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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1answer
298 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.
$$\langle \nabla^{2} x | y \rangle = \langle x | ...
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4answers
418 views
Which Schrodinger equation is correct?
In the coordinate representation, in 1D, the wave function depends on space and time, $\Psi(x,t)$, accordingly the time dependent Schrodinger equation is
$$H\Psi(x,t) = ...
4
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1answer
257 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
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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
309 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
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2answers
810 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
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2answers
216 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
130 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
142 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
363 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 ...
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2answers
271 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
110 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
392 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 ...
5
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4answers
289 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 ...
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2answers
188 views
Classical limit of a quantum system
If we have a one dimensional system where the potential
$$V~=~\begin{cases}\infty & |x|\geq d, \\ a\delta(x) &|x|<d, \end{cases}$$
where $a,d >0$ are positive constants, what then is ...
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5answers
752 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 ...
2
votes
4answers
575 views
solution of schrodinger equation - infinite solutions?
In Griffiths's introductory quantum mechanics book, it states that if $\Psi (x,t)$ is a solution to the Schrodinger's equation, then $A\Psi (x,t)$ must also be a solution, where A is any complex ...
3
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1answer
236 views
The eigenvalue of Schrodinger Equation
I'm a student majoring in Mathematics.But now I'm studying the KDV equation which uses Schrodinger Equation. My question is that in time-independent Schrodinger ...
2
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1answer
531 views
Calculating Ground State Energy in 1D Potential
Given potential $V(x) = Asec(x)$ for $x > 0$. I want to calculate the ground-state energy $E_0$ via the Schrödinger equation.
I'm completely stuck on this one. I've set up the time-independent ...
5
votes
3answers
153 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 = ...
2
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1answer
823 views
How to solve Schrodinger Equation - Tunnelling
I have to solve analitically the Schrodinger equation in one-dimension with a barrier of potential (tunnel effect):
$$ih \frac{d}{dt} U(x,t) = \left[ \left(-h^2 \frac{d^2}{dx^2} \right) + q V(x) ...
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4answers
491 views
Quantum mechanics as classical field theory
Can we view the normal, non-relativistic quantum mechanics as a classical fields?
I know, that one can derive the Schrödinger equation from the Lagrangian density
$${\cal L} ~=~ \frac{i\hbar}{2} ...
3
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1answer
304 views
Can we solve the particle in an infinite well in QM using creation and annihilation operators?
The particle in an infinite potential well in QM is usually solved by easily solving Schrodinger differential equation. On the other hand particle in the harmonic oscillator oscillator potential can ...
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1answer
718 views
How to calculate ground state wave function?
I have seen many ground state wave functions.
From where are they derived?
How can one calculate them?
Where can one find a list of all ground state wavefunctions discovered?
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5answers
390 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 ...
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3answers
488 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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1answer
245 views
Solving Schrödinger's equation for a specific potential
I am trying to solve this differential equation:
$$-\chi''(\epsilon)+\Big[\epsilon^2+\frac{2F}{hw}\sqrt{\frac{h}{hw}}\epsilon \Big]\chi(\epsilon)=\mu\chi(\epsilon) \tag1$$
This was found ...
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2answers
142 views
Show that the energy levels of a particle in a specific potential are $E_n=(n+\frac{1}{2})h\omega-\frac{1}{2}\frac{F^2}{m\omega^2}$ [closed]
A particle of mass m moves on the x-axis under the influence of the potential
$$V(x)=\frac{1}{2}m\omega^2x^2+Fx$$
Can anyone help me, using Schrödinger's equation in one dimension that the energy ...
6
votes
2answers
474 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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3answers
409 views
One dimensional Schrödinger equation equation with initial condition, finding the probability of the particle's future position
A particle of mass $m$ moves freely in the interval $[0,a]$ on the $x$ axis. Initially the wave function is:
$$f(x)=\frac{1}{\sqrt{3}}\operatorname{sin}\Big( \frac{\pi x}{a} ...
3
votes
3answers
359 views
Smoothness constraint of wave function
Is there anything in the physics that enforces the wave function to be $C^2$? Are weak solutions to the Schroedinger equation physical? I am reading the beginning chapters of Griffiths and he doesn't ...
2
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1answer
790 views
How to calculate time evolution of a wave function in an 1D infinite square well potential?
A particle in an infinite square well has an initial wavefunction
$$\psi (x,0) ~=~ Ax(a-x) \qquad \mathrm{for}\qquad 0\leq x\leq a.$$
Now the question is to calculate $\psi (x,t)$.
I have ...
1
vote
1answer
245 views
A quantum particle in a box (with a catch)
I am reading Shankar's Quantum Mechanics and I am looking at the case where there is one particle inside a box, where the potential is zero inside the wall and abruptly goes to infinity outside the ...
1
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1answer
260 views
Is it a total or an explicite time derivative in the Schrödinger equation?
I am always dubious when I need write Schrödinger equation: do I write $\partial / \partial t$ or $d/dt$ ?
I suppose it depends on the space in which it is considered. How?
4
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1answer
1k views
What inspired Schrödinger to derive his equation?
I have almost no background in physics and I had a question related to Schrodinger's Equation. I think, it is not really research level so feel free to close it, but I would request you to kindly ...
2
votes
2answers
309 views
Correct application of Laplacian Operator
Not a physicist, and I'm having trouble understanding how to apply the Laplacian-like operator described in this paper and the original. We let:
$$ \hat{f}(x) = f(x) + \frac{\int H(x,y)\psi(y) ...
2
votes
2answers
936 views
Sinusoidal vs exponential wave functions with Schrodinger's equation
When solving Schrodinger's equation, we end up with the following differential equation:
$$\frac{{d}^{2}\psi}{dx^2} = -\frac{2m(E - V)}{\hbar}\psi$$
As I understand it, the next step is to guess the ...
1
vote
3answers
698 views
Analytic solutions to time-dependent Schrödinger equation
Are there analytic solutions to the time-Dependent Schrödinger equation, or is the equation too non-linear to solve non-numerically?
Specifically - are there solutions to time-Dependent Schrödinger ...
4
votes
2answers
442 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? ...
2
votes
1answer
375 views
How far can you get (in quantum mechanics) with just commutation relations?
Clearly it is possible to derive a set of commutation relations from some Hamiltonian, and certainly they give useful and interesting invariants when investigating the behavior of quantum systems. ...
2
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1answer
162 views
Degeneracy and the Hamiltonian
How many linearly independent eigenfunctions can be associated with one degenerate eigenvalue of the Hamiltonian operator? (Is there a limit since it contains a 2nd order differential operator?) ...
2
votes
1answer
775 views
Probability current
Conservation of probability: Suppose a wavefunction has ${\partial \mathbb P \over \partial t} = -t f(x,t)$ and ${\partial j \over \partial x} = i f(x,t)$. How does it follow that ${\partial \mathbb P ...
2
votes
1answer
275 views
Superposition of wavefunctions
Suppose you have 2 normalized wavefunctions $\psi_1=Ne^{iax}e^{if(x)}e^{i\omega t}$ and $\psi_2=Ne^{-iax}e^{if(x)}e^{i\omega t}$ defined on $x\in [-x_0,x_0]$ and vanishes for $|x|>x_0$. What then ...
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1answer
644 views
Radial Schrödinger equation
I found a problem that says:
Show by direct substitution that $R_{10}$ is a solution of Schrödinger's radial equation.
AFAIK Schrödinger's radial equation is
...
5
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3answers
342 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.
...
2
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1answer
1k views
Degeneracies of the first excited state
I have a box with $x,y,z$ all ranging from 0 to $l$. It has $V(x)$=0 inside and =$\infty$ outside. By extending the 1D Schrodinger equation, I have that the allowed energy eigenvalues are ...
3
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1answer
197 views
Projection of states after measurement
Continuing from the my previous 2-state system problem, I am told that the observable corresponding to the linear operator $\hat{L}$ is measured and we get the +1 state. Then it asks for the ...
0
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1answer
267 views
Two-state system problem
Given a 2-state system with (complete set) orthonormal eigenstates $u_1, u_2$ with eigenvalues $E_1, E_2$ respectively, where $E_2>E_1$, and there exists a linear operator $\hat{L}$ with ...
3
votes
2answers
450 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 ...
