What is Square well: Definition and 223 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.
In a) I get that T should be largest where V_0 is least wide, because when V_0 is infinitely wide the particle would be fully reflected.
But I don't get how height in b) and energy levels height in c) correlates to T and R.
Is it because of their k? I get the opposite answer from the correct...
I have solved c), but don’t know how to solve the integral in d.
It looks like an integral to get c_n (photo below), but I still can’t figure out what to make of c) in the integral of d).
I also thought maybe you can rewrite c) into an initial wave function (photo below) with A,x,a but don’t...
I am trying to numerically solve (with Mathematica) a relativistic version of infinite square well with an oscillating wall using Klein-Gordon equation. Firstly, I transform my spatial coordinate ## x \to y = \frac{x}{L[t]} ## to make the wall look static (this transformation is used a lot in...
After watching this video: which explains why the wavefunction in an infinite square well is flattened, I tried running the calculation in both, what seems, the more more traditional way of using sin and by the method of, what seems to be, adding the wavefunction and its complex conjugate...
I took the w derivative of the wave function and got the following. Also w is a function of time, I just didn't notate it for brevity:
$$-\frac{\sqrt{2}n\pi x}{w^{3/2}}cos(\frac{n\pi}{w}x) - \frac{1}{\sqrt{2w^3}}sin^2(\frac{n\pi}{w}x)$$
Then I multiplied the complex conjugate of the wave...
I'm following Griffith's Modern Physics 2nd edition chapter 5.
I got to the part where we make ΨI(0) = ΨII(0) I get that
αCeα(0) = QAsin(Q(0)) - QBsin(Q(0)) => C = QA/α
But when I try to graph it, the region I distribution doesn't seem to equal the region II distribution at 0.
The book goes...
Obviously a particle inside an ISW of width L cannot have arbitrarily precise momentum because ΔP ≥ ℏ/2ΔX ≥ ℏ/2L. Therefore you cannot have a particle with arbitrarily low momentum, since that would require ΔP be arbitrarily small.
I need to show that a photon inside an ISW cannot have...
Here are the results from the python code:
Odd results:
Even results:
I tried to solve for energy using the equation:
I substituted the value for a as 4, as in the code the limit goes from -a to a, rather then 0 to a, and hence in the code a = 2, but for the equation it would equal to 4...
Hello! I have been recently studying Quantum mechanics alone and I've just got this question.
If the potential function V(x) is an even function, then the time-independent wave function can always be taken to be either even or odd. However, I found one case that this theorem is not applied...
Hi,
I think I'm having a bit of a brain fart...I'm messing with this numerical code trying to understand the 1-D time-independent Schrodinger's equation infinite square well problem (V(x) infinite at the boundaries, 0 everywhere else). If normalized Phi squared is the probability of finding...
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...
A particle of mass m is in the ground state on the infinite square well. Suddenly the well expends to twice it's original size (x going from 0 to a, to 0 to 2a) leaving the wave function monetarily undisturbed.
On answering, for ##\Psi_{n}## I got ##\Psi_{n}## = ##\sqrt{\frac{1}{a}}...
Attempt: I'm sure I know how to do this the long way using the definition of stationary states(##\psi_n(x)=\sqrt{\frac {2} {a}} ~~ sin(\frac {n\pi x} {a})## and ##\int_0^{{a/2}} {\frac {2} {a}}(1/5)\left[~ \left(2sin(\frac {\pi x} {a})+i~ sin(\frac {3\pi x} {a})\right)\left( 2sin(\frac {\pi x}...
I have always seen this problem formulated in a well that goes from 0 to L
I am confused how to use this boundary, as well as unsure of what a dimensionless hamiltonian is.
This is as far as I have gotten
I've tried to carry out the solution to this as a normal 2nd order Differential Equation
##\psi ##'' - ##-k^2 \psi ## = 0
Assume solution has form ##e^{\gamma x}##
sub this in form ##\psi## and get
##\gamma ^2## ##e^{\gamma x} ## + ##k^2 e^{\gamma x}## = 0
Solution is ##\gamma## = 0 or ##k^2##...
The problem is:
Solve the time independent Schrodinger Equation for infinite square well centered at origin. Show that the energy is same as in the original case(well between x=0 and x=L). Also show that the solution to the this case can be obtained by setting x to x-L/2 in ##\psi## in the...
Homework Statement
Construct the four lowest-energy configurations for particles of spin-##\frac{1}{2}## in the infinite square well, and specify their energies and their degeneracies. Suggestion: use the notation ##\psi_{n_1,n_2}(x_1, x_2) |s,m>##. The notation is defined in the textbook...
Homework Statement
My doubts are on c)
Homework Equations
$$< H > = \int \Psi^* \hat H \Psi dx = \frac{2}{a} \int_{0}^{a} sin (x\frac{\pi}{a}) \hat H sin (x\frac{\pi}{a}) dx$$
The Attempt at a Solution
I understand that mathematically the following equation yields (which is the right...
For this problem at t=0
Ψ(x,0)=Ψ1-Ψ3
Where Ψ1 and Ψ3are the normalised eigenstates corresponding to energy level 1 and 3 of the infinite square well potential.
Now for it's time evolution it will be Ψ1exp(-iE1t/ħ)- Ψ3exp(-iE3t/ħ)
And taking the time given in the question the time part of the...
Homework Statement
Homework Equations
For this question my ans. is coming option (3) since the time part of the wave comes out to be same for both the energy states which is (-1)^(-1/8) and (-1)^(-9/8) respectively (using exp(-iEt/ħ)).
But the correct option is given option (4).
Am I right...
Homework Statement
Consider the standard square well potential
$$V(x) =
\begin{cases}
-V_0 & |x| \leq a \\
0 & |x| > a
\end{cases}
$$
With ##V_0 > 0##, and the wavefunctions for an even state
$$\psi(x) =
\begin{cases}
\frac{1}{\sqrt{a}}cos(kx) & |x| \leq a \\...
Let's suppose I have a finite potential well: $$
V(x)=
\begin{cases}
\infty,\quad x<0\\
0,\quad 0<x<a\\
V_o,\quad x>a.
\end{cases}
$$
I solved the time-independent Schrodinger equation for each region and after applying the continuity conditions of ##\Psi## and its derivative I ended up with...
Homework Statement
Hello, I am stuck on four equations for which I must find the coefficients A,B,I,J.
I have tried using latex but the commands don't seem to work.Homework Equations
Four equations:
A+B = I+J
\frac{\alpha}{k}(J-I) = A - B
D = Ie^{ia(\alpha-k)} + Je^{-ia(\alpha + k)}
D =...
Hi,
I'm preparing for an exam, and I'm going over past papers. I've solved parts a & b of this question without any problems, however I'm finding it hard to understand part c.
I thought of shifting the boundary conditions so I'd have 0 and L in the place of ± L/2, but that would not work...
Homework Statement
A particle of mass m is moving in an infinite square well of width a. It has the following normalised energy eigenfunctions:
$$u_n (x) = \sqrt{\frac{2}{a}} sin(\frac{n \pi x}{a})$$ (1)
a) Give an expression that relates two orthogonal eigenfunctions to each other and use it...
Homework Statement
The wave function of a particle of mass m confined in an infinite one-dimensional square well of width L = 0.23 nm, is:
ψ(x) = (2/L)1/2 sin(3πx/L) for 0 < x < L
ψ(x) = 0 everywhere else. The energy of the particle in this state is E = 63.974 eV.
1) What is the rest energy...
I am not sure how is it possible that asymetric potential well does not have bond states if ##E<U_1<U_2##. In symmetric case solution always exists. Why this is a case?
Hello! I am trying to write a program that solves the Schrodinger Equation for a particle in an infinite square well. I did a lot of research regarding the methods that could be used to accomplish this. I am writing this program in Matlab. The method I am using is called the Shooting Method. In...
Homework Statement
At t < 0 we have an unperturbed infinite square well. At 0 < t < T, a small perturbation is added to the potential: V(x) + V'(x), where V'(x) is the perturbation. At t > T, the perturbation is removed. Suppose the system is initially in the tenth excited state if the...
A particle is in its ground state of an infinite square well of width a <xl i>=√2/a*sin(πx/a) and since it's an eigenstate of the Hamiltonian it will evolve as <xlα(t)>=√2/a*sin(πx/a)e^(-iE1t/ħ) where E=π2ħ2/2ma2
If the well now suddenly expands to witdh 2a
If the well suddenly expands to 2a...
Homework Statement
A particle of mass m, is in an infinite square well of width L, V(x)=0 for 0<x<L, and V(x)=∞, elsewhere.
At time t=0,Ψ(x,0) = C[((1+i)/2)*√(2/L)*sin(πx/L) + (1/√L)*sin(2πx/L) in, 0<x<L
a) Find C
b) Find Ψ(x,t)
c) Find <E> as a function of t.
d) Find the probability as a...
Homework Statement
A particle is in the n=1 state in an infinite square well of size L. What is the probability of finding the particle in the interval Δx = .006L at the point x = 3L/4?
Homework Equations
ψ(x) =√(2/L) sin(nπx/L)
The Attempt at a Solution
The problem states that because Δx is...
Hi,
I'm trying to work my way through some problems and am stuck on one for a symmetric infinite square well, of width 2a, so -a<x<+a. Since this is the symmetric case, the wavefunction should be a linear combination of the terms
(a)-½ cos (nπx/2a) for odd n,
(a)-½ sin (nπx/2a) for even n...
Homework Statement
ISW walls at 0 and L, wavefunction ψ(x) = { A for x<L/2; -A for x>L/2. Find the lowest possible energy and the probability to measure it?
Homework Equations
Schrodinger equation
ψ(x)=(√2/L)*(sin(nπx/L)
cn=√(2/a)∫sin(nπx/L)dx {0<x<a}
En=n2π2ħ2/2ma2
The Attempt at a...
Homework Statement
The global topology of a ##2+1##-dimensional universe is of the form ##T^{2}\times R_{+}##, where ##T^{2}## is a two-dimensional torus and ##R_{+}## is the non-compact temporal direction. What is the Fermi energy for a system of spin-##\frac{1}{2}## particles in this...
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 conditions \psi(x) = \psi(x+L)
In the fixed boundary case, we had \psi(0) = 0 which meant B=0 and...
How how can we calculate the future evolution of a particle after the infinite square well potential is (somehow) turned off, releasing it into a free state? Assuming that it was in the ground state before.
Homework Statement
Suppose that an infinite square well has width L , 0<x<L. Nowthe right wall expands slowly to 2L. Calculate the geometric phase and the dynamic phase for the wave function at the end of this adiabatic expansion of the well. Note: the expansion of the well does not occur at...
Homework Statement √[/B]
A particle in an infinite square well has the initial wave function:
Ψ(x, 0) = A x ( a - x )
a) Normalize Ψ(x, 0)
b) Compute <x>, <p>, and <H> at t = 0. (Note: you cannot get <p> by differentiating <x> because you only know <x> at one instance of time)Homework...
Homework Statement
Consider a one-dimensional, non-relativistic particle of mass ##m## which can move in the three regions defined by points ##A##, ##B##, ##C##, and ##D##. The potential from ##A## to ##B## is zero; the potential from ##B## to ##C## is ##\frac{10}{m}\bigg(\frac{h}{\Delta...
Homework Statement
An electron in a finite square well has 6 distinct energy levels. If the finite square well is 10nm long determine:
a) Approximate the possible values for the depth of the finite square well ##V_0##.
b) Using a well depth value in the middle of the results obtained from part...
Homework Statement
Is state ψ(x) an energy eigenstate of the infinite square well?
ψ(x) = aφ1(x) + bφ2(x) + cφ3(x)
a,b, and c are constants
Homework Equations
Not sure... See attempt at solution.
The Attempt at a Solution
I have no idea how to solve, and my book does not address this type...
I just noticed in reading Griffiths that he places the base of the infinite square well at a zero potential while he places the base of the finite square well at a negative potential -V_0, where V_0 is a positive, real number; is there any reason for this? I just started learning about them/am...
The quantum states ##\psi(x)## of the infinite square well of width ##a## are given by
##\psi(x) = \sqrt{\frac{2}{a}}\sin\Big(\frac{n \pi x}{a}\Big),\ n= 1,2,3, \dots##
Now, I understand ##n \neq 0##, as otherwise ##\psi(x)## is non-normalisable.
But, can't we get additional states for...
How do you know when to use exponentials and trig functions when solving for the wave function in a finite square well? I know you can do both, but is there some way to tell before hand which method will make the problem easier? Does it have something to do with parity?
Homework Statement
Consider a particle in an infinite square well potential that has the initial wave-function:
Ψ(x,0) = (1/√2) [Ψ_1(x) + Ψ_2(x)]
where Ψ_1(x) and Ψ_2(x) are the ground and first excited state wavefunctions. We notice that <x> oscillates in time. FIND the frequency of...