What is Particle in a box: Definition and 19 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. U

    I Quantum Mechanics Particle in a Box

    I need help .I did not A) E < V0 for T =? (passing coefficient ) B) E = V0 for T = ? C ) E > V0 for T =? A
  2. C

    MATLAB_Quantum Well_Finite Difference Method

    Homework Statement Hi, I am new to MATLAB and have an assignment where I have to construct a Hamiltonian matrix, apply boundary conditions, then find corresponding eigenvalues and eigenvectors for the electron in a box problem. I am stumped where to start. From our class we learned that you...
  3. Alan Ezra

    I Particle in a box and quantization of energy

    Greetings, In the scenario of a particle in an infinite potential well, there are discrete energy levels, i.e.##E=\hbar ^2 n^2 \pi ^2/ (2 m L^2)## where L is the width of the potential well, and n takes on positive integers. But what will happen if I put a particle of energy ##E_i## that is not...
  4. H

    Transition energy estimation based on bond length

    I come across this question in a textbook. Somehow my result is way off from the solution answer. I used the energy formula for particle in a box with n(Initial) = 22 and n(End) = 23, the calculated box length is 732 pm. I arrived at an answer of 39.3 nm. The answer from the answer book is but...
  5. FranciscoSili

    Partition Function of N particles in an assymetrical box

    Homework Statement Consider a gas sufficiently diluted containing N identical molecules of mass m in a box of dimensions Lx, Ly, Lz. Calculate the probability of finding the molecules in any of their quantum states. Calculate the energy of each quantum state εr, as a function of the quantum...
  6. Pushoam

    Particle in a box quantum mechanics

    Homework Statement Homework EquationsThe Attempt at a Solution ## E \propto n^2 ## ## \frac { \Delta E} E = \frac { 2n +1} {n^2} ## In the limit of large quantum number, ## \frac { \Delta E} E = \frac { 2} {n} ##, option (a).
  7. Pushoam

    Force exerted by a particle in a box on the boundary

    Homework Statement Homework EquationsThe Attempt at a Solution The energy eigen - value of a particle in a box is given by ## E_n = \frac { n^2 h^2}{8mL^2} ## . Now, applying classical mechanics , ## \frac { p^2}{2m} = \frac { n^2 h^2}{8mL^2} ## . ## p \propto \frac { 1} L ## , So...
  8. L

    Need help finding fermion energies and probabilities

    <Moved from a technical forum, therefore no template> For two non-interacting fermions confined to a 1d box of length L. Construct the antisymmetric wave functions (Slater determinant) and compare ground state energies of two systems, one in the singlet state and the other in the triplet state...
  9. tomdodd4598

    I Interpreting These '2 Particles in a Box' Plots

    Hey there, I am familiar with the mathematics of multi-particle systems, but have now moved on to trying to plot them. I was a little ambitious and thought I'd attempt to somehow plot energy eigenfunctions of two particles in a 2-D box. Obviously I immediately ran into the issue that there are...
  10. TheSodesa

    A relativistic electron in a potential box

    Homework Statement In a potential box (##L = 1.00pm##) an electron moves at a relativistic speed, meaning it's momentum can't be expressed as ##P = \sqrt{2mE}##. a) Using the uncertainty principle, show that the speed is indeed relativistic b) Derive an expression for the allowed energy states...
  11. D

    Find the minimum kinetic energy of two electrons in a 1D box

    Homework Statement Problem: Consider a "crystal" consisting of two nuclei and two electrons arranged like this: q1 q2 q1 q2 with a distance d betweem each. (q1=e, q2=-e) a) Find the potential energy as a function of d. b) Assuming the electrons to be restricted to a one-dimensional...
  12. S

    I Heisenberg Uncertainty Principle & Wave Function

    Considering how Heisenberg's uncertainty principle is applied to a top-hat wave function: This hyperphysics page shows how you can go about estimating the minimum kinetic energy of a particle in a 1,2,3-D box: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/uncer2.html You can also...
  13. michaelmolli

    Particle in One-Dimensional Box Problem [Quantum Mechanics]

    Homework Statement a) Determine the ratio (Em/En) between two energy states of a particle in a one dimensional box of length l. b) Show that this is consistent with the non-relativistic low-energy limit. The attempt at a solution I have figured out a) using the de broglie wave-particle duality...
  14. P

    Energy eigenstates of a particle in a one dimensional box.

    Homework Statement A One dimensional box contains a particle whose ground state energy is ε. It is observed that a small disturbance causes the particle to emit a photon of energy hν=8ε, after which it is stable. Just before emission a possible state of the particle in terms of energy...
  15. fricke

    Particle in a box with the finite depth

    For particle in a box with the finite depth, is it traveling wave? or standing wave? I am confused with its ability to pass through the potential walls that is classically forbidden area which makes me think it is traveling wave. But for particle in a box with infinite potential, I understand...
  16. L

    Probability of finding a particle in a 1D box

    Homework Statement If a one-dimensional box is 1 nm long, what is the probability of finding the particle between the following limits? (a) x = 0 nm and x = 0.05 nm (b) x = 0.55 nm and x = 0.65 nm Homework Equations ψ = (2/L)½ sin(πx/L) The Attempt at a Solution (I do chemistry and I'm really...
  17. P

    Wavefunctions that don't satisfy Schrödinger equation

    If there exists some normalized wavefunction ##\psi## that is not a solution to the Schrödinger equation (1D), what does this mean? You can still presumably use the square of the wave function to ascertain the probability it exists at some interval in space, but does it provide any other useful...
  18. B

    Why does infinite potential well give rise to standing wave?

    For a particle in a box that is described with a wave function, why can there only be a standing wave when there is an infinite potential well? From my understanding, the infinite potential well makes it impossible for the particle to tunnel through the barrier and so the wave function cannot...
  19. R

    Probability for a particle in a box

    Homework Statement A particle in a box of width 2a is in a state \psi=\frac{1}{\sqrt{2a}} for |x| less than or equal to a and 0 for |x| greater than a. What is the probability of finding the particle in [-b, b] inside the box? What is the probability of finding the particle with momentum p...