What is Finite square well: Definition and 45 Discussions
The finite potential well (also known as the finite square well) is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a "box", but one which has finite potential "walls". Unlike the infinite potential well, there is a probability associated with the particle being found outside the box. The quantum mechanical interpretation is unlike the classical interpretation, where if the total energy of the particle is less than the potential energy barrier of the walls it cannot be found outside the box. In the quantum interpretation, there is a non-zero probability of the particle being outside the box even when the energy of the particle is less than the potential energy barrier of the walls (cf quantum tunnelling).
The idea here (as I'm told) is to use the boundary conditions to get a transcendental equation, and then that transcendental equation can be solved numerically. So I'm making a few assumptions in this problem:
1. The potential ##V(x)## is even, so the wavefunction ##\psi(x)## is either even or...
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'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...
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...
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##...
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
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...
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...
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?
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...
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...
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?
Hello, in Griffith's section on the Finite Square Well, ##\psi(x)## (what is the name of this anyway?, I know ##\Psi(x,t)## is called the wave function but how do I call just ##\psi(x)##?)
Anyways,
The solution is
For x < a and x > a, the terms that are infinite as x approaches infinity are...
hi guys i need some help with the iteration made in a numerical solution for the eigenstates in a finite square well potential using the effective length approximation
First, i find the eigenstate for a infinite square well, then i define the related alpha and i use it to define an effective...
Homework Statement
Consider a particle of mass m in the ground state of a potential well of width 2 a and depth.
the particle was in the ground state of the potential well with V0 < Vcritical, which means the well is a narrow one.
At t = 0 the bottom of the potential well is shifted down to Vo'...
Ok here's a potential I invented and am trying to solve:
V =
-Vo in -b<x<b
and 0 in -a<x<-b , b<x<a where b<a
and ∞ everywhere elseI solved it twice and I got the same nonsensical transcendental equation for the allowed energies: \frac{-k}{\sqrt{z_0 - k^2}} \frac{e^{2kb} +...
Hello everyone,
I am reading about the Finite Square Well in Griffiths Quantum Mechanics Text. Right now, I am reading about the case in which the particle can be in bound states, implying that it has an energy E < 0. After some derivations, the author comes across the equation
\tan z =...
if I have a transcendental equation such as this one: tan(l a) = -l / sqrt (64/a^2 - l^2 ) Where
l=sqrt(2m(E+V) /hbar^2 ) and 'a' is the width of a finite square well, how can I solve this equation in terms of both l and a. I have successfully graphed the two sides of the equation...
Reading from http://quantummechanics.ucsd.edu/ph130a/130_notes/node150.html
Again we have assumed a beam of definite momentum incident from the left and no wave incident from the right.
Why is the above statement made?
What does the reflected wave mean? There is now all why reflected...
Homework Statement
Hello.
Imagine a particle bound in a square well potential of potential energy
V0 if |x| > a
0 if |x| < a
The wave function of the particle is: (ignoring the time dependency)
-A*exp(kx) if x<-a
B*sin(3*pi*x/4a) if |x|<a
A*exp(-kx) if x>a
where k =...
Homework Statement
http://img842.imageshack.us/img842/4917/physp6.jpg
I am trying to solve the above problem. However, I am supposed to solve it with the following values:
U=54.7eV
L=0.2nm
Particle is an electron, so:
m=9.109E-13kg=0.511eV/c^2
Essentially I am supposed to...
Homework Statement
A particle of mass m is in the potential
V(x) = \left\{
\begin{array}{rl}
\infty & \text{if } x < 0\\
-32 \hbar / ma^2 & \text{if } 0 \leq x \leq a \\
0 & \text{if } x > a.
\end{array} \right.
(a) How many bound states are there?
(b) In the highest energy...
Homework Statement
Already defined that for a 1D well with one finite wall the eigenvalue solutions are given by
k cot(kl) = -α
Show the eigenvalue solutions to well with both walls finite is given by
tan(kl) = 2αk / (k^2 - α^2)
Well is width L (goes from 0 to L) with height V_0...
Homework Statement
Solve Explicitly the first two eigenfunctions ψ(x) for the finite square wave potential V=V0 for x<a/2 or x>a/2, and V=0 for -a/2<x<a/2, with 0<E<V0.
Homework Equations
See image
The Attempt at a Solution
See image. After modeling an in class example, my classmates and i...
Homework Statement
A finite square well 2.0fm wide contains one electron. How deep must the well be if there are only two allowed (bound) energy levels for the electron?
Homework Equations
(1) E = [ a^2 * hbar^2 ] / 2m
(2) u = sqrt [2m(E+Vo)] / hbar
The Attempt at a Solution
Use...
I need to find B in terms of F in a finite square well potential
I started with
-Ae^(-i*K*a) - Be^(i*K*a) = Csin(k2*a) - Dcos(k2*a)
and
Ae^(-i*K*a) - Be^(i*K*a) = i*K*k2 [C*cos(k2*a) - D*sin(k2*a)]
where
C = [sin(k2*a) + i*(K/k2)cos(k2*a)]*Fe^(i*K*a)
D = [cos(k2*a)-...
For a finite one-dimensional square potential well if a proton is bound, how many bound energy states are there?
If m = 1.67*10^(-27) kg a = 2.0fm and the depth of the well is 40MeV.
Now I know the energy levels are
En = (n^2 * h^2) /(8ma^2) = (n^2*pi*2)/4 * (2hbar^2)/(ma^2)
but I am...
Homework Statement
1. Mixed Spectrum
The finite square well has a mixed spectrum or a mixed set of basis functions. The set of
eigenfunctions that corresponds to the bound states are discrete (call this set {ψ_i(x)}) and
the set that corresponds to the scattering states are continuous...
Homework Statement
What is the probability, that the particle is in the first third of the well, when it is in the ground state?
Homework Equations
\Psi(x)=Asin((n*pi)/L)
A=(2/L)1/2
The Attempt at a Solution
so probablility is related to the wave function by \Psi2
so i...
Consider double finite square well with
-Vo; -(a+b/2) < x < -b/2
V(x) = 0; -b/2 < x < b/2
-Vo; b/2 < x < a + b/2
0; otherwise
Sketch the ground wave function Psi(n=1) and the first excited state Psi(n=2) for
1. b = 0
2. b~a
3. b >> a
So...
I am trying to prove that there is always one bound state for a finite square well using variational method, and I am stuck. I've tried using e^(-bx^2) as my trial wave function, but I am left with E(b)=(hbar^2)b/2m - V, where V is the depth of the well. In this equation, taking the derivative...
Homework Statement
Determine the transmission coefficient for a rectangular barrier. Treat seperately the three cases E<Vo, E>Vo, and E=Vo.Homework Equations
V(x)= +Vo if -a<xa
V(x)= 0 otherwise
Transmission coefficient=(amplitude of transmited wave)2/(amplitude of incoming wave)2
I am also...
Homework Statement
I have to show that the delta function bound state energies can be derived from the finite square well potential.
Homework Equations
The wave functions in the three regions for the finite square well. (See wikipedia)
The Attempt at a Solution
1. I start from the...
Homework Statement
Based on the finite potential well defined by the following equations, how many bound states are there, which of these states are even and which are odd, and what are their energies?
V(x)= 0 for x\leq-l/2 and x \geq +l/2
V(x)=-\hbar^{2}/ma^{2}
Homework...
Homework Statement
Hi all.
When talking about the finite square well with a potential V > 0 for - A < x < A, I have never seen an example of bound states (i.e. E<0). They only treat examles with scattering states (i.e. E>0). Is there any reason for this? My book (Griffith's Intro. to QM)...
In the problem of a finite potential well, we search for bound states, in which E<V.
Say the potential is defined to be 0 outside of the well, and -V0 inside it.
Analyzing the case when -V0 < E < 0, one finds that the probability of finding the particle outside the well is not zero. This...
Hi,
My Quantum textbook loves to skip the algebra in its derivations. It claims that the solution for 'T', the transmission coefficient of the wave function (E>0, ie. unbound) is
T = \frac{2qke^{-2ika}}{2qkcos(2qa)-i(k^{2}+q^{2})sin(2qa)}
Its prior step is to offer two equations (which match...
Hello,
I have a quick question: While working on a problem involving a particle in a harmonic oscillator potential, I had to compute the average kinetic energy at t=0. My question is: would that average kinetic energy be the same or different if the particle was in the same state, but in a...
I am a second year physics student and have been set a homework assignment to solve a one dimensional time independant schrodinger equation in a finite square well using microsoft excel.
I understand the physics behind the situation but am not exactly sure how to use microsoft excel to solve...
Hey,
An electron is in a finite square well of 1 Å so the question is to find the values of the well's depth V0 that have exactly two state ?
How to proceed with this - finding the eigenvalues En = \hbar^2\pi^2 / 2ma^2
Thanks in advance
I read through the derivation of bound and scattering states for a finite square well. The logic made sense to me, but I am not entirely sure how to accommodate an arbitrary initial wave function (with mean E < 0). Afterall, there are only a finite number of bound states. My guess was that the...
Hi,
I have a problem with the finite square well. I have to analyze the odd bound states of the finite square well,
V(x)=
\begin{cases}
-V_0 & \text{for } -a<x<a\\
0 & \text{otherwise}
\end{cases}.
Specifically, I have to examine the limiting cases (wide, deep well and narrow...
I'm trying to normalize the even wave functions for the finite square well. The wave function is:
\psi(x)=
\begin{cases}
Fe^{\kappa x} & \text{for } x< a\\
D\cos(lx) & \text{for } -a\leq x \leq a\\
Fe^{-\kappa x} & \text{for } x> a
\end{cases}
How can I determine D and F? When I...