What is Step potential: Definition and 37 Discussions
In quantum mechanics and scattering theory, the one-dimensional step potential is an idealized system used to model incident, reflected and transmitted matter waves. The problem consists of solving the time-independent Schrödinger equation for a particle with a step-like potential in one dimension. Typically, the potential is modelled as a Heaviside step function.
I am struggling with how to go about this; in particular, I'm not sure I understand what state is being alluded to when Ballentine says "For an electron that approaches the surface from the interior, with momentum ##\hbar k## in the positive ##x## direction, calculate the probability that it...
Homework Statement
I am trying to figure out the average penetration depth into a finite step potential, similar to this thread:
https://www.physicsforums.com/threads/qm-barrier-penetration.104641/
But instead of just estimating the depth as ##\frac{1}{\kappa}##, i would like to calculate it...
Homework Statement
I am being asked to show that the wave function ψ and dψ/dx are continuous at every point of discontinuity for a step potential. I am asked to make use of the Heaviside step function in my proof, and to prove this explicitly and in detail.
Homework Equations...
Homework Statement
A particle with mass m and spin 1/2, it is subject in a spherical potencial step with height ##V_0##.
What is the boundary conditions for this eigenfunctions?
Find the degeneracy level for the energy, when it is ##E<V_0##
Homework Equations
Radial equation
\begin{equation}...
Homework Statement
I'm a pharmacologist and I have a modern physics course to do. This is not my field and I'm completely lost... We were given this problem to do. Thanks a lot in advance.
Consider a potential where
U(x) = 0 for x ≤ 0
U(x) = -3E for x > 0
Consider a particle of energy E...
At t=0 we have a particle incident from the left on a step potential (where V(x) = V0 for x ≥ 0 and V(x) = 0 otherwise). The particle has energy of 5/4 V0 and the question asks to find the probability that it will be on the right (ie. x > 0) as t→∞.
I understand how to solve this problem and...
Homework Statement
Consider the one-dimensional Schrodinger equation for the step potential, that is for U(x) = 0 for x<0, and for , . Consider a particle with mass m and energy E < U. Assume the particle is initially at x<0.
a) Calculate the penetration depth Δx at which the probability...
Homework Statement
V(x) = -V ( if x > 0 )
0 ( if x <= 0)
( V is positive constant)
the transmission coefficient is not simply F2 / A2 ( with A the incident amplitude and F the transmitted amplitude ), because the transmitted wave travels at a different speed.
Calculate...
Homework Statement
A woman is walking along a road. She has a mass of 52 kg and is walking at 1 m/s.
(a) She is not paying careful attention and is walking straight towards the wall of a nearby building. Assume that the wall is infinitely hard and that she can be described as a plane wave (a...
I am able calculate the solution for the TISE relating to the interval to the right of a step potential; however, I am unclear as to what this solutions represents physically for this scattering problem.
So, starting with
(-ħ2/2m)d2/dx2Ψ2(x) + V(x)Ψ(x)2 = EΨ(x)2
where V(x)Ψ(x)2 =...
Is it the case that there is no general solution for the step potential with E = V0?
This seems as though it must be the case as there are only certain energy eigenvalues which would be allowed and, therefore, we cannot for all possible values for the potential barrier find allowed energy...
I'm working on a basic potential step problem from Griffiths QM. Its problem 2.34 in the textbook. I am having trouble understanding the transmission coefficient in this problem. I realize that I can calculate the reflection coefficient and just do 1 minus the result but I am looking for...
Consider the Schrodinger equation with the step-function potential
V(x)=\begin{cases}
0, & x<0 \\
U, & x>0
\end{cases}\qquad .
A pulse with E>U comes in from the left with unit amplitude and undergoes partial reflection. The reflection has an amplitude (ignoring phase) that is given by the...
Undergraduate Quantum Mechanics problem. However the course hasn't gone as far to include R or T so I'm assuming there must be a way to solve this without needing to know about those.
Homework Statement
Asked to show that \psi(x)=A\sin(kx-\phi_0) is a solution to the 1D-time-independent...
Hi,
I am stuck with a problem which effectively boils down to this: Given the eigenstates of a Hamiltonian with a step potential in the x direction
H=-\hbar^2/2m \nabla^2 + V_0 \Theta(x)
\psi(q)_{in}=cos(qx)-\frac{\sqrt{K_{V_0}^2-q^2}}{q}sin(qz) \qquad x<0...
Homework Statement
We have a particle in a step potential. Consider it as a plane wave traveling left to right and hitting a step potential V at x=0. Assuming it behaves like a fluid, what is the force it exerts on the wall in terms of F'[0], where F'[x] is the derivative of the wave function...
Consider the case for a particle with energy E in a region where :
V = 0 , x<0; V=Vo, x>0;
for the first region when we write Schrodinger’s equation(Time independent) it will be
-h^2/2m d2ψ/dx2 = Eψ
and for second region it was
-h^2/2m d2ψ/dx2 = (Vo-E)ψ;
My doubt is that won't...
I understand the two cases dealt with when we consider the step potential. The two cases being when E>U (i.e the total energy is less than the max potential of the potential barrier) and when E<U. What is the physics of the situation when E=U? Is it a sort of a hybrid situation between the two...
Homework Statement
I'm working through a step potential and I am confused as to why one of the coefficients doesn't go to zero.
V(x) = 0 when x < 0;
V(x) = V_not when x > 0;
a. Calculate reflection coefficient when E < V_not
I can solve the reflection part, it is a step towards it...
I am reading Shankar's "Principles of Quantum Mechanics" and am up to the part where he uses Schrodinger's equation to derive the wave function for various 'simple' scenarios in one spatial dimension.
The first few were fine but his presentation of the step potential problem (specifically...
Homework Statement
I have a step potential where E < V0. I also have to do this for E >V0 but I think after we do the first part it'll make sense.
Plot the behavior of the wave function in crossing the region at x = 0, considering
four values of the energy. So I'll end up with four...
Homework Statement
Write out the spatial portion of the wavefunction for a particle coming from the right with energy (E) greater than the step potential (U) and passing over the step.Homework Equations
The Attempt at a Solution
I just want to know if this is the right set of equations for this...
Homework Statement
Consider the potential:
V(x)
=0 x<0
=V1 0<x<a
=V2 x>a
Where 0<V1<V2 and E>V2
Evaluate the transmission coefficient for the incident wave into the region x>a
Homework Equations
T= 4k0k1/((k0+k1)^2)
The Attempt at a Solution
I have not attempted this. It...
Hi everybody! I'm studying the simple case of the solution of the Schrodinger equation for a step potential "[URL .[/URL] As my professor states , the transmission coefficient is 0 when the energy of the particle is E<V.
I really don't get how this result is not a contradiction with the fact...
Homework Statement
I am a beginning quantum mechanics student, and I need to solve a journal paper that uses the step potential fairly simply. I've been searching for (literally) days, but everything seems to be far too complicated. Can someone please point me in a direction, or offer some...
Homework Statement
E>V_1 & V_2
So it's a step potential wave, setup. Let's set it up along the x axis. At x=0, there is the first step where it is V_1. At x=a, there is another step, where V_2>V_1. Show that the transmission coefficient is...
Hi. I've been trying to understand the phenomenon of step potential, when energy of particle E is higher than the potential V.
Then we have solution on both sides of boundary in the form of wave functions...
Is the reflection coefficient in this case simply (E-V)/E ??
Can anyone show me a...
Homework Statement
Consider particles incident on a one dimensional step function potential, V(x)=Vphi(x) with energy E>V. Calculate the reflection coefficent for particles incident from the right (Barrier height=V at x>0, Barrier height=0 at x<0). Consider the limits as E tends to V and E...
Homework Statement
Hey, I found this interesting case in the tunneling problem.
How do we calculate the transmission probability when the
energy of the approaching particle is equal to the height of the potential barrier?
I.e E=Vo.
Homework Equations
Same equations as in other...
Homework Statement
A step potential is defined by: V(x) = 0 ( for for x <0 ) and V(x) = V0 ( for x >0 ). Hence the step occurs at the origin.
Suggest a real physical situation, that might correspond to this idealized problem
Homework Equations
None needed.
The Attempt at a Solution...
Considering a step potential of V(x) = o when x<o and V(x) = Vo when x>o so step occurring at origin of x axis.
Write down in words the strategy for solving it.
Answer:
Solve the time-independent schrodinger equation for V=o when x<o and find the solution for the free particle wave function...
I am taking a course in solid state electronics at the university and I was thinking about one detail in the calculation of k. In a simple situation like a step potential we say that k=sqrt(2m*abs(E-V))/hbar . Now my question is What happens if the electron approaching the barrier is having an...
In the left side of the barrier, the potential energy V(x)= 0, while on the right side of the barrier, V(x) = V. Given that the total energy of the particle in such a system has a total energy E < V..
a. What are my acceptable solutions?
On the left side:
Should I include the cos kx...
Hi...I've come across this question on a past exam, and I can't seem to resolve it!
A particle is incident on a step potential at x=0. The total energy of the particle E, is less then the height of the step, U. The particle has wavefunctions
$\begin{array}{l}
\psi (x) = \frac{1}{2}\{ (1 +...
Consider a potential step located at x=0, such that u=0 to the left of the step (for x<0), and u = U to the right of the step (x>0).
A particle with total energy E > U traveling from left to right has:
\psi_1(x) = Ae^{ik_1x} + Be^{-ik_1x} \;\textrm{for}\; x\le0 \;\textrm{and} \;k_1...