22. Quantum mechanics IV: Measurement theory, states of definite energy

The professor begins by referencing an email from students, Jerry Wang and Emma Alexander, who question the absence of time in the wave function ψ(x) and its relation to time. The professor clarifies that the current discussion is limited to a single instant in time, analogous to Newtonian mechanics where the position and momentum at a single instant can predict future states. The wave function ψ(x) is a complete description of a quantum system at a moment, and the probability density for finding a particle is given by the square of the wave function's absolute value. The professor also discusses the relationship between the wave function and momentum, highlighting the complex nature of quantum mechanics.

The discussion shifts to the specifics of quantum states and the probability of measuring certain momentum values. The professor explains the mathematical representation of these probabilities using coefficients of functions ψ_p(x), which are related to the allowed values of momentum. The integral of the complex conjugate of the wave function with respect to x gives the probability amplitude A(p) for a given momentum. An example is provided to illustrate the concept, with the professor using cosine functions to demonstrate how the momentum values and their probabilities can be determined. The importance of normalization is also emphasized, ensuring that the integral of the probability density over all space equals one.

The professor addresses a student's confusion regarding the wave function ψ(x) and its relationship with momentum. It is clarified that ψ(x) is a function that describes the system at a given instant, while ψ_p(x) represents a particle with a definite momentum. The importance of distinguishing between the two is emphasized. The professor also explains that the probability amplitude A_p does not have to be a real number, but its absolute square, which gives the probability, is real and positive. The concept of normalization and the role of the integral in calculating probabilities are further elaborated upon.

The professor delves into the process of normalizing the wave function to ensure that the integral over all space equals one, which is a requirement for a valid quantum state. Using the example of a cosine function, the professor demonstrates how to rescale the wave function and how the momentum values can be determined from the coefficients of the function. The concept of a particle being in a superposition of states with definite momentum is introduced, and it is explained that measuring the momentum will result in the wave function collapsing to a single momentum value with a certain probability.

The professor explains the postulate of quantum mechanics regarding measurement, stating that upon measuring the momentum, the wave function collapses to a single term corresponding to the measured value. It is emphasized that the act of measurement changes the system. The concept of the wave function's collapse is further explored, noting that if the position is measured, the wave function becomes a spike at that position, and all other information is lost. The professor also touches on the idea that the wave function's collapse is instantaneous and that re-measuring the same property immediately would yield the same result.

The professor discusses the uncertainty principle, noting that it is not mentioned as a separate postulate because it is a mathematical consequence of the wave function's properties. The principle states that the product of the uncertainties in position (Δx) and momentum (Δp) must be greater than or equal to ℏ/2. The professor connects this to the impossibility of having a well-defined position and momentum simultaneously, which is a fundamental aspect of quantum mechanics.

The focus shifts to energy quantization, with the professor explaining that atoms can only have certain discrete energy levels. The energy difference between these levels corresponds to the energy of a photon emitted or absorbed during transitions between these states. This energy difference is given by the formula E(n=3) - E(n=1), which can be related to the frequency and wavelength of the photon. The concept of energy quantization is central to understanding atomic structure and behavior, as well as the interaction of atoms with light.

The professor presents the Schrödinger equation for a free particle, which has no potential energy, and solves it. The solution to the equation is given in terms of exponential functions, which are related to the momentum of the particle. The professor explains that the momentum is quantized, taking on discrete values determined by an integer n, and that this quantization leads to quantized energy levels for the particle. The solution involves the concept of degeneracy, where multiple states can have the same energy.

The professor contrasts quantum theory with classical theory regarding the momentum and energy of a particle. In classical mechanics, a particle with energy E would have a definite speed and direction, whereas in quantum mechanics, a particle can exist in a superposition of states with positive and negative momentum. The professor also addresses a student's concern about the mass of the particle and clarifies that there are no restrictions on the mass for the validity of the quantum mechanical equations.

The professor discusses the concept of a particle in a one-dimensional box, also known as a quantum well or infinite potential well, where the particle is confined by infinitely high potential barriers. The allowed wave functions and energy levels are quantized, with the wave function being non-zero only within the box and vanishing outside. The energy levels are given by the formula ℏ²n²π²/(2mL²), where n is an integer, and the corresponding wave functions are sine functions. This example illustrates the quantization of energy in quantum mechanics and is fundamental for understanding more complex quantum systems.