infinite potential well

• Infinite Potential Well


  A particle confined to a fixed region of space, with potential energy U(x) = 0 inside the well and U(x) = ∞ outside the well

  
• How a particle behaves when confined

  1. Particle's energy is quantised

  2. Particle's wave function behaves like a wave on a string of fixed length

  
• Infinite Potential Well


  • Particle does not exist outside the well, so wave function vanishes at the boundaries
  • Only certain wavelengths can be supported, corresponding to quantised energy levels

  
• Wave function


  The mathematical function that describes the quantum state of an object
• The wave function must be 0 at the walls of the well, which tells us that the electron only exists inside the well</b>

  
• Normalisation constant


  The constant that ensures the total probability of finding the particle somewhere in the well is 1
• The energy of the particle is quantised and scales as n^2, where n is the quantum number

  
• Energy changes of a particle in an infinite potential well


  1. Absorption of a photon causes transition to higher energy state

  2. Emission of a photon causes transition to lower energy state

  3. Photon energy equals energy difference between states

  
• Finite Potential Well


  Potential energy U(x) = 0 inside the well and U(x) = U0 outside the well, where 0 < x < L

  
• Behaviour of quantum particle in finite potential well

  1. Solve Schrodinger's equation in 3 regions: x<0, 0<x<L, x>L

  2. Wave function extends into regions outside the well

  3. Wave function is continuous at boundaries and has continuous slope

  
• Tunnelling


  Phenomenon where particle has non-zero probability of being found outside the potential well, even if its energy is less than the potential energy barrier
• There are specific allowed energy levels for a particle in a finite potential well, found by solving the boundary condition equations

  
• If particle energy exceeds the potential energy of the well, it behaves like a free particle with no quantised energy levels