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.

View More On Wikipedia.org
Recent contents Top users
  1. P

    Probability for particle in infinite square well

    Homework Statement A particle is confined between rigid walls separated by a distance L=0.189. The particle is in the second excited state (n=3). Evaluate the probability to find the particle in an interval of width 1.00 pm located at a)x=0.188nm b)x=0.031nm c)x=0.79nm What would be the...
  2. D

    Minimum value of V_0 for Odd Solutions in Square Well Potential

    Homework Statement Consider a particle in a square well potential: $$V(x) = \begin{cases} 0, & |x| \leq a \\ V_0, & |x| \geq a \end{cases} $$ We are interested in the bound states i.e. when ##E \leq V_0##. (1) Show that the even solutions have energies that satisfy the transcendental...
  3. C

    Ground state in an infinite square well with length doubling

    Homework Statement Assume a particle is in the ground state of an infinite square well of length L. If the walls of the well increase symmetrically such that the length of the well is now 2L WITHOUT disturbing the state of the system, what is the probability that a measurement would yield the...
  4. N

    Infinite Square Well (Conceptual)

    Homework Statement Say, for example, a wave function is defined as 1/sqrt(2)[ψ(1)+ψ(2)] where ψ are the normalized stationary state energy eigenfunctions of the ISQ. Now, say I make a measurement of position. What becomes of the wavefunction at a time t>0 after the position measurement (i.e...
  5. Robsta

    Calculating variance of momentum infinite square well

    Homework Statement Work out the variance of momentum in the infinite square well that sits between x=0 and x=aHomework Equations Var(p) = <p2> - <p>2 $$ p = -i\hbar \frac{{\partial}}{\partial x} $$ The Attempt at a Solution I've calculated (and understand physically) why <p> = 0 Now I'm...
  6. wood

    Infinite Square Well Homework: Solutions & Analysis

    Homework Statement Consider an infinite square well defined by the potential energy function U=0 for 0<x<a and U = ∞ otherwise Consider a superposed state represented by the wave function ## \Psi(x,t)## given at time t=0 by $$\Psi(x,0) = N \{(-\psi_1(x) + (1+ i)\psi_2(x)\}$$ 1. Assume that...
  7. gfd43tg

    Two independent particles in an infinite square well

    Homework Statement Homework Equations The Attempt at a Solution a) For this part, I know for distinguishable particles, the expectation value of the square distance $$\langle (x_{1}^{2} - x_{2}^{2}) \rangle = \langle x^{2} \rangle_{2} + \langle x^{2} \rangle_{3} - 2 \langle x \rangle_{2}...
  8. W

    Time Dependent Wavefunction in Infinite Square Well

    Homework Statement A particle of mass m is confined to a space 0<x<a in one dimension by infinitely high walls at x=0 and x=a. At t=0, the particle is initially in the left half of the well with a wavefunction given by, $$\Psi(x,0)=\sqrt{\dfrac{2}{a}}$$ for 0<x<a/2 and, $$\Psi(x,0)=0$$ for a/2...
  9. gfd43tg

    Finite square well ##\psi(x)## solution for ##-a < x < a##

    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...
  10. gfd43tg

    Infinite Square Well Homework: Solutions

    Homework Statement Homework EquationsThe Attempt at a Solution (a) $$ \int_{0}^{a} \mid \Psi (x,0) \mid^{2} \hspace {0.02 in} dx = 1 $$ $$ \int_{0}^{a} \mid A[ \psi_{1}(x) + \psi_{2}(x) ] \mid^{2} \hspace {0.02 in} dx = 1 $$ Since the ##\psi_{1}## and ##\psi_{2}## are orthonormal (I don't...
  11. B

    Finite square well potential numerical solution

    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...
  12. J

    Solve Infinite Square Well: Homework Statement

    Homework Statement The wording of the question is throwing me off. It is a standard inf. pot. well problem and we are given the initial position of the particle to be in the left fourth of the box, \Psi(x,0)=\sqrt{\frac{4}{a}} We are asked to a) write the expansion of the wave function in...
  13. P

    Finite square well, excited states

    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'...
  14. Q

    Electron in double square well

    Homework Statement Consider an electron subject to the following 1-D potential: U(x) = -U_0 \left( \delta(x+a) + \delta(x-a) \right) where U_0 and a are positive reals. (a) Find the ground state of the system, its normalized spatial wavefunction and the parameter κ related to the ground...
  15. P

    Bound States of Infinite Square Well

    Hi all, So I was recently set straight on the fact that bound state does *not* necessarily mean E<0 but rather is the statement that E<V(+/- infinity). So how do we apply this definition to the infinite square well where the potential at +/- infinity vanishes, and yet the bound states have...
  16. U

    Infinite Square well with a Finite square well inside

    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} +...
  17. B

    Finite Square Well: Deriving Eq. (1)

    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 =...
  18. M

    Infinite square well, Probability of measurement of particle's energy

    Homework Statement Homework Equations The Attempt at a Solution I have managed to do the first 3 parts of the questions. The last two 4 markers are the ones I am having difficulties with. I have tried using the expansion postulate which states the wavefunction is equal to the...
  19. C

    Infinite square well with barrier in the middle

    Homework Statement Show that the energy levels of a double square well V_{S}(x)= \begin{cases} \infty, & \left|x\right|>b\\ 0, & a<\left|x\right|<b\\ \infty, & \left|x\right|<a \end{cases} are doubly degenerate. (Done) Now suppose that the barrier between -a and a is very high, but finite...
  20. LunaFly

    Time Dependent Wave Function for Particle in Infinite Square Well

    Homework Statement A particle is in a bound state of the infinite square well. It is in a state represented by the following wavefunction, written here at t=0: ψ(x)= -√(2/3)√(2/L) * sin (3πx/L) + i*√(1/3)√(2/L) * sin (2πx/L) (a)Write the full time-dependent wavefunction for this state...
  21. S

    Wave function in infinite square well, with potential step

    Homework Statement A Particle energy A trapped in infinite square well. U(x)=0 for 0<x<L and U(x)=U0 for L<x<2L. find the wave function of the particle when A) E>U0 B) E<U0 C) E=U0. Homework Equations 1-D time independent Schrodinger equation. The Attempt at a Solution I have...
  22. kmm

    Should the Normalization Constant be Positive or Complex?

    In finding solutions to the time independent Schrodinger equation we have to normalize \psi to find the constant A. So we get \int_{0}^{a} |A|^{2} sin^{2}(kx) dx = |A|^2 \frac{a}{2}=1 For A we then get |A|^2 = \frac{2}{a} . Griffiths says that this only determines the magnitude of A but...
  23. C

    Normalizing a State Function for an Infinite Square Well

    Homework Statement Normalize: \Psi_1 (x,t) = N_1 \cos(\frac{\pi x}{L}) e^{-\frac{iE_1t}{\hbar}} Where N_1 and E_1 are the normalization constant and energy for the ground state of a particle in an infinite square well. Homework Equations Normalization Condition: \int_\infty^\infty P(x,t)...
  24. U

    Delta wall and infinite square well potentials ,and 2 other questions

    Consider the following potential function: V=αδ(x) for x=0 and V=∞ for x>a and x<-a , solve the shroedinger equation for the odd and even solutions. solving the shroedinger equation I get ψ(x)=Asin(kx) +Bcos(kx) for -a<x<0 and ψ(x)=Asin(kx) +Bcos(kx) for 0<x<a is it...
  25. U

    Transcendental equation from a finite square well potential

    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...
  26. J

    Normalization of the infinite square well.

    I have been going through my textbook deriving equations in preparation for my test on QM tomorrow. I noticed in the infinite square well that i was unable to complete the normalization. My textbook, Griffiths reads : (integral from 0 to a) ∫|A|^2 * (sin(kx))^2 =|A|^2 * (a/2) =1 Therefore...
  27. F

    Two dimensional Square well and parity

    Homework Statement A particle is placed in the potential (a 2 dimensional square well) V(x) = (0 for -a/2 <= x =< a/2 and -a/2 <= y =<a/2, infinity for x>a/2, x<-a/2 and y>a/2, y<-a/2) The hamiltonian commutes with the parity operator P, Pψ(x,y) = ψ(-x,-y) = λψ(x,y), where the eigenvalue λ...
  28. E

    Implementation of the Numerov Method for the 1D square well

    I want to solve the Schrodinger via the Numerov Method but I had some troubles. I'm programing in C++, so here is my code: #include<cstdlib> #include<iostream> #include<cmath> using namespace std; double x_min=-4.0 , x_max=4.0; int N=2000; double...
  29. P

    Most probable position of electron in an infinitely deep square well

    Homework Statement An electron in an infinitely deep square well has a wave function that is given by http://postimg.org/image/s159u7ynt/ What are the most probable positions of the electron?The Attempt at a SolutionI got the values as L/6, L/2, 5L/6 by finding the x values that made the wave...
  30. O

    Bound state of a square well, no allowed bound state mean?

    Homework Statement Show in the graph ,there will be no allowed bound states with odd-parity if the well depth is less than ${V_min}$ Find ${V_min}$ in terms of k and a.where a is the half of the well width. What does no allowed bound state mean? Homework Equations $cotz=-pa/z$ where p^2...
  31. O

    Bound state of finite square well, why do we make this statement?

    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...
  32. J

    Classical Square Well: Hamiltonian Form & Elastic Collision

    My understanding is that a classical idealized particle, moving in one dimension, with momentum p and kinetic energy T comes into contact with an infinite step-function potential V, there will be an (instantaneous) elastic collision - the particle's momentum becomes -p, so its energy remains...
  33. T

    Energy of an Infinite Square Well

    Hey guys, this is my first post so go easy on me. I was looking over the simple case of a 1D particle restrained inside an infinite square well potential ("particle in a box") and was having some difficulty understanding the relationship between the energy states and the expectation value for...
  34. Q

    An infinite square well problem

    Homework Statement Particle in well: V(x)=0 for |x|<\frac{L}{2} V(x)=∞ for |x|>\frac{L}{2} initial wave function \Psi(x,0)=\frac{1}{√L}[cos\frac{\pi*x}{L}+ i*sin\frac{2*\pi*x}{L}] a) calc P(p,t) (momentum prob density) Homework Equations Anything from Griffiths QM The Attempt at a...
  35. P

    Finding momentum distribution for particle in square well

    Homework Statement A particle of mass ##m## is trapped between two walls in an infinite square well with potential energy V(x) = \left\{ \begin{array}{cc} +\infty & (x < -a), \\ 0 & (-a \leq x \leq a), \\ +\infty & (x > a).\end{array} \right. Suppose the wavefuntion of the particle at time...
  36. J

    Infinite square well transitions

    Homework Statement A particle, mass m propagates freely in a box, length L. The energy states are: ϕ_n(x) = (2/L)^(1/2)sin(n∏x/L) and energies E_n = n^2∏^2/(2mL^2) at time t=0 the system is in state ϕ_1 and the perturbation V=kx is applied (k constant) and turned off at t=T...
  37. L

    Wave function and infinite square well potential

    Homework Statement An electron in a one-dimensional infinite square well potential of length L is in a quantum superposition given by ψ = aψ1+bψ2, where ψ1 corresponds to the n = 1 state, ψ2 corresponds to the n = 2 state, and a and b are constants. (a) If a = 1/3, use the normalization...
  38. A

    Suppose you have an electron in the infinite square well

    Suppose you have an electron in the infinite square well. The system is completely isolated from the rest of the world and has been its entire lifetime. Do we then know that the wave function describing the electron is an eigenstate of the Hamiltonian? The question arose because I was given a...
  39. G

    Infinite Square Well - Particle in linear combination of states

    A particle of mass m is trapped in a one-dimensional infinite square well running from x= -L/2 to L/2. The particle is in a linear combination of its ground state and first excited state such that its expectation value of momentum takes on its largest possible value at t=0.I know the process of...
  40. D

    QM - Transmission coefficient for square well

    Homework Statement A steady stream of 5 eV electrons impinges on a square well of depth 10 eV. The width of the well is 7.65 * 10^-11 m. What fraction of electrons are transmitted?Homework Equations The following equation for the transmission coefficient, T, is given: T = [1 + \frac{V_0 ^2...
  41. K

    8 Electrons in a 3-D Infinite Square Well w/ Spin

    Homework Statement A cubical box whose sides are length L contains eight electrons. As a multiple of $$\frac{h^2}{2mL^2}$$ what is the energy of the ground state of the eight electrons? Assume the electrons do not interact with each other but do not neglect spin. Homework Equations...
  42. 7

    Particle in an infinite square well - interval -d/2<x<d/2

    Homework Statement Particle is in an infinite square well of width ##L## on an interval ##-L/2<x<L/2##. The wavefunction which describes the state of this particle is of form: $$\psi = A_0\psi_0(x) + A_1\psi_1(x)$$ where ##A_1=1/2## and where ##\psi_0## and ##\psi_1## are ground and first...
  43. Jalo

    Finite square well potential energy

    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 =...
  44. C

    Probability of energy measurement in an infinite square well

    Homework Statement Consider a particle in 1D confined in an infinite square well of width a: $$ V(x) = \begin{cases} 0, & \text{if } 0 \le x \le a \\ \infty, & \text{otherwise} \end{cases} $$ The particle has mass m and at t=0 it is prepared in the state: $$ \Psi (x,t=0) = \begin{cases} A...
  45. C

    Infinite Square Well, finding Psi(x,t)

    Homework Statement A particle in the infinite square well has the initial wave function ## \Psi (x, 0) = Ax(a-x), (0 \le x \le a) ##, for some constant A. Outside the well, of course, ## \Psi = 0 ##. Find ## \Psi (x,t) 2. Homework Equations : Equation [1.0]:## \displaystyle c_n =...
  46. W

    Electron in one dimensional infinite square well

    An electron in the ground state of a one-dimensional infinite square well of width 1.10 nm is illuminated with light of wavelength 600 nm. Into which quantum state is the electron excited? ok so I first calculated the engery of the electron in the first ground state of the square well...
  47. phosgene

    Wavefunction of infinite square well potential between -L<x<L

    Homework Statement Solve for the wavefunctions and energy levels of an infinite square well potential extending between -L<x<L. Hint: It may be worth noting that for a potential symmetric in x, then the observed probability density must also be symmetric in x, i. |ψ(x)|2 = |ψ(-x)|2. Homework...
  48. A

    Quantum Mechanics - Finite Square Well - Graphical Solution

    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...
  49. B

    Energy levels of a 3 dimensional infinite square well

    Homework Statement Calculate the wavelength of the electromagnetic radiation emitted when an electron makes a transition from the third energy level, E3, to the lowest energy level, E1. Homework Equations E_n = \frac{\left (n_{x}^{2}+n_{y}^{2}+n_{z}^{2} \right) \pi^{2}...
  50. T

    Infinite Square Well for Bosons in an optical lattice

    I'm working on a research project and was wondering what you could use to experimentally create a periodic infinite square well (dirac comb?) in a direction orthogonal to a different potential, say a periodic potential. To help you understand what I'm trying to do picture a grid of atoms and...
Back
Top