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Particle in a 1-dimensional box

    Version as of 04:25, 19 May 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 psi 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). If you're wondering why it has to go to infinity at the walls, this is simply a boundary condition to ensure that the particle stays 100% inside the box. A single particle cannot contain all the energy is the universe, right?

    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
    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.

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