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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4answers
756 views
In quantum mechanics, given certain energy spectrum can one generate the corresponding potential?
A typical problem in quantum mechanics is to calculate the spectrum that corresponds to a given potential.
Is there a one to one correspondence between the potential and its spectrum?
If the ...
14
votes
6answers
950 views
Why can we treat quantum scattering problems as time-independent?
From what I remember in my undergraduate quantum mechanics class, we treated scattering of non-relativistic particles from a static potential like this:
Solve the time-independent Schrodinger ...
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votes
1answer
1k views
Schrödinger Equation
I am reading up on the Schrödinger equation and I quote
Because the potential is symmetric under $x\to-x$, we expect that there will be solutions of definite parity.
Could someone kindly explain ...
10
votes
2answers
683 views
Schrodinger equation in spherical coordinates
I read a paper on solving Schrodinger equation with central potential, and I wonder how the author get the equation(2) below. Full text.
In Griffiths's book, it reads
...
10
votes
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 ...
9
votes
3answers
670 views
Schrodinger's equation (explanation to non physicist)
For a report I'm writing on Quantum Computing, I'm interested in understanding a little about this famous equation. I'm an undergraduate student of math, so I can bear some formalism in the ...
8
votes
5answers
828 views
The Many Body problem
(This is a simple question, with likely a rather involved answer.)
What are the primary obstacles to solve the many-body problem in quantum mechanics?
Specifically, if we have a Hamiltonian for a ...
8
votes
4answers
472 views
Quantum mechanics as classical field theory
Can we view the normal, non-relativistic quantum mechanics as a classical fields?
I know, that one can derive the Schrödinger equation from the Lagrangian density
$${\cal L} ~=~ \frac{i\hbar}{2} ...
8
votes
1answer
382 views
Interpretation of the Random Schrödinger Equation
I should preface this by admitting that my physics background is rather weak so I beg you to keep that in mind in your responses. I work in mathematics (specifically probability theory) and a paper ...
7
votes
2answers
395 views
Solving the Schrödinger equation for the double-slit experiment
I'm not sure if this is the right place to ask a question about the Schrödinger equation, but I'll take my chances anyway. Basically, I would like to know how one can set up a potential function that ...
6
votes
1answer
1k views
Eigenfunctions v.s. eigenstates
Is there a difference between "eigenfunction" and "eigenstate"? They seem to be used interchangeably in texts, which is confusing. My guess is that an "eigenfunction" has an explicit ...
6
votes
2answers
449 views
Why does iteratively solving the Hartree-Fock equations result in convergence?
[ Cross-posted to the Computational Science Stack Exchange: http://scicomp.stackexchange.com/questions/1297/why-does-iteratively-solving-the-hartree-fock-equations-result-in-convergence ]
In the ...
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votes
0answers
223 views
exponential potential $ \exp(|x|) $
For $a$ being positive what are the quantization conditions for an exponential potential?
$$ - \frac{d^{2}}{dx^{2}}y(x)+ ae^{|x|}y(x)=E_{n}y(x) $$
with boundary conditions $$ y(0)=0=y(\infty) $$
I ...
5
votes
4answers
281 views
Examples of exact many-body ground state wavefunction
Is there any non-trivial many-body system for which the exact solution to Schrödinger's equation is known? (By non-trivial, I mean a system with particle-particle interactions.) Perhaps something like ...
5
votes
2answers
786 views
Matrix Representations of Quantum States and Hamiltonians
I am a high school student trying to teach himself quantum mechanics just for fun, and I am a bit confused. As a fun test of my programming/quantum mechanics skill, I decided to create a computer ...
5
votes
1answer
196 views
Nomenclature of radial solutions to the Schrodinger Equation
For the free particle with quantum number $l=0$, the regular solution to the radial Schrodinger equation is $R_0 (\rho)=\frac{\sin{\rho}}{\rho}$ while the irregular solution is $R_0 ...
5
votes
3answers
251 views
Rationale for writing wave function as product of independent wave functions
When solving Schrödinger's equation for a 3D quantum well with infinite barriers, my reference states that: $$\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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votes
5answers
528 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 ...
5
votes
3answers
145 views
Time Varying Potential, series solution
Suppose we have a time varying potential $$\left( -\frac{1}{2m}\nabla^2+ V(\vec{r},t)\right)\psi = i\partial_t \psi$$ then I want to know why is the general solution written as $\psi = ...
5
votes
2answers
2k views
Solving one dimensional Schrodinger equation with finite difference method
Consider the one-dimensional Schrodinger equation
$$-\frac{1}{2}D^2 \psi(x)+V(x)\psi(x)=E\psi(x)$$
where $D^2=\dfrac{d^2}{dx^2},V(x)=-\dfrac{1}{|x|}$.
I want to calculate the ground state ...
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votes
1answer
779 views
5
votes
3answers
464 views
What is the relationship between Schrödinger equation and Boltzmann equation?
The Schrödinger equation in its variants for many particle systems gives the full time evolution of the system. Likewise, the Boltzmann equation is often the starting point in classical gas dynamics.
...
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votes
2answers
127 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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votes
1answer
177 views
What type of PDE are Navier-Stokes equations, and Schrödinger equation?
What type of PDE are Navier-Stokes equations, and Schrödinger equation?
I mean, are they parabolic, hyperbolic, elliptic PDEs?
5
votes
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 ...
5
votes
3answers
334 views
Nonlinear dynamics beneath quantum mechanics?
Yesterday I asked whether the Schroedinger Equation could possibly be nonlinear, after reviewing the answers and material given to me in that thread I feel like my question were adequately answered.
...
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votes
5answers
741 views
Hydrogen radial wave function infinity at r=0
When trying to solve the Schrödinger equation for hydrogen, one usually splits up the wave function into two parts:
$\psi(r,\phi,\theta)= R(r)Y_{l,m}(\phi,\theta)$
I understand that the radial part ...
4
votes
1answer
111 views
Temporal part of Quantum Wavefunction
I was hoping that someone could give me the more fundamental reason that we take as the temporal part of a quantum wavefunction the function $e^{-i\omega t}$ and not $e^{+i\omega t}$? Clearly ...
4
votes
2answers
183 views
Why must the angular part of the Schrodinger Equation be an eigenfunction of L^2?
I was reading about the solution to the Schrodinger Equation in spherical coordinates with a radially symmetric potential, $V(r)$, and the book split the wavefunction into two parts: an angular part ...
4
votes
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 ...
4
votes
2answers
555 views
Radial Schrodinger equation with inverse power law potential
Recently I read a paper about solving radial Schrodinger equation with inverse power law potential.
Consider the radial Schrodinger equation(simply set $\mu=\hbar=1$):
...
4
votes
2answers
299 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 ...
4
votes
2answers
673 views
On numerically solving the Schrödinger equation
I just read a paper 'A pocket calculator determination of energy eigenvalues' by J Killingbeck
(1979).
Link: http://iopscience.iop.org/0305-4470/10/6/001
I have some questions about section 2.
Why ...
4
votes
2answers
2k views
Zero probability of finding an electron in the nucleus
One and the same electron in a p orbital and taking part in a common π (pi) bond has two lobes visualized as connecting through the nucleus. There is however zero probability of finding an electron at ...
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votes
2answers
87 views
Do we always ignore zero energy solutions to the (one dimensional) Schrödinger equation?
When we solve the Schrödinger equation on an infinite domain with a given potential $U$, much of the time the lowest possible energy for a solution corresponds to a non-zero energy. For example, for ...
4
votes
1answer
194 views
Gross-Pitaevskii equation in Bose-Einstein condensates
I was hoping someone might be able to give a approachable explanation of the Gross-Pitaevskii equation. All the sources I've been able to find seem to concentrate on the derivation, and I don't have ...
4
votes
1answer
1k views
What inspired Schrödinger to derive his equation?
I have almost no background in physics and I had a question related to Schrodinger's Equation. I think, it is not really research level so feel free to close it, but I would request you to kindly ...
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votes
2answers
426 views
Use of Operators in Quantum Mechanics
I understand the form of operators in use for quantum mechanics such as the momentum operator:
$$\hat{\text{P}}=-ih\frac{d}{dx}$$ My question is in what ways can I use it and what am I getting back? ...
4
votes
2answers
338 views
Quantum Mechanics: Show that the expectation value of angular momentum does not change with time
The potential is given by $V\left(\left\|(x,y,z)\right\|\right)$, so $[\hat{L}, \hat{H}] = 0$.
Using the definition of $\langle \hat{L} \rangle$ and the time-dependent Schrödinger equation, show that ...
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(- ...
4
votes
5answers
383 views
How isolated must a system be for it's wave function to be considered not collapsed?
As an undergrad I was often confused over people's bafflement with Schodinger's cat thought experiment. It seemed obvious to me that the term "observation" referred to the Geiger counter, not the ...
4
votes
2answers
91 views
Solve the angular part of Schrodinger equation numerically
I would like to solve the angular part (the one for what is usually called the $\theta$ angle) of a time-independent 3D Schrodinger equation
$$
\frac{\mathrm{d}}{\mathrm{d}x}\left[ (1-x^2) ...
3
votes
4answers
675 views
Where is spin in the Schroedinger equation of an electron in the hydrogen atom?
In my current quantum mechanics, course, we have derived in full (I believe?) the wave equations for the time-independent stationary states of the hydrogen atom.
We are told that the Pauli Exclusion ...
3
votes
2answers
446 views
Infinite square well
1. Given that for an infinite square well problem, $\psi(x,0)=\frac{6}{a^3}x(a-x)$, I can show by Fourier transform that the probability of measuring $E_n$ for $n$ even is 0. But is there a physical ...
3
votes
2answers
282 views
Galilean invariance of the Schrodinger equation
I am only asking this question so that I can write an answer myself with the content found here:
http://en.wikipedia.org/wiki/User:Likebox/Schrodinger#Galilean_invariance
and here:
...
3
votes
2answers
108 views
Time-dependence in LCAO
I would like to study time-dependence (TD) in linear combinations of atomic orbitals (LCAO).
The Hückel method enables quick and dirty determination of MOs for suitable systems (view link for ...
3
votes
1answer
133 views
How does a state in quantum mechanics evolve?
I have a question about the time evolution of a state in quantum mechanics. The time-dependent Schrodinger equation is given as
$$
i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle
$$
I am ...
3
votes
3answers
346 views
Smoothness constraint of wave function
Is there anything in the physics that enforces the wave function to be $C^2$? Are weak solutions to the Schroedinger equation physical? I am reading the beginning chapters of Griffiths and he doesn't ...
3
votes
1answer
731 views
Light waves and Schrödinger probability waves
Ok, bearing in mind that I only have a brief understanding of quantum mechanics (no formal education, only from reading about concepts in books), so I could be way off here, I have a question ...
3
votes
1answer
138 views
Schrödinger equation for a harmonic oscillator
I have came across this equation for quantum harmonic oscillator
$$
W \psi = - \frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi
$$
which is often remodelled by defining a new ...
