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.

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  1. B

    Half Finite Square Well Problem

    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...
  2. 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...
  3. 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...
  4. 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...
  5. 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...
  6. 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.
  7. 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...
  8. 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...
  9. 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)...
  10. C

    Solving for Eigenvalues in a Finite Square Well with Both Walls Finite

    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...
  11. B

    Find the wave function of a particle bound in a semi-infinite square well

    Homework Statement Consider the semi-infinite square well given by V(x) = -V0 < 0 for 0≤ x ≤ a and V(x) = 0 for x > a. There is an infinite barrier at x = 0 (hence the name "semi-infinite"). A particle with mass m is in a bound state in this potential with energy E ≤ 0. Solve the Schrodinger...
  12. S

    Solve two eigenfunctins for a Finite Square Well

    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...
  13. A

    Nucleon in Inifinte Square Well

    Homework Statement Assuming an infinite square well of radius 2.8E-13cm, find the normalized wave functions and the energies of the four lowest states for a nucleon. 2. The attempt at a solution I want to say that the wave function is \psi (x) = \sqrt{\frac{2}{a}} sin(\frac{n \pi}{a} x)...
  14. 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...
  15. P

    How Deep Should a Finite Square Well Be for Two Electron Energy Levels?

    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...
  16. 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...
  17. J

    Time Dependant One Dimensional Square well

    Given the wave function: Ψtot(x,t) = ((√2)/2)ψ3e^(-(iE3t)/h) + ((√2)/2)ψ5e^(-(iE5t)/h) @ |x|≤a/2 and ψtot(x,t) = 0 @ |x|≥a/2 where a=100nm , E=(((h)^2)((kn)^2))/2m , kn=pi*n/a , & T1 = 2pih/E1 How would I find the Period of the wave function in terms of T1??
  18. S

    Schrodinger equation for one dimensional square well

    Homework Statement the question as well as the hint is shown in the 3 attachments Homework Equations The Attempt at a Solution i know how to normalize an equation, however i do not understand what the hint is saying, or how to do these integrals, any guidance would be greatly...
  19. 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...
  20. I

    Quantum Mechanics Square Well Potential Problem

    Homework Statement A beam of neutrons (m=1.675x10-27kg) is incident on a nucleus. Consecutive transmission maxima are observed for beam energies of 1.15, 23.656, and 50.254 MeV. Treating the nucleus as a one-dimensional square-well potential: (a) What is the width of the potential? (ans. 10...
  21. S

    Finite square well potential question Constants

    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)-...
  22. S

    Finite square well potential question

    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...
  23. 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...
  24. 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} =...
  25. C

    How Does Resizing an Infinite Square Well Affect Ground State Probability?

    Homework Statement we have a particle in an infinite square well from x=0 to x=L/2 Then it says that we suddenly move the right hand side of the wall to x=L and then it asks to find the probability that the particle is in the ground state of the widened well. The Attempt at a Solution...
  26. 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]=?
  27. V

    How Do You Apply Orthonormality and Completeness in Quantum Finite Square Wells?

    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...
  28. 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}...
  29. M

    Expectation Values for momentum and a particle in a square well

    Homework Statement Calculate the expectation values of p and p2 for a particle in state n=2 in a square well potential. Homework Equations \Psi(x,y) = (2/L)*sin(n1\pix/L)*sin(n2\piy/L) p= -i\hbar\partial/\partialx The Attempt at a Solution \int\Psip\Psidxdy...
  30. 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...
  31. E

    QM: The double square well potential

    QM: The "double square well" potential Homework Statement Consider the "double square well" potential below. Qualitively (no calculations) how do the energies of the ground state and the first excited state vary as b goes from zero to infinity (i.e. the two wells become further and further...
  32. 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...
  33. K

    Probability inside finite square well

    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...
  34. 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...
  35. 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...
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. R

    Wave function of infinite square well but with time dependence included

    Homework Statement Find the wavefunction for an infinite well, walls are at x=0 and x=L(include the time dependence) The Attempt at a Solution I don't understand what it's meant by include the time dependence. Can I just find the time-independent wavefunction and then multiply it by...
  41. T

    10 Electrons in an Infinitely Deep 1D Square Well

    Homework Statement An infinitely deep one-dimensional potential well has a width of 1 nm and contains 10 electrons. The system of electrons has the minimum total energy possible. What is the least energy, in eV, a photon must have in order to excite a ground-state electron in this system to the...
  42. B

    Infinite Square Well - Quantum Mechanics

    SPECIFICALLY SEE POST 8 AND AFTER PLEASE Hi so that I can get the help for the specific problem I am working on I will set the question up and include all the steps that I can get and work out. The end question will be about quantized energy levels. This is for a maths module. I am...
  43. M

    Schrodinger Equation : Infinite Square Well

    Hi, I am having trouble understanding an example from a textbook I am reading on the Schrodinger equation. The example deals with an infinite square well in one dimension. With the following properties: V = 0\,where -a \leq x \leq a V = \infty\,|x| \geq a Where V is the potential. The...
  44. W

    Energy in infinite square well

    Homework Statement Find the energy of a particle of mass m in an infinite square well with one end at x=-L/2 and the other at x=L/2. Homework Equations Schrodinger Equation The Attempt at a Solution To save time, I won't type the solving of the differential equation which results...
  45. C

    Intro Quantum: Expanding infinite square well

    Homework Statement Griffiths Intro to Quantum, problem 2.38: A particle of mass m is in the ground state of the infinite square well. Suddenly the well expands to twice its original size: the right wall moving from a to 2a, leaving the wave function (momentarily) undisturbed. The energy of...
  46. M

    Uncertainty product of infinite square well

    For the ground state of a particle moving freely in a one-dimentional box 0\leqx\leqL with rigid reflecting end-points, the uncertainty product (del x)(del p) is 1 h/2 2 sqrt{2}h 3 >h/2 4 h/sqrt{3} I used (del x)^2 =<x^2>-<x>^2 and (del p)^2 =<p^2>-<p>^2 Using the wavefunction of...
  47. A

    Infinite Square Well with an Adiabatic Evolution

    Homework Statement I'm trying to find the geometric phase for the adiabatic widening of the infinite square well. Griffiths defines the geometric phase to be: \gamma=i* \int^{w2}_{w1}<\psi_{n}|\frac{d\psi_{n}}{dR}>dR Where R is the aspect of the potential that is changing and w1, w2 are the...
  48. Q

    Solving seperable wavefunction in 2D infintie square well using parity operator

    Homework Statement You are given in a earlier stage of this problem that the wavefunction is separable, ie.) \Psi(x,y) = X(x)Y(y) The problem asks you to solve for the wavefunction of a particle trapped in a 2D infinite square well using Parity. ie.) solve \Psi(-x,-y) = \Psi(x,y) and...
  49. D

    Infinite square well with delta potential

    Homework Statement I have infinite square well which has a potential V(x)=\frac{\hbar^2}{m}\Omega\delta(x) in x=0, and is 0 in the interval x\in[-a,a]Homework Equations Schrodinger eq.The Attempt at a Solution I solved the time independant Schrodinger eq. by integration around x=0 by some...
  50. P

    Infinite Square Well Eigenstates

    Homework Statement The eigenstates of the infinite square well are not energy eigenstates and are not momentum eigenstates. Homework Equations The Attempt at a Solution I don't understand how this can be? If the eigenstates of the infinite square well are energy eigenstates...
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