What is Infinite square well: Definition and 149 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.

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  1. 2

    Variance in position for the infinite square potential well?

    [Note from mentor: this thread originated in a non-homework forum, therefore it doesn't use the standard homework template] ------------------------------------------ This exercise pops up in the Cavendish Quantum Mechanics Primer (M. Warner and A. Cheung) but I can't seem to figure it out. So...
  2. 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...
  3. Z

    How Do You Determine the Ground State Energy in a Spherical Infinite Well?

    Homework Statement A particle of mass ##m## is constrained to move between two concentric hard spheres of radii ##r = a## and ##r = b##. There is no potential between the spheres. Find the ground state energy and wave function. Homework Equations $$\frac{-\hbar^2}{2m} \frac{d^2 u}{dr^2} +...
  4. 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...
  5. 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} +...
  6. 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...
  7. 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...
  8. 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...
  9. 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...
  10. 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...
  11. 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)...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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...
  19. 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...
  20. 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...
  21. 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...
  22. 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...
  23. 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 =...
  24. 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...
  25. 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...
  26. 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}...
  27. 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...
  28. 8

    Group velocity in infinite square well

    ello everybody, how can I calculate the group velocity of a wave package in an infinite square well? I know only how it can be calculated with a free particle, the derivation of the dispersion relation at the expectation value of the moment. But in the well, there are only discrete...
  29. D

    Potential Function of Infinite Square Well - Help Needed!

    Please I'm new here, and would need your help with identifying what sort of potential function is described by the following expression: V(x) = 0 for |x| < 1, =1 at x = \pm 1, and =\infty for |x|>1. (Note that: \pm is plus (+) or minus (-) sign). Could it be referred to as the infinite...
  30. A

    Expected values in infinite square well

    Ok...this must sound stupid, because i didn't found answer on the web and on my books...but i am having trouble with the infinite square well. I want to calculate <x>. V(x)=0 for 0<=x<=a <x>=\frac{2}{a}\int^{a}_{0} x \sin^2(\frac{n\pi}{a}x)dx Doing integration by parts i got to...
  31. F

    Infinite square well expectation value problem

    Homework Statement A particle in an infinite box is in the first excited state (n=2). Obtain the expectation value 1/2<xp+px> 2. The attempt at a solution Honestly, I don't even know where to begin. I assumed V<0, V>L is V=∞ and 0<V<L is V=0 I tried setting up the expectation...
  32. R

    What is the correct method for solving the infinite square well energy problem?

    Hi I have attached my attempt of solving the infinite square well for Energy. The value I get is different from that of the book, also in the attachment, Kindly explain if my answer is correct given the fact that I proceeded step by step and used no tricks. Thank you.
  33. ElijahRockers

    Infinite Square Well (Quantum Mechanics)

    Homework Statement An electron is trapped in an infinitely deep potential well 0.300nm in width. (a) If the electron is in its ground state, what is the probability of finding it within 0.100nm of the left-hand wall? (b) Repeat (a) for an electron in the 99th excited state (n=100). (c) Are...
  34. ElijahRockers

    Finding the eigen function for an infinite square well (quantum mechanics)

    Homework Statement Quantum mechanics is absolutely confusing me. A proton is confined in an infinite square well of length 10-5nm. Calculate the wavelength and energy associated with the photon that is emitted when the proton undergoes a transition from the first excited state (n=2) to the...
  35. M

    Infinite Square Well Electron Jumps from n=4 to ground state

    Homework Statement An electron is trapped in an infinite square-well potential of width 0.5 nm. If the electron is initially in the n=4 state, what are the various photon energies that can be emitted as the electron jumps to the ground state?Homework Equations ΔE=13.6(1/nf2-1/ni2)...
  36. S

    Infinite Square Well and Energy Eigenstate question

    Hi all, just studying for my final exam and needed a little clarification on this. Our prof did an example: Consider a particle of mass m moving in the nth energy eigenstate of a one-dimensional infinite square well of width L. What is the uncertainty in the particle's energy? He said the...
  37. S

    Schrodinger and Infinite Square Well hell

    Schrodinger and Infinite Square Well... hell Homework Statement Show that Schrodinger Equation: \frac{d^{2}\psi(x)}{dx^{2}}+k^{2}\psi(x)=0 has the solution \psi(x)=A\sin(kx) Homework Equations k=\frac{\sqrt{2mE_{tot}-E_{pot}}}{\hbar} The Attempt at a Solution I already know that...
  38. F

    Infinite square well with finite potential energy inside

    Assume that you have a one dimension box with infinite energy outside, and zero energy from 0 to L. Then my understanding of the Schrodinger equation is that the equation inside will be: -h^2/2m*d2/dx2ψ = ihd/dtψ And the energy eigenstates are given by ψ(x,t) = e-iwt*sin(kx) where k = n*π/L...
  39. C

    Double infinite square well, energies

    Homework Statement We're modelling an ammonia maser with a double infinite square well defined by: V(x) = \begin{cases} V_{0} & |x| < b - \frac{a}{2}\\ 0 & b-\frac{a}{2} < |x| < b+\frac{a}{2}\\ \infty & |x| \geq b + \frac{a}{2} \end{cases} I have had no trouble with the assignment up until...
  40. E

    Finding the momentum-space wave function for the infinite square well

    Homework Statement Find the momentum-space wave function for the nth stationary state of the infinite square well. Homework Equations Nth state position-space wavefunction: \Psi_n(x,t) = \sqrt(\frac{2}{a})sin(\frac{n\pi}{a}x)e^{-iE_nt/\hbar}. Momentum operator in position space: \hat{p} =...
  41. J

    Wavefunction in an infinite square well

    Homework Statement A wavefunction in an infinite square well in the region -L/4≤x≤3L/4 is given by ψ= Asin[(πx/L)+δ] where δ is a constant Find a suitable value for δ (using the boundary conditions on ψ) Homework Equations The Attempt at a Solution Asin[(πx/L)+δ]=?
  42. C

    QM: Infinite Square Well -a/2 to a/2

    I have read similar threads about this problem but I wasn't able to make progress using them. Homework Statement Consider an infinite square-well potential of width a, but with the coordinate system shifted so that the infinite potential barriers lie at x=\frac{-a}{2} and x=\frac{a}{2}...
  43. E

    (Quantum Mechanics) Infinite Square Well

    Hi, I'm stuck in this Griffiths' Introduction to QM problem (#2.8) Homework Statement A particle in the infinite square well has the initial wave function \Psi(x,0) = Ax(a-x) Normalize \Psi(x,0) Homework Equations \int_{0}^{a} |\Psi(x)|^2 dx = 1 The Attempt at a Solution...
  44. M

    Paradox: The infinite square well vs. the Uncertainty Principle

    I've come across an apparent paradox in elementary quantum mechanics, and after a little Googling, haven't found a reference to it. Here goes, The 1-D infinite square well is a classic problem in introductory QM. We find that the position-space eigenfunctions of the Hamiltonian (the "allowed...
  45. P

    Multiple electrons in an infinite square well

    Homework Statement suppose you put 5 electrons into an infinite square well. (a) how do the electrons arrange themselves to achieve the lowest total energy? (explain with help of diagram) (b) give an expression for this energy in terms of electron mass, well width L and planks constant The...
  46. M

    What is the significance of the infinite square well width in quantum mechanics?

    I am new to quantum mechanics so I am just trying to get an understanding of the infinite square well. I have been reading a lot of material and I see a lot of times that the barriers of the well say -L/2 and L/2. I know that outside the well to the left is -infinity and to the right is...
  47. M

    Wavevector in infinite square well

    Right guys, I want to get this one straight... We have all seen the simple infinite square well a million times. From it, we can get the condition for the k-vector of the electron that k = n.pi / L Now, I also come across all the time that k = 2n.pi / L When do we use which boundary...
  48. K

    Solving Infinite Square Well: Find Probability of Electron in 0.15nm

    Homework Statement An electron is trapped in a 1.00 nm wide rigid box. Determine the probability of finding the electron within 0.15nm of the center of the box (on either side of the center) for a) n = 1 Homework Equations Int[-0.15nm, 0.15nm] psi^2 dx The Attempt at a Solution I...
  49. S

    What are the possible solutions for the TISE in the infinite square well model?

    Homework Statement As part of my homework, I am solving the TISE for the infinite square well model. The potential is zero for |x| =< a and infinite otherwise. Homework Equations The Attempt at a Solution For |x| >= a, the wavefunction is zero. For |x| =< a, there are...
  50. K

    Particle in an infinite square well

    Homework Statement Consider a point particle of mass m contained between two impenetrable walls at +/- 2a. The potential V(x) between the walls is zero. Assume that at time t=0 the state of the particle is described by the wave function \Psi(x) = A\frac{1+cos(\frac{2*\pi*x}{a})}{2} for...
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