What is Infinite potential well: Definition and 63 Discussions
In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example, a particle trapped inside a large box can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important. The particle may only occupy certain positive energy levels. Likewise, it can never have zero energy, meaning that the particle can never "sit still". Additionally, it is more likely to be found at certain positions than at others, depending on its energy level. The particle may never be detected at certain positions, known as spatial nodes.
The particle in a box model is one of the very few problems in quantum mechanics which can be solved analytically, without approximations. Due to its simplicity, the model allows insight into quantum effects without the need for complicated mathematics. It serves as a simple illustration of how energy quantizations (energy levels), which are found in more complicated quantum systems such as atoms and molecules, come about. It is one of the first quantum mechanics problems taught in undergraduate physics courses, and it is commonly used as an approximation for more complicated quantum systems.
TL;DR Summary: A particle of mass m, placed in an infinite rectangular one-dimensional potential well that confines it in the segment between x=-a/2 and x=a/2
Hi guys, I need help with this exercise which reads: a particle of mass m, placed in an infinite rectangular one-dimensional potential...
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 first normalized the given wavefunction and found the value of n that satisfies the normalization condition. I then used E = <E> = pi^2* h_bar^2* n^2/(2*m) to get the expectation value of energy. Assuming that this was the right process, I'm now trying to find <E^2> using the same equation...
I am guessing time-energy uncertainty relation is the way to solve this. I solved the Schrodinger equation for both the regions and used to continuity at ##x=-a, 0,a## and got ##\psi(-a<x<0) = A\sin(\kappa(x+a))## and ##\psi(0<x<a) = -A\sin(\kappa(x-a))## where ##\kappa^2 = 2mE/\hbar^2##...
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...
Some questions:
Why is this even a valid wave function? I thought that a wave function had to approach zero as x goes to +/- infinity in all of space. Unless all of space just means the bounds of the square well.
Since we have no complex components. I am guessing that the ##\psi *=\psi##.
If...
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?
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...
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...
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...
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
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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...
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...
Homework Statement
Find the ground and first excited state eigenfunctions of for the 1D infinite square well with boundaries -L/2 and +L/2
Homework Equations
$$\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi(x) = E\psi(x)$$
The Attempt at a Solution
Okay so I know how to solve it and...
For a particle in a box that is described with a wave function, why can there only be a standing wave when there is an infinite potential well? From my understanding, the infinite potential well makes it impossible for the particle to tunnel through the barrier and so the wave function cannot...
Homework Statement
An electron is confined to a narrow evacuated tube. The tube, which has length of 2m functions as a one dimensional infinite potential well.
A: What is the energy difference between the electrons ground state and the first excitied state.
B: What quantum number n would the...
Homework Statement
Sketch the difference of probability distributions at the two times. Does the energy change with time?
The potential well suddenly disappears, what is the form of the wavefunction?
Homework Equations
The Attempt at a Solution
Part (a)
At t = 0, the probability...
What happens to ψ in a infinite potential well when the width is suddenly reduced to half its previous value ?
Will this instantly adjust ψ to the new size of the well or will it take some time to confine itself in this new well ? And is there a possibility of quantum tunneling here?
Homework Statement
Part (a): Find wavefunction and energy levels.
Part (b): Find a possible wavefunction. Is this wavefunction unique?
Part (c): What is the probability of finding it in the ground state?
Part (d): What's the probability of finding it in the second excited state?
Homework...
Infinite potential well "proposal"
Homework Statement
An experimental physicist submits a proposal to a granting agency requesting support to construct an infinite potential well analogous to the one shown in Figure 3.5. Specifically, the proposal is to build a well with L = 1mm, inject some...
Why we don't have acceleration in quantum mechanics. For example why particle in infinite potential well can not accelerate. For example dimension of well is ##L## and ##L=\frac{at^2}{2}##, where ##a## is acceleration.
Hello guys,
I need some serious help for the solution of a problem in Q.M, I'm not so sure if I deal with it properly..
Consider an infinite potential well with the traits:
V(x):∞, for x>a and x<-a...
As we know, the 1d infinite potential well has a stationary state. The function that depends on x onky is a sin function.
However, I don't understand the concept in this question. I have the answer of this question and this is not a homework. I am not asking for the answer so please don't put...
Homework Statement
Hey guys:) Maybe you will be able to help me with this problem i got as an assignment for my quantum mechanics course, it goes as follows:
a particle of mass m moves in the potential
for x<0 infinity,
for 0<x<a -U,
for a<x<b 0,
for b<x infinity.
a) Sketch the...
For potential well problem for well with potential which is zero in the interval ##[0,a]## and infinite outside we get ##\psi_n(x)=\sqrt{\frac{2}{a}}\sin \frac{n\pi x}{a}##. If I want to get this result for well with potential which is zero in the interval ##[-\frac{a}{2},\frac{a}{2}]## and...
In one dimensional problem of infinite square potential well wave function is ##\phi_n(x)=\sqrt{\frac{2}{L}}\sin \frac{n\pi x}{L}## and energy is ##E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}##. Questions: What condition implies that motion is one dimensional. Did wave function describes motion of...
I'm a little confused about the electron wavelength in an infinite potential well.
It is my understanding that the maximum wavelength that the electron can achieve is 2 times the length of the potential well.
As the eigenvalue increases, does the wavelength change?
I believe that the...
nvm i figured it out. it was not in reference to n=4. equation used would be wavelength = 2L/n
Homework Statement
An electron is in an infinite potential well of width L. Which is not an allowed deBroglie wavelength for the electron to have when n=4?
wavelength(k) = 3L, 2L, L/2, or L/3Homework...
Homework Statement
A particle of mass m is confined (in one dimension) to the region 0 ≤ x ≤ a by a potential which is zero inside the region and infinitely large outside.
If the wavefunction at time t = 0 is of the form
ψ (x,0) = Ax(a - x) inside the region
ψ (x, 0) = 0 outside the...
Homework Statement
For the case n=1, calculate the probability that the particle is found in within the region a/4<x<3a/4 (n is the energy level, a is the width of the infinite potential well). Compare this result with the case n=8 and with the classical result.
Homework Equations...
Homework Statement
Assume that inside an infinite potential well there are 2 identical particles that doesn't interactuate between themselves and that have spin 1/2 (for instance electrons).
1)Write down the Schrödinger's equation associated with such a system. Write the eigenfunctions in...
Homework Statement
particle moving in the semi infinite potentail well set up and solve SE for the system assume E<VoHomework Equations
(-h2/2m) d2\psi/dx2 +v(x)\psi=E\psiThe Attempt at a Solution
so in reagion one its infinite so \psi=0. reagion 2 is what i am confused about. looking throught...
Hello again. Thank you guys. You have been great help...
I have another one:
Given a potential well- 2a is it's width, and in the middle - there is a delta potential:
V(x)= \frac {\hbar^2} {2m} \frac {\lambda} {a} \delta(x)
I am looking for the odd solution to this problem.
I thought...
problem is:
(a)write down the spatial or orbital for two-non interacting particles, with the same mass, in a one dimensional well, where the potential energy is zero for 0<x<2a and infinite anywhere else.
(b)What are the energies of the four lowest energy levels for the system in units of...
Homework Statement
An electron is confined to an infinite potential well of width L. Find the force it exerts on the walls of the well in the lowest energy state:
a) Estimate the force using uncertainty principle
b) Calculate the force exactly for the ground-state wavefunction
Homework...
By considering the wavefunctions within the potential described below, determine the incident, reflected and transmitted amplitudes of each part of the wavefunction at the step boundary (as necessary).
For x<0, V = infinity.
For 0<x<a, V = 0
For a<x<L, V = V1
For x>L, V = infinity...
Homework Statement
A particle in the infinite potential well in the region 0 < x < L is in the state
\psi(x) = \begin{cases}
Nx(x-L) & \text{ if } 0<x<L \\
0 & \text{ if } otherwise
\end{cases}
a) Determine the value of N so that the state is properly normalised
b) What is the...
Homework Statement
This is a problem from Merzbacher.
Assuming a particle to be in one of the stationary states of an infinitely high one-dimensional box, calculate the uncertainties in position and momentum, and show that they agree with the Heisenberg uncertainty relation. Also show that...
For the second, third, fourth, and fifth levels of the three-dimensional cubical box, find the energies in terms of the quantity Eo=π2*h2/(2mL2), where m is the particle mass and L is the box's sidelength.
ok so i have a question on infinite potential wells... if you have the energy state when n=3, does that mean that you have to have the energy state when n=2 and n=1?
Ok basically the Q we are given is:
A particle of mass m is trapped in an infi nitely deep one-dimensional potential well...
I have an exam later today, and I would really apreciate it if someone could check my work or at least point out flaws in my process here. Thanks!
Homework Statement
At x=0, a proton with a kinetic energy of 10 eV is traveling in the x direction (potential energy = 0). At x=1nm, it...
infinite potential well and the uncertainty principle
the solution for Schroedinger equation in infinite potential well satisfy the following
energy levels:
where l is the width of the well.
E can't be zero since then \psi=0 so there isn't any particle in the well . i read in...
Homework Statement
The nth energy level for a particle of mass m confined in an infinite potential well is given by:
where L is the width of the well and h is Planck’s constant. Assuming that the uncertainty in the particle’s momentum is equal to the momentum itself, show that the...
Homework Statement
Hi, this is basically a particle in a box problem but V(x) is sinusoidal inside the box.
V(x)=-Asin(pi x/a) 0<x<a
V(x)=Infinity otherwise
A: depth of the well
In the question, it asks me to plot the first 4 eigenfunctions.
The Attempt at a...