Unbound States. today. A review about the discussions we have had so far on the Schrödinger equation. Quiz 10.21 Topics in Unbound States: The potential step. Two steps: The potential barrier and tunneling. Real-life examples: Alpha decay and other applications.
Unbound States today A review about the discussions we have had so far on the Schrödinger equation. Quiz 10.21 Topics in Unbound States: The potential step. Two steps: The potential barrier and tunneling. Real-life examples: Alpha decay and other applications. A summary: Particle-wave propagation. Thurs.
Review: The Schrödinger equation The free particle Schrödinger Equation: The plane wave solution completely defined. The momentum of this particle: undefined. The location of this particle: From and We understand this equation as energy accounting And this leads to the equation that adds an external potential The Schrödinger Equation Solve for with the knowledge of , for problems in QM.
Review: the time independent Schrödinger equation and the two conditions for the wave function When the wave function can be expressed as We have found and the time independent Schrödinger Equation: The solution of this equation is the stationary states because The probability of finding a particle does not depend on time: Normalization: Two conditions The wave function be smooth the continuity of the wave function and its first order derivative.
Review: Solving the Schrödinger equation. Case 1: The infinite potential well Equation and Solution: Energy and probability • Standing wave. • The QM ground-state. A bound state particle cannot be stationary, although its wave function is stationary. • Energy ratio at each level: n2. • With very large n, QM CM.
The change Solving the Schrödinger equation. Case 2: The finite potential well The change Equations:
Penetration depth: Solutions: Energy quantization: Review: Solving the Schrödinger equation. Case 2: The finite potential well
Solve for wave function and energy level Review: Solving the Schrödinger equation. Case 3: The simple harmonic oscillator This model is a good approximation of particles oscillate about an equilibrium position, like the bond between two atoms in a di-atomic molecule.
Energy are equally spaced, characteristic of an oscillator Review Solving the Schrödinger equation. Case 3: The simple harmonic oscillator Wave function at each energy level Gaussian
Wave function: Wave function: Energy levels: Energy levels: Energy levels: Penetration depth: Compare Case 1, Case 2 and 3:
Re-write it in the form Right moving. Why? Right moving (along positive x-axis) wave of free particle: Left moving wave of free particle: New cases, unbound states: the potential step From potential well to a one-side well, or a step: Free particle with energy E. Standing waves do not form and energy is no quantized. We discussed about free particle wave function before. Which is:
The Schrödinger Equation: When When When Solution: The potential step: solve the equation Initial condition: free particles moving from left to right. Why no reflection here? Trans. Inc. Refl. Ans: no 2nd potential step on the right. Normalization and wave function smoothness Are we done? What do we learn here? Any other conditions to apply to the solutions?
The potential step: apply conditions Smoothness requires: Express B and C in terms of A: Transmission probability: Reflection probability:
Transmission probability: Reflection probability: Reference: for normal incidence, light transmission probability: n air Reflection probability: The potential step: transmission and reflection Now using the definitions of k and k’:
The Schrödinger Equation: When When When Solution: Inc. Refl. The potential step: solve the equation Initial condition: free particles moving from left to right.
Express B in terms of A: One can prove: Reflection probability: Penetration depth: The potential step: apply conditions Smoothness requires: Transmission probability:
Review questions • The plane wave solution of a free particle Schrödinger Equation is Can you normalize this wave function? • Try to solve for the wave function and discuss about transmission and reflection for this situation:
Preview for the next class (10/23) • Text to be read: • In chapter 6: • Section 6.2 • Section 6.3 • Section 6.4 • Questions: • How much do you know about radioactive decays of isotopes? Have you heard of alpha decay, beta decay and gamma sources? • Have you heard of the “tunneling effect” in the EE department (only for EE students)? • What is a wave phase velocity? What is a wave group velocity?
Homework 8, due by 10/28 • Problem 25 on page 187. • Problem 34 on page 188. • Problem 15 on page 224. • Problem 18 on page 224.