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    nanoHUB U Strachan L1.4

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    Abdullah Alghamdi
    This document discusses quantum mechanics and electronic structure, specifically quantum wells and quantization. It begins by introducing the concept of an infinite potential well and solving the Schrodinger equation for this system. Only certain energy values, or energy levels, are allowed based on the boundary conditions. The energy levels are quantized based on an integer index. Transitions between these quantized energy levels can correspond to absorption or emission of photons with specific frequencies. Quantum confinement in small molecules or nanostructures similarly leads to quantized energy levels.

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    nanoHUB U Strachan L1.4

    Uploaded by

    Abdullah Alghamdi
    23 views8 pages
    This document discusses quantum mechanics and electronic structure, specifically quantum wells and quantization. It begins by introducing the concept of an infinite potential well and solving the Schrodinger equation for this system. Only certain energy values, or energy levels, are allowed based on the boundary conditions. The energy levels are quantized based on an integer index. Transitions between these quantized energy levels can correspond to absorption or emission of photons with specific frequencies. Quantum confinement in small molecules or nanostructures similarly leads to quantized energy levels.

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    Lecture notes

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    nanoHUB-U-Strachan-L1.4

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    This document discusses quantum mechanics and electronic structure, specifically quantum wells and quantization. It begins by introducing the concept of an infinite potential well and solving the Schrodinger equation for this system. Only certain energy values, or energy levels, are allowed based on the boundary conditions. The energy levels are quantized based on an integer index. Transitions between these quantized energy levels can correspond to absorption or emission of photons with specific frequencies. Quantum confinement in small molecules or nanostructures similarly leads to quantized energy levels.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    23 views8 pages

    nanoHUB U Strachan L1.4

    Uploaded by

    Abdullah Alghamdi
    This document discusses quantum mechanics and electronic structure, specifically quantum wells and quantization. It begins by introducing the concept of an infinite potential well and solving the Schrodinger equation for this system. Only certain energy values, or energy levels, are allowed based on the boundary conditions. The energy levels are quantized based on an integer index. Transitions between these quantized energy levels can correspond to absorption or emission of photons with specific frequencies. Quantum confinement in small molecules or nanostructures similarly leads to quantized energy levels.

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    From Atoms to Materials:

    Predictive Theory and Simulations


    Week 1: Quantum Mechanics and Electronic Structure
    Lecture 1.4: Quantum Well, Quantization and Optical Processes

    Ale Strachan
    strachan@purdue.edu
    School of Materials Engineering &
    Birck Nanotechnology Center
    Purdue University
    West Lafayette, Indiana USA

    1
    First example: infinite potential well
    “Particle in a box or quantum well”
    Potential Energy

    To find the possible states (WFs) of the


    electron: solve Schrödinger Eq:
     2 ∂2 
     − + V ( x )ψ ( x ) = Eψ ( x )
     2m ∂x
    2

    x x=0 x=L

    What is the probability of the electron being outside [0:L] ?

    Schrödinger Eq. inside well:

    2 ∂2
    − ψ (x ) = Eψ (x )
    2m ∂x 2

    x=0 x=L

    Ale Strachan – Atoms to Materials 2


    Infinite Potential Well
    Solve the following differential equation:
    2 ∂2
    − ψ (x ) = Eψ (x )
    2m ∂x 2
    Energy

    Boundary
    ψ (x = 0) = 0 ψ (x = L ) = 0 condition
    x x=0 x=L

    Let’s try this function:

    Still have to satisfy the boundary conditions


    Ale Strachan – Atoms to Materials 3
    Infinite Potential Well
    We have a solution but it need to satisfy
    the boundary conditions
    Energy

    ψ (x = 0) = 0 ψ (x = L ) = 0
    x x=0 x=L

    Wavefunctions Energies

    Ale Strachan – Atoms to Materials 4


    Infinite Potential Well: quantization
    • Only some values of k and energy are allowed (quantized)
    • Ground state energy is not zero!

    Wave functions
     nπ 
    16
     2π 2 ψ n (x ) = A sin  x
    2mL2  L 

     2π 2
    Energy

    9
    2mL2

     2π 2
    4
    2mL2
     2π 2 Each WF is shifted up
    1
    2mL2 according to their energy
    x/L

    • The more wavy a WF is, the higher its energy (remember kinetic energy is
    proportional to the gradient of the WF squared)

    Ale Strachan – Atoms to Materials 5


    Infinite Potential Well: optical properties
    Photons are particles with energy proportional to their frequency

      2π 2 
     
    2 
     2mL  Absorption:
     2π 2 A photon can only be absorbed if it
    16
    2mL2 carries the energy required to promote
    an electron to an excited state
     2π 2
    Energy

    9
    2mL2

     2π 2 Emission:
    4
    2mL2
    An excited electron with energy En)
    π
    2 2
    can relax to an empty, lower energy
    1
    2mL2 state (En’) and emit a photon with
    x/L frequency:

    Ale Strachan – Atoms to Materials 6


    Optical properties small molecules
    cyanine
    π electrons in conjugated molecules
    are basically free to move around:

      2π 2  2
    pinacyanol ∆E =  
    2 
    (
    N ex − N 2
    GS )
     2 mL 

    dicarbocyanine

    G. M. Shalhoub, J. Chem. Edu., 74, 1317 (1997)


    http://www.files.chem.vt.edu/chem-ed/quantum/pib1.html
    Ale Strachan – Atoms to Materials 7
    Quantum confinement

    Ale Strachan – Atoms to Materials 8

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