from?
Where $m$ is the mass and $h$ is Planck's constant
I understand the variables... but I'm unsure of the application of
$8\pi^...
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1answer
44 views
Time evolution of a free particle with a given initial state [closed]
My homework problem reads:
Consider a free particle in one dimension. Write an expression for the wavefunction $\psi(x, t)$ given an initial state $\psi_0(x) = Ae^{-ax^2}$ at $t = 0$, where $A$ is ...
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2answers
54 views
Help understanding the solution of the Schrodinger equation in a finite well [closed]
Given a finite well like this I found that the TISE has the general solutions. $$\psi_{1}(x)=A_{+}e^{ikx}+A_{-}e^{-ikx} \qquad x\in[-L,L] $$
$$\psi_{2}(x)=B_{+}e^{k^{'}x}+B_{-}e^{-k^{'}x} \qquad x\in(-...
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5answers
858 views
Why do we nondimensionalize the Schrödinger equation when solving the quantum harmonic oscillator?
I read about how to solve the Schrödinger equation for the quantum harmonic oscillator in one dimension. It started with the Schrödinger equation,
$$
\frac{p^2}{2m}\psi(x, t)+\frac{1}{2}m\omega^2x^2\...
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0answers
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Would the time dependent wave equation of a free particle be the same in 4D Euclidean spacetime as it is in 4D Minkowski spacetime?
In 4d minkowski spacetime the time dependent wave equation of a free particle is $$\Psi(x,t) = \left({a \over a + i\hbar t/m}\right)^{3/2} e^{- {x^2\over 2(a + i\hbar t/m)} }.$$ I notice that there ...
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1answer
55 views
“General” for time evolution of quantum state
I am reading a book in which at some point they find the time-evolved wavefunction $\phi_0(\mathbf{r},t)$ from the static $\phi_0(\mathbf{r})$.
They say that "employing the Heisenberg time evolution ...
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1answer
39 views
Question about fundamental states on an finite well
My question is the following, when we search for the bound states a finite well potential we have solutions symmetric and antisymmetric so we get two families of solutions. In this case, the ...
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1answer
57 views
How does the ground state of the quantum Ising model relate to Schrodinger equation?
The Hamiltonian
$$H = -\sum_{i\in V} h_i \sigma_i^z -\sum_{(i,j)\in E} J_{ij} \sigma_i^z\sigma_j^z - \Gamma\sum_{i\in V} \sigma_i^x$$
is kind of the cost function of the quantum annealing optimization ...
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1answer
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How to evaluate the matrix element of coulomb repulsion term between electrons in an atom suing spherical harmonics multipole expansion?
This is a lecture notes take from the following link on numerical calculation of atomic physics:http://www.phys.ubbcluj.ro/~lnagy/pdf/1curs.pdf
I am trying to evaluate the two electron matrix element ...
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4answers
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Electron travelling through a step potential from $V_0$ to 0
Most of the time you discuss the step potential case when $E>V_0$, you consider an electron (or a beam of electrons) travelling from a region of space $x<0$ in which $V=0$, to a region $x>0$ ...
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2answers
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What are the energy eigenvalues of a particle subject to the potential $V(x)=mg|x|$?
I am considering a particle within a potential given by $$V(x)=mg|x|$$ and am attempting to find the energy eigenvalues of the system.
Taking $V(x)$ to be defined piecewise, I've solved the ...
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2answers
80 views
Scattering by a Delta Function Well in 1D [closed]
Let us consider a scattering process by a delta function well in 1D:
$$
V(x) = -\alpha \, \delta(x), \quad \alpha > 0.
$$
I solve the Schrödinger equation for the scattering states and get the ...
0
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1answer
123 views
Schrodinger Equation [duplicate]
I was wondering if anyone knows the origins of how Schrodinger arrived to his equation? And can it be derived from Newtonian mechanics? How did Schrodinger form the equation out of his MIND?
I also I ...
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1answer
44 views
Difference, in terms of completeness, between the Dirac well and barrier
I was in my undergraduate QM lecture and we just finished with the Dirac barrier. My question is as follows: We know that the Dirac well’s complete set of solutions requires one bound state and an ...
2
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1answer
59 views
Solving Schrödinger equation by neural networks - trial function explanation
I'm reading this paper about solving Schrödinger equation using the combination of genetic algorithm and neural networks.
But one part confuses me - the author defines his trial function, i.e. the ...
2
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1answer
119 views
Proving Gauge invariance of Schrodinger Equation
I am trying to proof explicitly that Schrodinger equation:
$$ i\hbar \partial_t \psi = \big[ -\frac{1}{2m}\big(\frac{\hbar}{i}\nabla-q\vec{A}\big)^2+qV \big]\psi$$
remains the same under the ...
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1answer
279 views
2D isotropic quantum harmonic oscillator: polar coordinates
This may come a bit elemental, what I was working on a direct way to find the eigenfunctions and eigenvalues of the isotropic two-dimensional quantum harmonic oscillator but using polar coordinates:
$...
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2answers
75 views
Are there any general results about the nodes of energy eigenfunctions in higher dimensions?
A well-known result of quantum mechanics is that for a single particle in one dimension in a bounding potential $V(x)$ that goes to $+\infty$ as $x \to \pm \infty$, the energy eigenfunctions are ...
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2answers
117 views
What's the time derivative of the Annihilation operator?
I've been dealing with annihilation operator recently where you can see related information
Time derivative of the state vector as expressed in abstract Hilbert space vs. as a wavefunction
How to ...
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1answer
38 views
Wavefunction of a shifted radial harmonic oscillator [duplicate]
Suppose we know the solution of Schrodingers equation for a radial potential $V(r)$. Then the energy eigenstates are $\psi(r,\theta,\phi) = \frac 1ru(r)Y_\ell^m(\theta,\phi)$ where the radial ...
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1answer
61 views
Power series solution for a shifted spherical harmonic oscillator
I'm trying to solve the Schrodinger equation for a radial Harmonic oscillator whos equilibrium point has been shifted away from the origin, i.e. $V(r) = V_0(r-1)^2$. The standard approach is to make ...
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1answer
51 views
Harmonic Oscillator Trial Wavefunction
I was learning today about trial wave functions for a harmonic oscillator.
We learnt that the solution to Schrödinger equation for a harmonic oscillator is a Gaussian curve, i.e.
$$
f(x) = e^{-x^2} ....
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1answer
20 views
How does the image of $\sqrt{2/L} \ \sin\left({k_n x}\right)$ satisfy the boundary conditions for the infinite square well?
I understand mathematically how $\sqrt{2/L} \ \sin\left({k_n x}\right)$ satisfies the boundary conditions for the infinite square well in terms of the fact that $\psi(0) = \psi(a) = 0$, and excuse the ...
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1answer
41 views
What is the shape of the initial universal wavefunction?
In the many-worlds theory and Bohmian mechanics, the universe has an "initial" "universal" wavefunction which then evolves according to the Schrodinger's equation and determines the future of the ...
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1answer
71 views
What is the state of particle at time $t$ if at $t=0$ it is in an eigenstate of $\hat{A}$, and $\hat{A}$ commutes with $\hat{H}$?
EDIT: added (assuming $\lambda$ to be non-degenerate). Based on the specifics of the question, we don't in fact know whether this is the case, so it may be that $\left|\lambda\right>$ is not an ...
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1answer
31 views
Can the spin of a spin-$1/2$ particle flip in a time-varying magnetic field?
Given that the state of the particle at time $t=-\infty$ is $\left|S_z^+\right>$, a magnetic field of the form $\mathbf{B}=B \tanh(t/\tau) \hat{\mathbf{z}}$, the Hamiltonian is $H=-\vec{\mu}\cdot\...
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2answers
78 views
General Solution to $\psi$ in the Time-Independent Schrodinger Equation
I am reading Griffith's Intro to Quantum Mechanics and when explaining the Time Independent Schrodinger Equation, he says the general solution to $d\varphi/dt$ is:
$$\varphi (t)=e^{-iEt/h}$$
Even ...
