Lecture 6: Time Evolution and the Schrödinger Equation

The script begins with an introduction to the video content provided under a Creative Commons license, highlighting the support for MIT OpenCourseWare, which offers high-quality educational resources for free. The professor then prompts for questions from previous lectures, addressing inquiries about the first exam and the association of operators with observables in quantum mechanics. The professor emphasizes the distinction between operating on a wave function with an operator and the act of measurement, clarifying that the former does not imply measurement and is a common misconception that will be further elaborated in subsequent lectures.

This paragraph delves into the foundational aspects of quantum mechanics, focusing on the role of wave functions in defining the configuration of a system and the association of observables with operators. The professor introduces the concept of eigenfunctions, which are special functions that, when acted upon by their corresponding operator, return the same function multiplied by a constant. The eigenfunctions of the momentum operator are discussed as an example, illustrating how any function can be expanded as a superposition of these eigenfunctions, a principle that will be further explored in the context of Fourier expansions.

The superposition principle is discussed in detail, explaining that any two valid wave functions can be combined to form another valid physical configuration. The audience is reminded that the eigenfunctions of operators corresponding to physical observables form a basis, allowing any function to be expressed as a linear combination of these basis functions. The paragraph also touches on the interpretation of the wave function, specifically the meaning of expansion coefficients within the context of probability and measurement outcomes.

The lecture shifts focus to the time evolution of quantum systems, introducing the concept of the collapse of the wave function upon measurement and the deterministic nature of the time evolution governed by the Schrodinger equation. The professor emphasizes the linearity of quantum evolution, contrasting it with the nonlinearity typically found in classical mechanics. The discussion also touches on the implications of continuous observation on the evolution of a quantum system, hinting at the Quantum Zeno problem.

This section explores the properties of the Schrodinger equation, including its linearity and unitarity, which ensures the conservation of probability. The professor addresses the audience's question about the time reversal of the Schrodinger equation, explaining that while the equation allows for both forward and backward integration in time, the act of measurement introduces a non-deterministic collapse of the wave function, leading to an apparent contradiction in the definitions of time evolution in quantum mechanics.

The professor discusses the non-local nature of quantum mechanics, addressing the audience's question about the locality of the time evolution of the wave function. It is clarified that the energy operator in the Schrodinger equation does not inherently possess locality, which has implications for the understanding of quantum mechanics in relation to special relativity. The conversation also revisits the superposition principle, highlighting the importance of distinguishing between the linearity of the Schrodinger equation and the superposition of states postulated in quantum mechanics.

The script addresses the mathematical possibility of creating discontinuous wave functions through the superposition of an infinite number of continuous functions, acknowledging the sneaky nature of such mathematical constructions. However, it is emphasized that in a physical context, such discontinuities are extremely rare and typically irrelevant, as physical systems tend not to exhibit the divergent behavior that could lead to such mathematical anomalies.

This section delves into the philosophical implications of quantum mechanics, particularly the tension between the probabilistic nature of measurement outcomes and the deterministic evolution of the wave function described by the Schrodinger equation. The professor critiques the Copenhagen interpretation, which encourages physicists to focus on computational predictions without delving into the deeper meanings or interpretations of quantum phenomena, labeling it as a 'cop out' and highlighting the dissatisfaction among some physicists with this approach.

The professor introduces the theory of decoherence as a more satisfying narrative to reconcile the probabilistic and deterministic aspects of quantum mechanics. Decoherence suggests that the act of measurement involves the interaction of a quantum system with a macroscopic apparatus, effectively 'washing out' quantum effects and leading to the observed collapse of the wave function. This theory will be explored in more detail in later lectures, offering a potential resolution to the paradox of quantum measurement.

The script concludes with a discussion on the nature of measurement in quantum mechanics, emphasizing that while a system may be described by a superposition of states, upon measurement, it is found in a definite state. The audience is reassured that there is no ambiguity in the outcome of a measurement, and the concept of a system being in a superposition does not lead to a state of confusion or uncertainty about the result obtained.

The focus shifts to the practical aspect of solving the Schrodinger equation, highlighting the importance of specifying the energy operator for a given system. The professor outlines different strategies for solving the equation, including brute force, extreme cleverness, and numerical methods. The emphasis is on understanding the physics of the system and the value of numerical solutions in learning about the problem and validating against experimental data.

The script explores the concept of stationary states in quantum mechanics, which are states that remain in an energy eigenstate with a constant phase change over time. The professor explains that the probability density of finding a particle in a stationary state does not change over time, and neither do the expectation values of position or energy. This leads to the understanding that stationary states are characterized by their invariance in the face of time evolution.

Building on the concept of stationary states, the professor demonstrates how to construct a general solution to the Schrodinger equation using the linearity of the equation. By taking a superposition of energy eigenstates, each evolving independently over time, the professor shows that the general solution can be expressed as a sum of these eigenstates, each multiplied by a phase factor that depends on the energy of the state. This approach reduces the problem of solving the Schrodinger equation to determining the expansion coefficients for the initial condition of the system.

The script addresses the physical relevance of energy eigenstates, acknowledging that while no real-world system is exactly in an energy eigenstate due to the continuous interaction with the environment, the concept is still valuable. The discussion turns to the accuracy of energy measurements, introducing the time-energy uncertainty relation, which implies that to measure energy with infinite precision, one would have to wait an infinite amount of time, a practical impossibility.

The professor draws a connection between the general solution to the Schrodinger equation and the Fourier theorem, highlighting the principle of superposition for the momentum operator. The eigenfunctions of the momentum operator are identified as plane waves, and the Fourier theorem is described as an application of the superposition principle for momentum, allowing any function to be represented as an integral involving these eigenfunctions and their corresponding expansion coefficients.

The script concludes with a discussion on the eigenfunctions of the position operator, which are Dirac delta functions. The professor explains that while these functions are not smooth or normalizable, they can be represented as limits of sequences of reasonable functions. This allows for the expansion of arbitrary functions as superpositions of these eigenfunctions, with the expansion coefficients being the function itself evaluated at the given position.