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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2answers
271 views
What does the general solution of the Schrodinger equation represent for the particle in a box problem?
For the particle in an infinitely deep potential well, I have an intuitive picture of the separable solutions of the Schrodinger equation as being the wavefunctions for the different allowed energy ...
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1answer
69 views
Schrödinger operator with a potential defined implicitly
let be the problem
$$ -\frac{d^{2}}{dx^{2}}y(x)+f(x)y(x)=E_{n}y(x)$$
however we have a problem, we do not know the potential but its inverse
$$ f^{-1}(x)=g(x) $$
we know $ g(x) $ but not $ f(x) $ ...
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2answers
265 views
Meaning of instantaneous probability densities in time dependent wavefunctions
For a time dependent wavefunction, are the instantaneous probability densities meaningful? (The question applies for instances or more generally short lengths of time that are not multiples of the ...
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2answers
87 views
What is the spectrum of energies for the potential $ a^{x} $?
Given a certain potential $ a^{x} $ with positive non-zero 'a' are there a discrete spectrum of energy state for the Schrodinger equation
$$ \frac{- \hbar ^{2}}{2m} ...
2
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1answer
292 views
Derivation of Bloch's theorem
I'm having a problem following a derivation of Bloch's theorem, looking at a one dimensional lattice with $N$ nodes and spacing a, we impose periodic boundary conditions, meaning that the ...
2
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1answer
303 views
Solving time dependent Schrodinger equation in matrix form
If we have a Hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector
$$\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t))$$
With Hamiltonian $H$ given by
$$H=\hbar\omega
...
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0answers
168 views
Force of a particles on a Potential Barrier [closed]
A particle confined by a potential wall exerts some pressure on it. More specifically, suppose that the particle moves in this potential:
$$V(x) ~=~\left\{ \begin{array}{lcc}\text{finite ...
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1answer
1k views
Bound States in a Double Delta Function Potential [closed]
Let $V(x) = −u \delta(x) - v \delta(x − a)$ where $u, v > 0$ correspond to a potential with two $\delta$ wells. Let $v > u$. If $a$ is very large, there is certainly a bound state: the particle ...
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1answer
141 views
Inhomogenous schrodinger equation
Please help me out in solving this inhomogeneous Schrodinger equation in Cylindrical co-ordinates [You may suggest if I have to go for mathematics]:
$$
\ddot R + \frac1r\dot ...
5
votes
3answers
251 views
Rationale for writing wave function as product of independent wave functions
When solving Schrödinger's equation for a 3D quantum well with infinite barriers, my reference states that: $$\psi(x,y,z) = \psi(x)\psi(y)\psi(z) \quad\text{when}\quad V(x,y,z) = V(x) + V(y) + V(z) = ...
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0answers
182 views
Finding transcendental equation for the energy of a particle in delta potential well near infinite potential barrier [closed]
I'm having trouble finding the transcendental equation for a particle in a delta potential settled near an infinite potential wall. The potential is given by
$$
V(x) =
\begin{cases}
\infty & x ...
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1answer
70 views
Why minibands are formed in superlattices?
In a single, finite quantum well, there are energy levels defined by the eigenstates - the solutions of the Schroedinger's Equation. The corresponding wavefunctions leak to the barrier because of its ...
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2answers
336 views
Schrödinger's equation, time reversal, negative energy and antimatter
You know how there are no antiparticles for the Schrödinger equation, I've been pushing around the equation and have found a solution that seems to indicate there are - I've probably missed something ...
7
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2answers
391 views
Solving the Schrödinger equation for the double-slit experiment
I'm not sure if this is the right place to ask a question about the Schrödinger equation, but I'll take my chances anyway. Basically, I would like to know how one can set up a potential function that ...
4
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1answer
79 views
What are relativistic and radiative effects (in quantum simulation)?
I'm reading about Quantum Monte Carlo, and I see that some people are trying to calculate hydrogen and helium energies as accurately as possible.
QMC with Green's function or Diffusion QMC seem to be ...
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2answers
492 views
Barrier in an infinite double well
I am stuck on a QM homework problem. The setup is this:
(To be clear, the potential in the left and rightmost regions is $0$ while the potential in the center region is $V_0$, and the wavefunction ...
2
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1answer
357 views
Schrödinger equation with complex potential
In 1 dimension what is the solution of the Schrödinger equation with potential
$$ V(x) = V_r + i V_i $$
Potentials are constant.
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1answer
85 views
Why is amplitude of a wavefunction to propagate from $q$ to $q'$ governed by $e^{-\frac{i}{\hbar}HT}$ unitary operator?
In the textbook Quantum Field Theory by A. Zee, it says:
In quantum mechanics, the amplitude to propagate from a point $q_i$ to a point $q_f$ in time $T$ is governed by the unitary operator ...
2
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2answers
428 views
What math is needed to understand the Schrödinger equation?
If I now see the Schrödinger equation, I just see a bunch of weird symbols, but I want to know what it actually means. So I'm taking a course of Linear Algebra and I'm planning on starting with PDE's ...
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2answers
187 views
How to write Schrodinger equation when a particle with some spin quantity and orbital angular momentum
Quantum mechanics: Suppose that there is a particle with orbital angular momentum $|L|$. But the particle also has spin quantity $|S|$. The question is, how do I reflect this into Schrodinger ...
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5answers
525 views
Derivation of Schrodinger equation for a system with position dependent effective mass
How to derive the Schrodinger equation for a system with position dependent effective mass? For example, I encountered this equation when I first studied semiconductor hetero-structures. All the books ...
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2answers
152 views
$\nabla$ and non-locality in simple relativistic model of quantum mechanics
In Wavefunction in quantum mechanics and locality, wavefunction is constrained by $H = \sqrt{m^2 - \hbar^2 \nabla^2} $, and taylor-expanding $H$ results in:
$$ H = \dots = m\sqrt{1 - \hbar^2/m^2 ...
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1answer
343 views
Solution to Klein-Gordon equation always valid?
We know that there is a relativistic version of Schrodinger equation called Klein-Gordon equation. However, it has some problems and due to these problems, there is Dirac equation that handles these ...
2
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1answer
233 views
Wavefunction in quantum mechanics and locality
Every wavefunction of a form $\Psi(x)$ can be described as a superposition of multiple free particle solutions.
We can see the following Fourier transform:
$$ \psi(x) = \int e^{ik\cdot x} \psi(k) dk ...
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1answer
160 views
Complete set and Klein-Gordon equation
In http://www.physics.ucdavis.edu/~cheng/teaching/230A-s07/rqm2_rev.pdf, it says that when there is some external potential, the Klein-Gordon equation is altered, and it says the following:
The ...
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1answer
304 views
Electron Incident On A Finite Potential Barrier
This is problem 2.8.3 from Miller's Quantum Mechanics For Scientists And Engineers. I'm getting stuck when I try to figure out the wave equation on the right-hand side of the barrier. The original ...
2
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1answer
217 views
Explanation of equation that shows a failed approach to relativize Schrodinger equation
I'm reading the Wikipedia page for the Dirac equation:
$\rho=\phi^*\phi\,$
......
$J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)$
with the conservation of probability ...
1
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1answer
226 views
How to obtain Dirac equation from Schrodinger equation and special relativity?
I'm reading the Wikipedia page for the Dirac equation:
The Dirac equation is superficially similar to the Schrödinger
equation for a free massive particle:
A) ...
0
votes
1answer
82 views
a positive potential as $ x \rightarrow \infty $
let us suppose i can calculate the asymptotic of any potential $ V(x) $ in one dimension , and that i manage to prove that $ V(x) \ge 0 $ as $ x \rightarrow \infty $
could i conclude taht if or big ...
5
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2answers
125 views
Is there a time delay during tunnelling?
A particle hitting a square potential barrier can tunnel through it to get to the other side and carry on. Is there a time delay in this process?
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3answers
352 views
Can we have discontinuous wavefunctions in the Infinite Square well?
The energy eigenstates of the infinite square well problem look like the Fourier basis of L2 on the interval of the well. So then we should be able to for example make square waves that are an ...
3
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1answer
329 views
Finding $\psi(x,t)$ for a free particle starting from a Gaussian wave profile $\psi(x)$
Consider a free-particle with a Gaussian wavefunction,
$$\psi(x)~=~\left(\frac{a}{\pi}\right)^{1/4}e^{-\frac12a x^2},$$
find $\psi(x,t)$.
The wavefunction is already normalized, so the next thing to ...
3
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1answer
321 views
Even and Odd States of a 1D finite potential well
Is it possible for a particle trapped in a 1D finite potential well to evolve from a even state to an odd state and vice-versa? Why?
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1answer
318 views
How does one solve the Schroedinger equation for a 2D, time-dependent harmonic potential?
This is the Schroedinger equation with a particular 2D harmonic potential:
$$\begin{multline}i\hbar\frac{\partial}{\partial t}\Psi(x_1,x_2,t) = \\ \Biggl[-\frac{\hbar^2}{2m}\nabla^2 + \frac{1}{2} ...
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0answers
340 views
Scattering on delta function potential
Suppose a particle has energy $E>V(+/-\infty)=0$, then the solutions to the Schrodinger equation outside of the potential will be $\psi(x)=Ae^{i k x}+Be^{-i k x}$.
How can one show or explain that ...
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1answer
83 views
Where can I find hamiltonians + lagrangians?
Where would you say I can start learning about Hamiltonians, Lagrangians ... Jacobians? and the like?
I was trying to read Ibach and Luth - Solid State Physics, and suddenly
(suddenly a Hamiltonian ...
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1answer
267 views
The Hermiticity of the Laplacian (and other operators)
Is the Laplacian operator, $\nabla^{2}$, a Hermitian operator?
Alternatively: is the matrix representation of the Laplacian Hermitian?
i.e.
$$\langle \nabla^{2} x | y \rangle = \langle x | ...
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4answers
399 views
Which Schrodinger equation is correct?
In the coordinate representation, in 1D, the wave function depends on space and time, $\Psi(x,t)$, accordingly the time dependent Schrodinger equation is
$$H\Psi(x,t) = ...
4
votes
1answer
251 views
Is momentum conservation for the classical Schrödinger equation due to non-relativistic or due to some more exotic invariance?
I had no problem appliying the Neothers theorem for translations to the non-relativistic Schrödinger equation
$\mathrm i\hbar\frac{\partial}{\partial t}\psi(\mathbf{r},t) \;=\; \left(- ...
3
votes
2answers
108 views
Time-dependence in LCAO
I would like to study time-dependence (TD) in linear combinations of atomic orbitals (LCAO).
The Hückel method enables quick and dirty determination of MOs for suitable systems (view link for ...
4
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2answers
296 views
Does String theory say that spacetime is not fundamental but should be considered an emergent phenomenon?
Does String theory say that spacetime is not fundamental but should be considered an emergent phenomenon?
If so, can quantum mechanics describe the universe at high energies where there is no ...
5
votes
2answers
781 views
Matrix Representations of Quantum States and Hamiltonians
I am a high school student trying to teach himself quantum mechanics just for fun, and I am a bit confused. As a fun test of my programming/quantum mechanics skill, I decided to create a computer ...
3
votes
2answers
203 views
Many-worlds: how often is the split how many are the universes? (And how do you model this mathematically.)
When I read descriptions of the many-worlds interpretation of quantum mechanics, they say things like "every possible outcome of every event defines or exists in its own history or world", but is this ...
2
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2answers
124 views
can we apply WKB method for curved space times
let be the Hamiltonian of a surface $ H= g_{a,b} p^{a}p^{b} $ (Einstein summation assumed) my question is if although the space time is curved then can we use the WKB approximation to get the quantum ...
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4answers
132 views
Why do we consider the evolution (usually in time) of a wave function?
Why do we consider evolution of a wave function and why is the evolution parameter taken as time, in QM.
If we look at a simple wave function $\psi(x,t) = e^{kx - \omega t}$, $x$ is a point in ...
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1answer
340 views
Bound states for sech-squared potential
I'm working on an introductory qm project, hope somebody has the time to help me (despite the length of this post), it will be highly appreciated.
My goal is to determine the bound states and their ...
10
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2answers
1k views
Is the Schrödinger equation derived or postulated?
I'm an undergraduate mathematics student trying to understand some quantum mechanics, but I'm having a hard time understanding what is the status of the Schrödinger equation.
In some places I've read ...
1
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2answers
253 views
Can we impose a boundary condition on the derivative of the wavefunction through the physical assumptions?
Consider the Schrödinger equation for a particle in one dimension, where we have at least one boundary in the system (say the boundary is at $x=0$ and we are solving for $x>0$). Sometimes we want ...
5
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1answer
108 views
Apparent contradiction between calculations and intuition?
I am rather confused because it would seem that mathematical conclusions I have drawn here goes against my physical intuition, though both aren't too reliable to begin with.
We have a potential step ...
8
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1answer
381 views
Interpretation of the Random Schrödinger Equation
I should preface this by admitting that my physics background is rather weak so I beg you to keep that in mind in your responses. I work in mathematics (specifically probability theory) and a paper ...
