shrodinger

• Bohr model


  Can describe the observed spectra of the hydrogen atom by quantising the possible electron energy levels
• Bohr model


  Also explains hydrogen like atoms that contain only one electron
• Wave particle duality


  Particles can behave as waves with a de Broglie wavelength

  
• Matter and light waves


  Can be considered as probability waves that can be used to infer knowledge about the particle in question
• Schrödinger Equation


  1. Find the wavefunction of a free particle

  2. Understand the conditions a wavefunction must satisfy to be a solution of the Schrödinger equation

  
• Wave


  A disturbance from a normal or equilibrium condition that propagates without the transport of matter
• Assumptions about waves


  • The wave is dependent on x and t, the displacement from normal is in y and the wave travels at a constant speed v
  • The wave also does not change shape or lose energy

  
• Wave equation


  𝑦(𝑥, 𝑡) = 𝑓(𝑥 − 𝑣𝑡), where f is an arbitrary function
• Wavefunction


  A solution to the one dimensional wave equation
• Wavefunction


  • 𝑦(𝑥, 𝑡) = 𝐴 sin(2𝜋/𝜆)𝑥 − 𝑣𝑡
  • 𝑦(𝑥, 𝑡) = 𝐴 sin(𝑘𝑥 − 𝜔𝑡)

  
• Angular wave number


  𝑘 = 2𝜋/𝜆 in 𝑚−1

  
• Angular frequency


  𝜔 = 2𝜋𝑓 = 2𝜋𝑣/𝜆 in 𝑠−1

  
• Schrödinger Equation


  1. Find the wave function for the quantum particle

  2. Describes the spatial and time evolution of a wave function

  
• Time-independent Schrödinger Equation


  Describes conservation of mechanical energy
• Kinetic energy


  𝐾 = 𝑝^2/2𝑚 = 1/2 𝑚𝑣^2
• Momentum


  �� = ℏ𝑘
• 𝜕^2𝜓/𝜕𝑥^2 = −𝑘^2��(𝑥) = −𝑝^2/ℏ^2 𝜓(𝑥)
• Time-Independent Schrödinger Equation


  −ℏ^2/2𝑚 𝜕^2𝜓/𝜕𝑥^2 + 𝑈(𝑥)𝜓 = 𝐸𝜓
• Solving the Schrödinger Equation

  1. 𝜕^2𝜓/��𝑥^2 + 8𝑚𝜋^2/ℎ^2 (𝐸 − 𝑈(𝑥))𝜓 = 0

  2. ��(𝑥) = 𝐴𝑒^{𝑖𝑘𝑥} + 𝐵𝑒^{−𝑖𝑘𝑥}
• Time-dependent wave function


  𝜓(𝑥) = (𝐴𝑒^{𝑖𝑘𝑥} + 𝐵𝑒^{−𝑖𝑘𝑥})𝑒^{−𝑖𝜔𝑡}

  
• Conditions for a valid wavefunction

  • Normalizable
  • Goes to 0 at ±∞ and finite at 𝑥 = 0
  • Continuous in 𝑥 and single valued everywhere
  • 𝑑𝜓/𝑑𝑥 must be finite, continuous in 𝑥 and single valued everywhere

  
• Quantum particle


  A combination of the wave and particle models
• Ideal particle


  Point-like and localised in space
• Ideal wave


  Infinitely long and unlocalised in space
• Superposition


  Used to build a localised entity from a set of infinitely long waves
• Superposition and interference


  The resultant displacement is the sum of all the waves

  
• 𝑘𝑥 + 𝑖𝐵 sin 𝑘𝑥


  A solution to the time independent Schrödinger equation
• The Quantum Particle


  • The idea that an object can be either wave-like or particle-like
  • The uncertainty principle
• The idea of a quantum particle is a combination of the wave and particle models
• An ideal particle is treated as point-like and localised in space
• An ideal wave with a single frequency is assumed to be infinitely long and so unlocalised in space

  
• Superposition and interference


  1. Resultant displacement is the sum of all the waves

  2. If two sine waves are in the same wavelength and amplitude, they interfere

  
• Wave packet


  A localised wave function of the form 𝜓 𝑥 = 𝐴𝑒𝑖 𝑘1𝑥−𝜔1𝑡 + 𝐴𝑒�� 𝑘2𝑥−𝜔2𝑡 + ⋯ .

  
• Wave packet


  • Localised in space around 𝑥 = 0 but of finite extent with a spread ∆𝑥

  
• Deriving key features of the wave-packet

  1. Consider two waves with equal amplitude 𝐴 and frequencies 𝑓1 and 𝑓2 propagating at a wave speed 𝑣phase

  2. Use the principle of linear superposition to get 𝑦 𝑥, 𝑡 = 2𝐴 cos ∆𝑘

  2 𝑥 − ∆𝜔

  2 𝑡 cos 𝑘1+𝑘2

  2 𝑥 − 𝜔1+𝜔2

  2 𝑡

  
• Envelope (beat wave)


  2�� cos ∆𝑘

  2 𝑥 − ∆𝜔

  2 ��
• Group velocity


  ∆𝜔

  ∆𝑘
• Dispersion relationship


  𝜔 = ℏ

  2𝑚 𝑘2
• Energy and momentum for a particle of mass m


  𝐸 = 𝑝2

  2��
• de Broglie's and Einstein's relationships

  𝐸 = ℏ𝜔, 𝑝 = ℏ𝑘