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.
8
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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 ...
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 ...
2
votes
2answers
187 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 ...
4
votes
5answers
740 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
554 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
226 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
votes
1answer
519 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
144 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
votes
1answer
805 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):
$$ih \frac{d}{dt} U(x,t) = \left[ \left(-h^2 \frac{d^2}{dx^2} \right) + q V(x) ...
8
votes
4answers
469 views
Quantum mechanics as classical field theory
Can we view the normal, non-relativistic quantum mechanics as a classical fields?
I know, that one can derive the Schrödinger equation from the Lagrangian density
$${\cal L} ~=~ \frac{i\hbar}{2} ...
3
votes
1answer
299 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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votes
1answer
685 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?
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 ...
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.
...
1
vote
1answer
239 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 ...
-1
votes
2answers
140 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
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 ...
1
vote
3answers
400 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:
$$f(x)=\frac{1}{\sqrt{3}}\operatorname{sin}\Big( \frac{\pi x}{a} ...
3
votes
3answers
345 views
Smoothness constraint of wave function
Is there anything in the physics that enforces the wave function to be $C^2$? Are weak solutions to the Schroedinger equation physical? I am reading the beginning chapters of Griffiths and he doesn't ...
2
votes
1answer
762 views
How to calculate time evolution of a wave function in an 1D infinite square well potential?
A particle in an infinite square well has an initial wavefunction
$$\psi (x,0) ~=~ Ax(a-x) \qquad \mathrm{for}\qquad 0\leq x\leq a.$$
Now the question is to calculate $\psi (x,t)$.
I have ...
1
vote
1answer
240 views
A quantum particle in a box (with a catch)
I am reading Shankar's Quantum Mechanics and I am looking at the case where there is one particle inside a box, where the potential is zero inside the wall and abruptly goes to infinity outside the ...
1
vote
1answer
256 views
Is it a total or an explicite time derivative in the Schrödinger equation?
I am always dubious when I need write Schrödinger equation: do I write $\partial / \partial t$ or $d/dt$ ?
I suppose it depends on the space in which it is considered. How?
4
votes
1answer
1k views
What inspired Schrödinger to derive his equation?
I have almost no background in physics and I had a question related to Schrodinger's Equation. I think, it is not really research level so feel free to close it, but I would request you to kindly ...
2
votes
2answers
306 views
Correct application of Laplacian Operator
Not a physicist, and I'm having trouble understanding how to apply the Laplacian-like operator described in this paper and the original. We let:
$$ \hat{f}(x) = f(x) + \frac{\int H(x,y)\psi(y) ...
2
votes
2answers
896 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
669 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
424 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
371 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
votes
1answer
161 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
753 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
263 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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votes
1answer
625 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
votes
3answers
333 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
votes
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
votes
1answer
192 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
votes
1answer
263 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
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 ...
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 ...
1
vote
2answers
231 views
Inspecting the form of a wavefunction
Just a quick check: If given a time-independent wavefunction of the form $$\psi(x) = e^{ikx}f(x)$$, where $f(x)$ any arbitrary function of $x$ but one can't factor out another $e^{i\alpha x}$, ...
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 ...
1
vote
2answers
223 views
Schematic expression of the Schrodinger equation
it would be great if someone could help me understand the following quote regarding wavefunctions :)
"$$\psi(x)=\sum_n C_nu_n(x)+\int dE C(E)u_E(x)$$ The expression is schematic because we have ...
11
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 ...
20
votes
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 ...
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 $\mu=\hbar=1$):
...
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
...
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 ...
4
votes
2answers
671 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 ...
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 ...
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
1answer
729 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 ...
