Lecture 13: More on Scattering

The script begins with an introduction to the MIT OpenCourseWare initiative, which offers free educational resources under a Creative Commons license. The professor encourages donations to support this cause and directs interested parties to the website. The lecture then transitions into a discussion about coherence states, a topic previously introduced in optional problems related to the harmonic oscillator. The professor explains the basic concept of coherence states, which are Gaussian wave functions that remain coherent over time, and their significance in quantum mechanics.

This paragraph delves deeper into the properties of coherence states, particularly their behavior under time evolution. The professor discusses how a displaced Gaussian wave function, representing the ground state of a harmonic oscillator, evolves over time. Despite the complexity of its phase, the probability distribution of this state remains a translating Gaussian, a property that distinguishes it from generic wave functions. The paragraph also explores the concept of eigenstates of the annihilation operator and how they relate to coherent states.

The script continues to explore the behavior of coherent states in quantum mechanics, emphasizing their classical-like behavior despite being quantum in nature. The professor discusses how the momentum of a coherent state changes over time and how its phase is related to its spatial rate of change. The paragraph also touches on the concept of squeeze states, which are variations of coherent states with altered uncertainty relations, and their periodic nature.

The focus shifts to quantum scattering, where a particle with mass m is considered against a potential barrier. The classical prediction for particle transmission is contrasted with the quantum mechanical approach, which involves solving the Schrödinger equation for wave functions in the presence of a potential. The professor outlines the process of matching wave function solutions to describe scattering and the significance of transmission and reflection probabilities.

This paragraph discusses the detailed analysis of transmission and reflection probabilities in quantum scattering. The professor derives expressions for these probabilities when the energy of the particle is above the potential barrier. The transmission probability is shown to approach 1 as the energy becomes much larger than the potential, aligning with classical predictions. The relationship between the reflection and transmission coefficients is also explored.

The script presents a thought experiment comparing quantum mechanical scattering with classical physics. The professor uses the analogy of a car rolling up a hill to illustrate the difference between classical and quantum predictions. In classical physics, the car would either make it over the hill or not, with no possibility of reflection back down the hill. In contrast, quantum mechanics allows for the particle to be reflected even when it has enough energy to overcome the barrier, highlighting the probabilistic nature of quantum phenomena.

The professor scrutinizes the quantum transmission probability, addressing the question of why it is squared when calculating probabilities. The explanation involves the quadratic nature of probability density in relation to the wave function and the current. The paragraph also corrects a typo from previous notes and emphasizes the importance of checking calculations against physical reasonableness.

This paragraph examines the implications of quantum scattering for understanding the potential barriers that particles encounter. The professor suggests that by measuring transmission probabilities as a function of energy, one can deduce the shape of the potential barrier. The concept of phase shift is introduced as an essential aspect of this analysis, and the idea that scattering experiments can reveal the internal structure of a system is discussed.

The script concludes with a detailed investigation of how potential barriers affect scattering. The professor discusses the conditions for different types of scattering experiments and how they relate to the coefficients of the wave function. The importance of understanding the transmission and reflection coefficients is highlighted, as they contain all the information needed to characterize the scattering process.

The final paragraph provides a summary of the quantum scattering analysis, emphasizing the ability to deduce the form of a potential barrier based on transmission probabilities. The professor outlines the steps involved in the analysis and encourages the use of computer algebra systems to simplify the algebraic calculations. The potential of quantum mechanics to reveal the properties of systems through scattering experiments is highlighted.

The script ends with a humorous anecdote about a friend who misplaced her car and jokingly attributed its disappearance to quantum tunneling. The story serves as a light-hearted conclusion to the lecture, illustrating the tendency to attribute everyday occurrences to quantum phenomena, despite the macroscopic scale at which such effects are negligible.