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.
1
vote
1answer
179 views
Solving the 1-D Schrodinger equation for a free particle: Confused about 2 possible general solutions
I am following Griffiths' Introduction to Quantum Mechanics, as well as an online lecture that follows a different book, and both sources give different equations for the general solution of the 1-D ...
0
votes
0answers
49 views
Proving One-dimensional Wave with Classic Expression [closed]
A general form of a one-dimensional wave is ψ(x) = sin kx where k is a constant. Show that
this function is a solution to the classical expression
((d^2 ψ(x))/ dx^2)= -((2pi)/lambda)^2 ψ(x)
What is ...
1
vote
2answers
195 views
Hydrogen atom in quantum mechanics
I have problems following the calculations in Griffiths' Introduction to Quantum Mechanics (Chapter 4.2.1):
If you apply the Schrödinger equation to the Coulomb potential you get the following ...
2
votes
3answers
220 views
Is this interpretation of $\psi=\frac{1}{\sqrt{\pi a^{3}}}e^{-r/a}$ correct?
Apologies if this is stating the obvious, but I'm a non-physicist trying to understand Griffiths' discussion of the hydrogen atom in chapter 4 of Introduction to Quantum Mechanics. The wave equation ...
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(- ...
6
votes
0answers
224 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 ...
1
vote
0answers
53 views
Calculating the error by a small change of the potential in Schrodinger equation
In $\mathbb{R}^3$, consider the time-dependent (non-rel) Schrodinger equation with the potential energy $V(\mathbb{x})$. When a small change(e.g., just a small constant $\delta>0$) of V(x) is ...
1
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1answer
188 views
Does a particle in a spherically symmetric infinite square well potential exert a force on the inner and outer shell barrier?
For a particle in the potential:
$$V(r) =
\begin{cases}
0 & \text{a < r < b}\\
\infty & \text{otherwise.}
\end{cases}$$
Does this guy in the ground-state exert a force on the shells a ...
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 ...
2
votes
2answers
417 views
Two expressions for expectation value of energy
I was looking up expectation value of energy for a free particle on the following webpage:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/expect.html
It says that $E=\frac{p^2}{2m}$ and ...
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 ...
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 ...
5
votes
2answers
128 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?
5
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 ...
0
votes
1answer
143 views
Morse potential and chaos
I have heard that the Morse potential equation
$ -\frac{\hbar ^{2}}{2m} \frac{d^{2}}{dx^{2}}y(x)+ae^{bx}y(x)-E_{n}y(x)=0 $ (1)
is related to the two dimensional equation on the Poincare half plane ...
1
vote
2answers
279 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 ...
1
vote
1answer
161 views
exponential potential solution
let be the Schroedinguer equation
$$ - \frac{d^{2}}{dx^{2}}y(x)+ae^{cx}y(x)=E_{n} $$ (1)
here a and c are constants.
i know how to solve it from ...
1
vote
1answer
70 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) $ ...
0
votes
1answer
142 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]:
$$
\ddot R + \frac1r\dot ...
0
votes
2answers
269 views
Meaning of instantaneous probability densities in time dependent wavefunctions
For a time dependent wavefunction, are the instantaneous probability densities meaningful? (The question applies for instances or more generally short lengths of time that are not multiples of the ...
1
vote
2answers
88 views
What is the spectrum of energies for the potential $ a^{x} $?
Given a certain potential $ a^{x} $ with positive non-zero 'a' are there a discrete spectrum of energy state for the Schrodinger equation
$$ \frac{- \hbar ^{2}}{2m} ...
2
votes
1answer
293 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
votes
1answer
306 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
$$H=\hbar\omega
...
0
votes
4answers
403 views
Which Schrodinger equation is correct?
In the coordinate representation, in 1D, the wave function depends on space and time, $\Psi(x,t)$, accordingly the time dependent Schrodinger equation is
$$H\Psi(x,t) = ...
0
votes
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:
$$V(x) ~=~\left\{ \begin{array}{lcc}\text{finite ...
2
votes
1answer
1k views
Bound States in a Double Delta Function Potential [closed]
Let $V(x) = −u \delta(x) - v \delta(x − a)$ where $u, v > 0$ correspond to a potential with two $\delta$ wells. Let $v > u$. If $a$ is very large, there is certainly a bound state: the particle ...
8
votes
4answers
472 views
Quantum mechanics as classical field theory
Can we view the normal, non-relativistic quantum mechanics as a classical fields?
I know, that one can derive the Schrödinger equation from the Lagrangian density
$${\cal L} ~=~ \frac{i\hbar}{2} ...
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: $$\psi(x,y,z) = \psi(x)\psi(y)\psi(z) \quad\text{when}\quad V(x,y,z) = V(x) + V(y) + V(z) = ...
3
votes
4answers
675 views
Where is spin in the Schroedinger equation of an electron in the hydrogen atom?
In my current quantum mechanics, course, we have derived in full (I believe?) the wave equations for the time-independent stationary states of the hydrogen atom.
We are told that the Pauli Exclusion ...
0
votes
0answers
182 views
Finding transcendental equation for the energy of a particle in delta potential well near infinite potential barrier [closed]
I'm having trouble finding the transcendental equation for a particle in a delta potential settled near an infinite potential wall. The potential is given by
$$
V(x) =
\begin{cases}
\infty & x ...
3
votes
1answer
333 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 ...
1
vote
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 ...
1
vote
2answers
345 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 ...
1
vote
2answers
493 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 ...
4
votes
1answer
80 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 ...
2
votes
2answers
439 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 ...
2
votes
1answer
361 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.
1
vote
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 ...
1
vote
2answers
153 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:
$$ H = \dots = m\sqrt{1 - \hbar^2/m^2 ...
1
vote
2answers
188 views
How to write Schrodinger equation when a particle with some spin quantity and orbital angular momentum
Quantum mechanics: Suppose that there is a particle with orbital angular momentum $|L|$. But the particle also has spin quantity $|S|$. The question is, how do I reflect this into Schrodinger ...
2
votes
1answer
237 views
Wavefunction in quantum mechanics and locality
Every wavefunction of a form $\Psi(x)$ can be described as a superposition of multiple free particle solutions.
We can see the following Fourier transform:
$$ \psi(x) = \int e^{ik\cdot x} \psi(k) dk ...
0
votes
1answer
349 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 ...
1
vote
1answer
162 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 ...
2
votes
1answer
221 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
vote
1answer
305 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 ...
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 ...
1
vote
1answer
227 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 ...
3
votes
3answers
355 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
votes
1answer
324 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?
