23. Quantum Mechanics V: Particle in a Box

The professor begins by revisiting an incomplete topic from the previous class, emphasizing the importance of questions from students to understand their doubts. The main focus is on quantum measurement, specifically the process of measuring energy in a quantum system. The explanation involves wave functions and the necessity to find functions ψ_E(x) that represent states of definite energy. The probability of measuring a certain energy E is given by the absolute value squared of the corresponding coefficient. The process is analogous to measuring momentum, with the integral A_E representing the expectation value of the energy. The challenge is to determine these functions ψ_E(x), which depend on the potential energy and position in classical mechanics. Each problem has a unique formula for energy, and the Schrödinger equation must be solved for each potential to find the allowed energy states.

The professor elaborates on the process of solving the Schrödinger equation to find the functions of definite energy. The solutions are not straightforward and require mathematical solutions for each potential. The professor also discusses the example of a free particle with no potential, leading to the equation d^(2)ψ/dx^(2) = k^(2)ψ = 0, where k^(2) is derived from the energy. The solutions to this equation are exponential functions e^(ikx) and e^(-ikx), which are related to the momentum of the particle. The professor also connects the quantum mechanical energy to the classical mechanics concept, highlighting that the energy in quantum mechanics is quantized and depends on the momentum squared divided by twice the mass.

The professor explores the quantum mechanics of a particle in a ring, explaining how the particle's momentum is quantized and related to the energy. The allowed values of momentum are restricted to be multiples of 2Π/L, leading to quantized energy levels. This quantization results in the system absorbing and emitting light only at certain frequencies, which can be used to probe the energy differences in the system. The professor also discusses the hydrogen 21-centimeter line as an example of how atomic spectra can be used to understand the universe, including the motion of galaxies through redshift and blueshift observations.

The professor delves into the concept of superposition in quantum mechanics, using the example of a particle in a ring that can exist in a state of both clockwise and counterclockwise momentum simultaneously. This is contrasted with classical mechanics, where the particle would have a definite momentum in only one direction. The professor also discusses the 'particle in a box' model, where the particle has a definite energy but an uncertain momentum, leading to the particle being in a state of 'limbo'. The probability of finding the particle moving in a particular direction is proportional to the square of the corresponding coefficient in the wave function.

The professor discusses the quantum mechanics of a particle confined to a one-dimensional box with infinitely high walls. The particle's wave function must be zero at the walls, leading to quantized energy levels and associated wave functions that are sine functions. The professor explains that the wave function must vanish at the boundaries, which quantizes the allowed values of the wave number k and, consequently, the energy levels. The energy levels are given by the formula ℏ^(2)Π^(2)/(2mL^(2)) times n^(2), where n is a positive integer, and the wave functions are given by Bsin(nΠx/L), with B being a normalization constant. The professor also addresses why the n=0 state is not allowed, as it would result in a trivial wave function that is identically zero.

The professor explores the behavior of quantum particles in a finite potential well, contrasting it with classical mechanics. In the classical view, a particle with energy less than the potential barrier would be confined, but in quantum mechanics, there is a non-zero probability of the particle tunneling through the barrier due to its wave-like nature. This phenomenon, known as barrier penetration, means that no barrier is completely impenetrable in quantum mechanics. The professor uses the example of alpha decay to illustrate this concept, where an alpha particle inside a nucleus can occasionally tunnel out, causing the nucleus to decay. The lecture concludes with the survival tip of repeatedly attempting to overcome a barrier, as there is always a small chance of success in quantum mechanics.