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31
votes
4answers
4k views

Why does the Schrödinger equation work so well for the hydrogen atom despite the relativistic boundary at the nucleus?

I have been taught that the boundary conditions are just as important as the differential equation itself when solving real, physical problems. When the Schrödinger equation is applied to the ...
2
votes
1answer
49 views

Symmetric potential well different solutions

I have solved $H|\psi\rangle=E_{n}|\psi\rangle$ with $V(x)=0$ from $-a<x<a$ and $\infty$ otherwise. If I propose a solution of the form $\psi(x)=A_{n}e^{ikx}+B_{n}e^{-ikx}$ I arrive to the ...
-1
votes
1answer
120 views

Why is the $k$-space in multiples of $2\pi/L$?

So when you find the solution to the Schrödinger equation you get that the wave function can have $k=n\pi/L$, $n=1, 2,3 \dots $ The problem I have is that when calculating the density of states of a ...
1
vote
0answers
56 views

Green's function for infinite square well

The Green's function can be given in terms of left and right solutions. $G(x,x';k) = \frac{1}{W}\left(\Psi_{L}(x_{<})\Psi_{R}(x_{>})\right)$ But I don't understand how to determine these left ...
3
votes
1answer
89 views

Schrödinger Equation for a freely falling body near the surface of Earth

Near Earth's surface the Schrödinger equation of a freely falling particle takes the form, $$ \frac {-\hbar^2}{2m} \frac {d^2 \psi (y)}{dy^2} + mgy\psi (y) = E \psi (y). $$ Putting $k=\frac {\sqrt {...
0
votes
1answer
78 views

Wavefunction of a shifted radial harmonic oscillator [duplicate]

Suppose we know the solution of Schrodingers equation for a radial potential $V(r)$. Then the energy eigenstates are $\psi(r,\theta,\phi) = \frac 1ru(r)Y_\ell^m(\theta,\phi)$ where the radial ...
1
vote
1answer
28 views

How does the image of $\sqrt{2/L} \ \sin\left({k_n x}\right)$ satisfy the boundary conditions for the infinite square well?

I understand mathematically how $\sqrt{2/L} \ \sin\left({k_n x}\right)$ satisfies the boundary conditions for the infinite square well in terms of the fact that $\psi(0) = \psi(a) = 0$, and excuse the ...
1
vote
1answer
109 views

Boundary Condition for Dirac comb potential in solving independant Schrodinger Equation

The Periodic potential is And, the general solution is: Then, boundary condition at $x=a$ is: Where does $2\Omega u(a)$ comes from? I know that boundary condition is just 1) $U(x<a)(a)=U(x>...
2
votes
1answer
201 views

Infinite annular potential well. Trouble with solving Bessel equation to get eigenstates and energy

I have infinite annular potential well (scheme in the picture). Schrodinger equtation in the anullus (for $R_1 <r<R_2$ is $V=0$) with polar coordinates is \begin{equation} - \frac{ \hbar }{...
0
votes
1answer
44 views

Step potential well solve the other kind of given boundary condition

Unlike the textbook, I was trying to test a new set of boundary condition in step potential where probability density and momentum was continuous at the boundary $x=0$. Suppose $k_0=\sqrt{2m/\hbar^2E}...
1
vote
0answers
81 views

Copenhagen interpretation and argument of boundary condition [duplicate]

Quote from https://en.wikipedia.org/wiki/Copenhagen_interpretation#Principles "...The wavefunction evolves smoothly in time while isolated from other systems." However, by the studying, my feeling ...
1
vote
1answer
89 views

Half-line vs radial Schrodinger equation

Assume that a quantum particle is constrained to move along a semi-infinite fixed rigid rod. This system could be described as a particle on a half-line, if we introduce the cartesian coordinates so ...
3
votes
3answers
465 views

$\sin$ and $\cos$ components in symmetric infinite potential well problem

Consider an infinite potential well in one dimension with boundaries at $\pm a/2$. Can $\psi(x) = A \sin(kx) + B \cos(kx)$ for this system? The way it was answered was "mathematically acceptable but ...
2
votes
1answer
299 views

Is this expression for radial probability flux in Sakurai's Modern Quantum Mechanics wrong?

The section on Schrodinger's equation for central potentials in Sakurai's Modern Quantum Mechanics (p. 208, 2nd edition) contains the following expression for the radial probability flux, as part of ...
3
votes
4answers
500 views

Why do the energies of the infinite square well decrease as width of the well increases?

The stationary state wavefunctions for the infinite square-well of width $a$ are given by $$\psi_n(x)=\sqrt{\frac{2}{a}}\sin{(\frac{n\pi x}{a})}.$$ These correspond to energies, $$E_n=\frac{n^2\pi^2\...
0
votes
2answers
241 views

Solving the TISE for Infinite square well mathematical question

Consider the infinite square well situation where the potential is infinite at positions $|x| > a$ and $0$ otherwise. When solving the Time independent Schrodinger Equation (TISE) we can come to ...
0
votes
1answer
387 views

1D Time independent Schrodinger equation applied to a ring of Radius $R$

In the lectures we have been doing basic examples of Applying the TISE to determine the solutions of simple situations such as the finite and infinite square well and I understand how to find the ...
1
vote
1answer
134 views

Energy spectrum for a step potential

Most of the books tend to give this explanation that for a bound physical system, the energy and momentum eigen values have discrete spectrum and otherwise, they have a continuous spectrum, which I ...
1
vote
1answer
511 views

Discontinuity of wave function derivative

In general, apart from mathematical standpoint, physically what causes the discontinuity of the derivative of wavefunction at infinitely high discontuinity of potential, but not in the case of a ...
2
votes
1answer
80 views

Boundary conditions for the numerical particle in a box example

I want to solve the one-dimensional Schrödinger equation for the particle in a box example, and want to force the wavefunctions to zero on the boundaries. I am using the matrix, \begin{equation} \hat{...
0
votes
2answers
195 views

Does the quantization come from that the wave function $\psi$ should vanish at infinity?

For one dimensional non-relativistic quantum mechanics, the solutions to $\hat H\psi=E\psi$ seems not requiring the energy $E_n$ to contain the "$n$" term without specific boundary conditions. Does ...
1
vote
1answer
370 views

Boundary conditions on radial Schrodinger equation for $V(r)=A/r²-B/r$ potential

I have to solve the radial Schrodinger equation for a particle subjected to the potential: $$V(r)= \dfrac{A}{r^2}-\dfrac{B}{r}$$ Where $r$ is the radial component (spherical coordinates) and $A,B&...
0
votes
2answers
3k views

Periodic boundary condition for a particle in a box

I've been trying to solve the Schrödinger equation for a particle in a box of volume $V$ (length $L$) with periodic boundary conditions. The general solution for the equation yields, for one dimension,...
1
vote
1answer
102 views

Understanding boundary condition in this exercise

I am trying to do this quantum mechanics exercise: A particle with mass $m$ in the potential: $$V(x)= \left\{\begin{array}{lc}\infty& x\leq -a\\ \Omega\, \delta(x) & x>-a\end{array}\...
1
vote
2answers
486 views

Particle in 1D box. Different solutions for wavefunction

For a particle in a box $x$ ranging from $0$ to $L$ you get a solution of $\sqrt{2/L}\sin (n \pi x/L)$. But if you have a particle in a box $x$ ranging from $-L/2$ to $L/2$ infinite square well ...
3
votes
2answers
946 views

2D Schrodinger equation in polar coordinates - boundary conditions at the origin

When solving the Schrodinger equation in 2D polar coordinates, one has to deal with various Bessel functions. In the most simple example, the infinite circular potential well, the solutions to the ...
14
votes
1answer
820 views

How fast does a wavefunction vanish at infinity?

When solving one-dimensional quantum mechanical systems, I find myself very confused about the behavior of wavefunctions at infinity. Let's first impose three reasonable constraints: The potential ...
1
vote
2answers
484 views

Solving Schrödinger equation numerically with boundary condition

Suppose I want to solve the one-dimensional Schrödinger equation: $\frac{-\hbar^2}{2m} \frac{d^2\psi}{dx^2} + V\psi = E\psi$ This can be done by discretizing it on a one-dimensional grid and ...
0
votes
1answer
211 views

How to show that $\lim_{r\to0} U(r)=0$ in the radial differential equation?

We have the following differential equation: $$\left({- \hbar ^2 \over 2 \mu } \frac{d^2}{dr^2} + {\hbar^2 \ell(\ell+1) \hbar^2 \over 2 \mu r^2} + V(r) \right)U(r)= EU(r)$$ in order to find the ...
0
votes
1answer
569 views

Hard-wall boundary conditions

Hard-wall boundary conditions says that wave function = 0 on walls. And you can't really use running wave, must be stationary. How to prove that these boundary conditions leads to Δki=π/L? Where ki is ...
2
votes
1answer
142 views

Boundary conditions for Quantum Cascade Laser (QCL)

The Quantum Cascade Laser (QCL) is a semiconductor device for the generation of radiation in the MIR region of the electromagnetic spectrum. One period of the device consists of two regions, the ...
-1
votes
1answer
624 views

Infinite square well - periodic boundaries

If we have an infinite square well, I can follow the usual solution in Griffiths but I now want to impose periodic boundary conditions. I have $\psi(x) = A\sin(kx) + B\cos(kx)$ with boundary ...
8
votes
3answers
3k views

Interpretation of boundary conditions in time-independent Schrödinger equation

The time-independent Schrödinger equation: $$\ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V\psi = E\psi$$ is second order, so we should expect the solution to have two "degrees of freedom" which can ...
0
votes
1answer
1k views

Full solution of the quantum particle on a ring

In quantum particle in a ring problem, the general solution for the wavefunction, with $k = R \sqrt{2 m E / \hbar^2}$, $R$ being the ring radius, $c_{+, -}$ being constants, $E$ the energy, and $m$ ...
1
vote
1answer
46 views

Correlating two definitions of bound states in quantum mechanics

In Griffiths, he defines a bound state to be that stationary state for which the total energy E is such that $E<V(\pm\infty)$. Let $\psi(x)$ is a stationary state satisfying $E<V(\pm\infty)$ and ...
1
vote
1answer
44 views

Normalization of states of continuos spectra with complicated boundary conditions

Let's consider the following Schrödinger equation: $$\psi''(x)+k^2\psi(x)=0$$ with the following boundary condition $$\psi(0)+a\psi'(0)=0$$ $k$ is supposed to be larger that $0$. This equation is ...
2
votes
0answers
74 views

Perodic boundary conditions vs Dirichlet?

I have been working through several examples recently involving particles in boxes (when finding the partition function of an ideal gas for example or looking at photon gases). I have seen two ...
3
votes
0answers
208 views

Eigenvalue problem $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue in the complex plane of $x$ for one dimensional Schrodinger equation $$ −ψ''(x) − (ix)^ N ψ(x) = Eψ(x). $$ where $N$ can be any real number, the boundary condition $ψ(x) → 0$ ...
1
vote
2answers
885 views

Difference for boundary condition, particle in a box

When solving the simple problem of a free particle in a box of volume $V = L^3$, we can impose either periodic boundary conditions $\psi(0) = \psi(L)$ and $\psi '(0)= \psi'(L)$ either strict boundary ...
0
votes
1answer
421 views

Periodic boundary and differential equation in Quantum Mechanics

Consider 3d box of size $L$ with periodic boundary. Then the Schrodinger equation gives \begin{align} \frac{d^2 \Psi}{d x_i^2} = -k_i^2 \Psi \end{align} thus we can set the solution in the following ...
2
votes
3answers
614 views

Particle in a box: value for wave function $u(x)$ when potential $V(x)$ is infinity

The time-independent Schrödinger equation (TISE) is: $$ -\frac{\hbar^2}{2m}\frac{d^2 u(x)}{dx^2}+V(x)u(x)=Eu(x) \hspace{15pt}$$ where $E$ is a constant. Imagine now a infinity potential well as ...
0
votes
1answer
551 views

The boundary condition for delta function

Beginning with the Schr\"odinger equation for $N$ particles in one dimension interacting via a $\delta$-function potential $$(-\sum_{1}^{N}\frac{\partial^2}{\partial x_i^2}+2c\sum_{<i,j>}\delta(...
0
votes
2answers
3k views

Rectangular potential barrier

Take the usual rectangular potential barrier, that is: $$V(x)=0 \: \text{if} \: x<0 \: \text{or}\: \: x>a$$ $$V(x)=V_0 \: \text{if} \: 0\leq x \leq a.$$ I've looked at several notes and books ...
1
vote
3answers
3k views

Why is $ \psi = A \cos(kx) $ not an acceptable wave function for a particle in a box?

Why is $ \psi = A \cos(kx) $ not an acceptable wave function for a particle in a box with rigid walls at $x=0$ and $x=L$ where $$ k = \frac {(2mE)^{1/2}} {\hbar} \, ?$$ I had plugged the wave ...
1
vote
1answer
623 views

Boundary conditions of the radial Schrodinger equation

Consider the radial differential equation $$\bigg( - \frac{d^2}{dr^2} + \frac{(\ell+\frac{d-3}{2})(\ell+\frac{d-1}{2})}{r^2} + V(r) + m^2 \bigg) \phi_\ell (r) = \lambda\ \phi_\ell (r),$$ which I've ...
14
votes
2answers
1k views

Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?

Let $\Omega$ be a domain in $\mathbb{R}^n$. Consider the time-independent free Schrodinger equation $\Delta \psi = E\psi$.[*] Solutions subject to Dirichlet boundary conditions can be physically ...
5
votes
3answers
555 views

Why do we not require higher derivatives to match at boundary when solving the Schrödinger equation in a given potential?

When solving the time independent Schrödinger equation for a given potential in 1D, the main part of the solving involves matching boundary conditions. Usually, we require the value and the first ...
1
vote
1answer
351 views

Eigenvalues of the radial Schrödinger equation on a finite integration interval

There are numerous ways to estimate the eigenvalues of a radial Schrödinger equation, see http://arxiv.org/abs/math-ph/0703040 as an example. Anyhow, the formulas only cover the Schrödinger equations ...
0
votes
1answer
83 views

Losing a term for 3D radial schrodinger equation

I am trying to solve the Schrodinger equation For a potential $V(r)$ defined for $ 0<r<R$ as $$V(r)=-V_0 $$ and zero everywhere else. For wavefunction $u$ I can easily get to $$ u'' =-k^2u,$$ ...
3
votes
3answers
2k views

Quantum mechanics in electric field

Consider a charged particle with charge $q$ trapped in a box of length $L$ with finite constant potential $ V_0 $ on both ends. A constant (static) electric field of magnitude $F$ is applied from $- \...