Square well Definition and 223 Threads

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.

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  1. Q

    A particle in an infinite square well

    What I am lost about is b, rather the rest of B. I am not sure what it means by probability density and a stationary state.
  2. Graham87

    Quantum mechanics - finite square well

    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...
  3. Graham87

    Quantum mechanics - infinite square well problem

    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...
  4. F

    I Infinite Square Well with an Oscillating Wall (Klein-Gordon Equation)

    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...
  5. sarahjohn

    Infinite Square Well with Multiple Particles

    Using the equation En = (h2*n2 ) / (8*m*L2), I got that E1 = 0.06017eV but the answer is not correct.
  6. J

    I Two Ways of Calculating the Solution to the Infinite Square Well?

    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...
  7. R

    Adiabatic Approximation for Infinite Square Well

    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...
  8. T

    Finite Square Well, Ψ[SUB]III[/SUB] const related too Ψ[SUB]II[/SUB]?

    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...
  9. Kostik

    A Upper bound for wavelength of a photon inside an infinite square well

    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...
  10. Z

    Energy of a particle in an Infinite square well?

    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...
  11. Terrycho

    I The wave function in the finite square well

    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...
  12. obstinatus

    I (Baby QM) Analytic Solution to the Infinite Square Well Problem

    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...
  13. Zack K

    Infinite Square Well with polynomial wave function

    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...
  14. I

    Infinite Square Well Expansion: Mass m in Ground State

    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}}...
  15. M

    Find the probability of a particle in the left half of an Infinite Square well

    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}...
  16. Cocoleia

    Infinite square well, dimensionless Hamiltonian..

    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
  17. I

    Simple Finite Square Well Problem help *Ignore, made stupid mistake*

    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##...
  18. Kaguro

    Infinite square well centered at the origin

    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...
  19. A

    Spin-##\frac{1}{2}## particles in infinite square well

    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...
  20. Jozefina Gramatikova

    Classical mechanics: Square well with Bounded particle

    My question is can we have negative energy in classical mechanics? Also I would need help for finding the velocity in part b)
  21. JD_PM

    Infinite Square Well -- Instantaneous expansion of the Well

    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...
  22. A

    Time evolution of wave function in an infinite square well potential

    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...
  23. A

    Time evolution of wave function in an infinite square well potential

    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...
  24. D

    Approximations with the Finite Square Well

    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 \\...
  25. A

    I Finite square well bound states

    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...
  26. D

    Solve coefficients for four equations in square well

    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 =...
  27. R

    Quantum Mechanics - Question about the Finite Square Well

    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...
  28. C

    Instantaneous doubling of the Infinite Square Well Width

    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...
  29. G

    Probability Density in an infinite 1D square well

    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...
  30. L

    I Finite Square Well: Bond States and Asymmetric Potential Wells

    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?
  31. Bassa

    "Shooting Method" for simulating a Particle in an Infinite Square Well

    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...
  32. acdurbin953

    Time-Dependent Perturbation of a 1D Infinite Square Well

    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...
  33. B

    Infinite square well doubled with time

    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...
  34. Fetchimus

    Infinite Square Well homework problem

    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...
  35. Vitani11

    Probability of Finding a Particle in a Small Interval in an Infinite Square Well

    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...
  36. S

    Symmetric square well, wavefunction is weird

    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...
  37. D

    Solve Infinite Square Well Homework: Find Energy, Probability

    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...
  38. Q

    Quantum mechanics HW problem on infinite square well.

    1. ##<x>= \int_{0}^{a}x\left | \psi \right |^{2}dx## ##\psi (x)=\sqrt{\frac{2}{a}}\sin\frac{n\pi x}{a}## then ##<x>= \frac{2}{a} \int_{0}^{a}x \sin\frac{n\pi x}{a}dx## 2. Homework Equations 1) ##y=\frac{n\pi x}{a}## then ##dy=\frac{n\pi}{a}dx## and 2) ##y=\frac{n\pi x}{a}## then...
  39. R

    Fermions in infinite square well in compact geometry

    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...
  40. M

    I Infinite square well solution - periodic boundary conditions

    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...
  41. S

    I Time evolution after "turning off" square well

    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.
  42. C

    Adiabatic expansion of infinite square well

    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...
  43. M

    Expectation value and momentum for an infinite square well

    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...
  44. S

    Infinite Square Well, Potential Barrier and Tunneling

    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...
  45. P

    Electron in a Finite Square Well

    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...
  46. acdurbin953

    Is state an energy eigenstate of the infinite square well

    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...
  47. Z

    I Finite vs. Infinite Square Well potential base question

    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...
  48. S

    Additional quantum states of the infinite square well

    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...
  49. B

    Exponentials or trig functions for finite square well?

    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?
  50. Blitzmeister

    Infinite Square Well Frequency of Oscillation

    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...
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