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Particle in a 1-dimensional box

    Version as of 11:58, 19 Jun 2013

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    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  

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     that the particle can possess.  E represents allowed energy values and
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    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

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    where 

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     is the reduced Planck Constant and is equal to 
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    m is the mass of the particle
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         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:

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

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    where A, B, and k are constants
     

    Step 3: Define the wavefunction

    The solution to the Schrodinger 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.

    The probability to find the particle at x=0 or x=L is zero, remember? So, when x=0 sin(0)=0 and cos(0)=1; therefore, B must equal 0 to fulfill this boundary condition giving:

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    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:

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    Since 

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    , then 

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

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    Now we plug k into our wavefunction: 

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

    When x = L: 

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    This is only true when 

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    , where n = 1,2,3…

    Plugging this back in gives us:

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    To determine A, remember that the total probability of finding the particle is 100% or 1. When we find the probability and set it equal to 1, we are normalizing the wavefunction.

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    For our system, the normalization looks like:

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

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

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    Step 4: Determine the Allowed Energies

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

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    This is a very important result; it tells us that

    1. The energy of a particle is quantized
    2. The lowest possible energy is NOT zero

    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 look like this:

     

    Psi2_n1n2.JPG

    Important Facts to Learn from the Particle in the Box

    • The energy of a particle is quantized
    • The lowest possible energy for a particle is NOT zero (even at 0 K)
    • The square of the wavefunction is related to the probability of finding the particle in a specific position
    • The probability changes with increasing energy of the particle and depends on where in the box you look
    • 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/

     

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