ChemWiki

username password
More Comments Print page Email link New page Edit page

Particle in a 1-dimensional box

    Version as of 23:30, 19 May 2013

    to this version.

    Return to Version archive.

    View current version

    A particle in a 1-dimensional box is a fundamental quantum mechanical approximation describing the translational motion of a single particle confined inside an infinitely deep well from which it cannot escape. 

    1. 1. Introduction
    2. 2. Step 1: Define the Potential Energy V
    3. 3. Step 2: Solve the Schrödinger Equation
    4. 4. Step 3: Define the wavefunction
    5. 5. Step 4: Determine the Allowed Energies
    6. 6. What does all this mean?
    7. 7. Important Facts to Learn from the Particle in the Box
    8. 8. Questions

    Introduction

    The particle in a box problem is a common application of a quantum mechanical model to a simplified system consisting of a particle moving horizontally within an infinitely deep well from which it cannot escape.  The solutions to the problem give possible values of E and  

    HTTP Status: BadRequest(400) (click for details)
     that the particle can possess.  E represents allowed energy values and
    HTTP Status: BadRequest(400) (click for details)
    is a wavefunction, which when squared gives us the probability of locating the particle at a certain position within the box at a given energy level.

    To solve the problem for a particle in a 1-dimensional box, we must follow our Big, Big recipe for Quantum Mechanics:

    1. Define the Potential Energy, V
    2. Solve the Schrödinger Equation
    3. Define the wavefunction
    4. Define the allowed energies

    Step 1: Define the Potential Energy V

    ParticleInABox.GIF


    The potential energy is 0 inside the box (V=0 for 0<x<L) and goes to infinity at the walls of the box (V=∞ for x<0 or x>L).  We assume the walls have infinite potential energy to ensure that the particle has zero probability of being at the walls or outside the box.  Doing so significantly simplifies our later mathematical calculations as we employ these boundary conditions when solving the Schrödinger Equation.

    Step 2: Solve the Schrödinger Equation

    The time-independent Schrödinger equation for a particle of mass m moving in one direction with energy E is

    HTTP Status: BadRequest(400) (click for details)

    where 

    HTTP Status: BadRequest(400) (click for details)
     is the reduced Planck Constant where 
    m is the mass of the particle
    HTTP Status: BadRequest(400) (click for details)
         is the stationary time-independent wavefunction
    V(x) is the potential energy as a function of position
    E is the energy, a real number

    This equation can be modified for a particle of mass m free to move parallel to the x-axis with zero potential energy (V = 0 everywhere) resulting in the quantum mechanical description of free motion in one dimension:

    HTTP Status: BadRequest(400) (click for details)

    This equation has been well studied and gives a general solution of:

    HTTP Status: BadRequest(400) (click for details)
     

    where A, B, and k are constants.
     

    Step 3: Define the wavefunction

    The solution to the Schrödinger equation we found above is the general solution for a 1-dimensional system. We now need to apply our boundary conditions to find the solution to our particular system.

    According to our boundary conditions, the probability of finding the particle at x=0 or x=L is zero. When x=0 sin(0)=0 and cos(0)=1; therefore, B must equal 0 to fulfill this boundary condition giving:

    HTTP Status: BadRequest(400) (click for details)
     

    We can now solve for our constants (A and k) systematically to define the wavefunction.

    Solving for k
    Differentiate the wavefunction with respect to x:

    HTTP Status: BadRequest(400) (click for details)

    HTTP Status: BadRequest(400) (click for details)

    Since 

    HTTP Status: BadRequest(400) (click for details)
    , then 

    HTTP Status: BadRequest(400) (click for details)

    If we then solve for k by comparing with the Schrödinger equation above, we find: 

    HTTP Status: BadRequest(400) (click for details)

     

    Now we plug k into our wavefunction: 

    HTTP Status: BadRequest(400) (click for details)

     

    Solving for A
    To determine A, we have to apply the boundary conditions again. Recall that the probability of finding a particle at x = 0 or x = L is zero.

    When x = L: 

    HTTP Status: BadRequest(400) (click for details)

    This is only true when 

    HTTP Status: BadRequest(400) (click for details)
    , where n = 1,2,3…

    Plugging this back in gives us:

    HTTP Status: BadRequest(400) (click for details)

    To determine A, recall that the total probability of finding the particle inside the box is 1, meaning there is no probability of it being outside the box. When we find the probability and set it equal to 1, we are normalizing the wavefunction.

    HTTP Status: BadRequest(400) (click for details)

    For our system, the normalization looks like:

    HTTP Status: BadRequest(400) (click for details)

    Using the solution for this integral from an integral table, we find our normalization constant, A: 

    HTTP Status: BadRequest(400) (click for details)

    Which results in the normalized wavefunction for a particle in a 1-dimensional box:

    HTTP Status: BadRequest(400) (click for details)

    Step 4: Determine the Allowed Energies

    Solving for E results in the allowed energies for a particle in a box: 

    HTTP Status: BadRequest(400) (click for details)

    This is a very important result; it tells us that:

    1. The energy of a particle is quantized.
    2. The lowest possible energy of a particle is NOT zero.  This is called the zero-point energy and means the particle can never be at rest because it always has some kinetic energy.

    This is also consistent with the Heisenberg Uncertainty Principle: if the particle had zero energy, we would know where it was in both space and time.

    What does all this mean?

    The wavefunction for a particle in a box at the n=1 and n=2 energy levels look like this: 

    Psi_n1n2.JPG

    The probability of finding a particle a certain spot in the box is determined by squaring Psi. The probability distribution for a particle in a box at the n=1 and n=2 energy levels looks like this:

     

    Psi2_n1n2.JPG

    Notice that the number of nodes(places where the particle has zero probabily of being located) increases with increasing energy n.  Also note that as the energy of the particle becomes greater, the quantum mechanical model breaks down as the energy levels get closer together and overlap, forming a continuum.  This continuum means the particle is free and can have any energy value.  At such high energies, the classical mechanical model is applied as the particle behaves more like a continuous wave.  

    Important Facts to Learn from the Particle in the Box

    • The energy of a particle is quantized.  This means it can only take on discreet energy values.
    • The lowest possible energy for a particle is NOT zero (even at 0 K).  This means the particle always has some kinetic energy.
    • The square of the wavefunction is related to the probability of finding the particle in a specific position for a given energy level.
    • The probability changes with increasing energy of the particle and depends on the position in the box you are attempting to define the energy for
    • In classical physics, the probability of finding the particle is independent of the energy and the same at all points in the box

    Questions

    1. Draw the wave function for a particle in a box at the n = 4 energy level.
    2. Draw the probability distribution for a particle in a box at the n = 3 energy level.
    3. What is the probability of locating a particle of mass m between x = L/4 and x = L/2 in a 1-D box of length L? Assume the particle is in the n=1 energy state.
    4. Calculate the electronic transition energy of acetylaldehyde (the stuff that gives you a hangover) using the particle in a box model. Assume that aspirin is a box of length 300 pm that contains 4 electrons.
    5. Suggest where along the box the n=1 to n=2 electronic transition would most likely take place.

    Helpful Links
    Provides a live quantum mechanical simulation of the particle in a box model and allows you to visualize the solutions to the Schrödinger Equation:

    http://www.falstad.com/qm1d/

     

    This page viewed 38990 times
    The ChemWiki has 9238 Modules.

     

    Creative Commons License    UC Davis ChemWiki by University of California, Davis is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. 
    Permissions beyond the scope of this license may be available at copyright@ucdavis.edu. Terms of Use
    Powered by Mindtouch Core 2010