What is Potential well: Definition and 231 Discussions
A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy (kinetic energy in the case of a gravitational potential well) because it is captured in the local minimum of a potential well. Therefore, a body may not proceed to the global minimum of potential energy, as it would naturally tend to due to entropy.
Hi,
I'm trying to solve a transcendental equation. I would like all the values of E that solve this equation.
##k = -l \cdot Cot(la)##
However, using Nsolve or FindRoot, they give me a precision error. Hence, I'm trying this form.
##\sqrt{-e /(e+v)} = -Cot(la)##
FindRoot only give me an...
I have a nanoparticle of cadmium selenide with a diameter d. When it emits a photon with a wavelenght lambda, it happens because an electron jumps from the conduction band to the occupied band across a forbidden band. I can suppose that jump as a jump from a higher energy level (the conduction...
I feel that this problem can be directly answered from the E>0 case of the attractive Dirac delta potential -a##\delta##(x), with the same reflection and transmission coefficients. Can someone confirm this hunch of mine?
When we are talking about Bloch's theorem and also the tight-binding approximation, we can use them to help finding eigenstates of a system. However, I am so confused how to apply it in this case (below is my homework) and don't even know how to start it...
All I understand about the Bloch's...
Thank you for reading :bow:
Section 1
To find the energy states of the particle, we define the wave function over three discrete domains defined by the sets ##\left\{x<-L\right\}##, ##\left\{-L<x<L\right\}##, and ##\left\{L<x\right\}##. The time independent Schrodinder equation is...
To find the energy states of the particle, we define the wave function over three discrete domains defined by the sets ##\left\{x<-L\right\}##, ##\left\{L<x\right\}## and ##\left\{|x|<L\right\}##. The time independent Schr\"odinder equation is...
is it correct that the continuum states will be free particle states? and the probability will be |< Ψf | ΨB>|^2 . Where Ψf is the wave function for free particle and ΨB is the wave function for the bound state when the depth is B.
Do any of you know of an article or book chapter that discusses the difference between a discontinuous potential well of length ##2L##
##V(x)=\left\{\begin{array}{cc}0, & |x-x_0 |<L\\V_0 & |x-x_0 |\geq L\end{array}\right.##
and a differentiable one
##\displaystyle V(x) = V_0...
Using the boundary conditions where psi is 0, I found that k = n*pi/a, since sin(x) is zero when k*a = 0.
I set up my normalization integral as follows:
A^2 * integral from 0 to a of (((exp(ikx) - exp(-ikx))*(exp(-ikx) - exp(ikx)) dx) = 1
After simplifying, and accounting for the fact that...
I did some calculations for the ground state energy and wave function of a system of two electrons put in a finite-depth 2D potential well. Regardless of the shape of the potential well (square or circular), the expectation value of the electron-electron distance ##\langle r_{12}\rangle =...
Hello folks,
So my level of quantum knowledge is equivalent to what is covered in (year one) two short chapters introducing the topic in Knight's Physics for Scientists and Engineers. Ch. 39 introduces the idea of a wavefunction in a pretty simple way, and ch. 40 touches provides the basics of...
Hello there. I want to understand the mathematical idea behind boundaries that we write for a potential well. Why we use equally greater and smaller than let's say x between -4a and -2a but we only write x is less than -4a ? How to approach this idea with convergence theorem or Hilbert space...
Hello folks,
A bit stumped with the following question:
Consider a potential well with an infinite wall at x=o and a finite wall at x=a. The height at x=a is such that U0=2E1' where E1' is the energy of the particle's n=1 state in this semi-infinite well.
How can one show that E1' is lower...
I'm self studying so I just want to ensure my answers are correct so I know I truly understand the material as it's easy to trick yourself in thinking you do!
A particle of mass m is in a 1-D infinite potential well of width a given by the potential:
V= 0 for 0##\leq## x ##\leq## a
=...
If I calculate ## <\psi^0|\epsilon|\psi^0>## and ## <\psi^0|-\epsilon|\psi^0>## separately and then add, the correction seems to be 0 since ##\epsilon## is a constant perturbation term.
SO how should I approach this? And how the Δ is relevant in this calculation?
Okay so I begin first by mentioning the length of the well to be L, with upper bound, L/2 and lower bound, -L/2 and the conjugate u* = Aexp{-iz}
First I begin by writing out the expectation formula:
## \langle p \rangle = \int_{\frac{L}{2}}^{ \frac{L}{2} } Aexp(-iu) -i \hbar \frac{ \partial }{...
I want to compute the fraction of time both particles spend outside the finite potential well. All I can get is the probability to find them outside. The wavefunction outside the potential is:
$$\frac{d^2\psi}{dr^2} = -L^2 \psi$$
Where:
$$L = \sqrt{\frac{2mE}{\hbar^2}}$$
Solving the...
Homework Statement
Hello today I am solving a problem where an electron is trapped in a potential well. I have a solved Schrodinger's Equation. I am having problems in figuring out what the wave function should be. When I solved the equation I got a complex exponential. I know I cannot use the...
Greetings,
In the scenario of a particle in an infinite potential well, there are discrete energy levels, i.e.##E=\hbar ^2 n^2 \pi ^2/ (2 m L^2)## where L is the width of the potential well, and n takes on positive integers. But what will happen if I put a particle of energy ##E_i## that is not...
Homework Statement
CLASSICAL MECHANICS
[/B]Homework Equations
E=U+K[/B]The Attempt at a Solution
Guys, can you please help me with part b) ? I am not sure how to find the velocity. Thanks
For a particle trapped in a region of length L the de broglie wave for the 1st excited state is a pure sine wave from 0 to 2pi
for which the particle momentum can be calculated as 2h/L from de broglie relation
Whereas from energy quantisation relation p=nh/2L where n is the state integer,for...
Let's suppose I have a potential well: $$
V(x)=
\begin{cases}
\infty,\quad x<0\\
-V_0,\quad 0<x<R\\
\frac{\hbar^2g^2}{2mx^2},\quad x\geq R
\end{cases}
$$
If ##E=\frac{\hbar^2k^2}{2m}## and ##g>>1##, how can I calculate how much time a particle of mass ##m## and energy ##E## will stay inside...
What is the energy gap between the ground state (n=0) and the first excited state (n=1) of an electron trapped in a deep rectangular potential well of width 1Å?
Homework Statement
1D Potential V(x) = mw^2x^2/2, part of a harmonic oscillator.
Suppose that the spring can only be stretched, so that the potential becomes V=infinity for x<0. What are the energy levels of this system?
Homework EquationsThe Attempt at a Solution
I argued my way though this...
Homework Statement
I have a few questions I'd like to ask about this example. (C1 was already derived before the second part)
1. What does the line "The rest of the coefficients make up the difference" actually mean?
2. What does "As one might expect...because of the admixture of the...
In the 'Particle in a box' system, with the well being extremely narrow, why does the particle path have to follow certain energy levels compared to the classical system?
Thanks in advance.
Hi,
I'm trying to understand the bound states of a periodic potential well in one dimension, as the title suggests. Suppose I have the following potential, V(x) = -A*(cos(w*x)-1). I'm trying to figure out what sort of bound energy eigenstates you'd expect for a potential like this. Specifically...
Homework Statement
Using the equations given, show that the wave function for a particle in the periodic delta function potential can be written in the form
##\psi (x) = C[\sin(kx) + e^{-iKa}\sin k(a-x)], \quad 0 \leq x \leq a##
Homework Equations
Given equations:
##\psi (x) =A\sin(kx) +...
Hi guys!
I'm struggling with the following problem:
Consider two distinguishable (not interacting) particles in a quadratic 2 dimensional potential well. So
##
V(x,y)=\left\{\begin{matrix}
0,\quad\quad-\frac { L }{ 2 } \le \quad x\quad \le \quad \frac { L }{ 2 } \quad and\quad -\frac { L }{...
Homework Statement
I'm currently working on a homework set for my intermediate QM class and for some reason I keep drawing a blank as to what to do on the first problem. I'm given three potentials, V(x), the first is of the form {A+Bexp(-Cx^2)}, the others I'll leave out. I'm asked to draw the...
Homework Statement
An electron is enclosed in a potential well, whose walls are ##V_0 = 8.0eV## high. If the energy of the ground state is ##E = 0.50eV##, approximate the width of the well.
Answer: ##0.72nm##
Homework Equations
For an electron in a potential well, whose energy is less than...
Homework Statement
Determine what colors of visible light would be absorbed by electrons in an infinite well, N = 3.1 nm. The effective mass for an electron in GaAs is one-fifteenth of the standard electron mass.
Homework Equations
En = πh2/[2*N2*me/15]*n2
L = nλ/2
Ψ = √(2/L)sin(nπx/L)
The...
Given the equation ##\frac{d^2 \psi (x)}{{dt}^2}+\frac{2m}{{\hbar}^2}(E-V(x))=0## the general solution is:
$$\psi (x)=A_1 e^{ix \sqrt{\frac{2m}{{\hbar}^2}(E-V(x))}} +A_2 e^{-ix \sqrt{\frac{2m}{{\hbar}^2}(E-V(x))}}$$
If we have an infinite potential well: ## V(x)=\begin{cases} \infty \quad x\ge...
We know that the solutions of time-independent Dirac delta potential well contain bound and scattering states:
$$\psi_b(x)=\frac{\sqrt{mu}}{\hbar}e^{-\frac{mu|x|}{\hbar^2}}\text{ with energy }E_b=-\frac{mu^2}{2\hbar^2}$$
and
$$
\psi_k(x)=
\begin{cases}
A(e^{ikx}+\frac{i\beta}{1-i\beta}e^{-ikx})...
Homework Statement
I think this is a square well potential problem. The question asks me to sketch the ground-state probability density, for the following situation:
A quasielectron moves in a 'quantum dot' device. The potential V(x) = 0 for 0 ≤ x < L, and is infinite otherwise.
Homework...
Okay so I am trying to solve a delta potential well with an infinite potential wall on one side a distance a away from the well. The other side is open so I am confused about how to set up the problem. Here is a picture of my work so far and if anyone has an insight into this I'd appreciate some...
1. The problem statement.
for Infinite symmetric well -a/2 < x < a/2 in one dimension
show that wave function Ψ = Acos(kx) + Bsin(kx)
is not physically accepted solution although its mathematically accepted
Homework Equations
∫ψ(x)* ψ(x) dx=1
I Have tried to solve a problem about infinite potential well with a delta well in the middle, but I haven't the results and so I can't check if the proceeding is wrong...
My attempt solution:
The Schroedinger's Equation is:
##\psi''(x)=\frac{2m}{\hbar^2} (V(x)-E) \psi (x)##
so we have...
Homework Statement
[/B]
Five free electrons exist in a three-dimensional infinite potential well with all three widths equal to a 12 angstroms. Determine the Fermi energy level at T 0 K.
Homework Equations
E = [(h_bar*pi)2/(2*m*a2)]*(nx2 + ny2 + nz2)
The Attempt at a Solution
Tried using EF...
Homework Statement
The wave function for a particle in a infinitely deep potential well is at some point in time Φ(x) = Nx(a-x). In which probability gives the energy measurment a another value than E1 ,etc ground state
Homework Equations
1 = |cn|^2 = |<Φn|Ψ>|^2 (1)
The Attempt at a...
The ground state energy of a particle trapped in an infinite potential well of width a is given by (ħ2π2)/2ma2. So the momentum is given by (2mE)1/2 = ħπ/a. Since this is a precise value, doesn't that mean that we know momentum with 100% certainty? And if that is the case shouldn't the...
Hi All,
During our Quantum Mechanics class one of the students asked if it was possible for a particle to enter a region (just before a potential well) with a negative energy. The TA said that the energy could be negative, but if the potential well has it's bottom at -U (where U is some...
Homework Statement
The odd bound state solution to the potential well problem bears many similarities to the zero angular momentum solution to the 3D spherical potential well. Assume the range of the potential is 2.3 × 10^−13 cm, the binding energy is -2.9 MeV, and the mass of the particle is...