25. Quantum Mechanics VII: Summary of postulates and special topics

The professor begins by emphasizing the importance of understanding the postulates of quantum mechanics. These rules, based on experimental facts rather than mathematical deduction, form the foundation of the theory. The first postulate introduces the concept of a wave function, denoted as Y(x), which contains all the possible information about a particle in one dimension. The wave function must be normalized so that the integral of Y squared over all space equals one. The second postulate discusses the existence of a wave function for a state of definite momentum, Y_p(x), which is a specific function ensuring that measuring the momentum of a particle in this state yields a definite value, p. The professor also touches on the mathematical aspect of quantization of momentum in a ring and introduces the concept of a state of definite position, Y_x0, characterized by a spike function at x0.

The discussion continues with the exploration of special mathematical forms in quantum mechanics. The professor explains that taking the derivative of a function, Y(x), typically results in a new function, but in quantum mechanics, there are special functions that, when differentiated, result in a multiple of the same function. This leads to the concept of a state of definite position, described by a spike function at x0, which is unique because multiplying by x affects only the non-zero value at x0. The professor also introduces the idea that states of definite energy, E, are solutions to a specific equation involving the Hamiltonian operator and the potential, V(x). However, this is not elevated to a postulate due to its relation to the known states of definite momentum and position.

The fourth postulate deals with the probability of measuring a specific value, g, such as momentum or energy, within a quantum state described by a wave function Y(x). The probability is given by the square of the coefficient, |A_g|^2, obtained from the integral of the product of the conjugate of Y_g and Y. The fifth postulate addresses the collapse of the wave function upon measurement: if a measurement yields a specific value, g_0, the wave function collapses to just that term, Y_g0. This postulate highlights the unique feature of quantum mechanics where measurement affects the state of the system. The professor also clarifies that states of definite momentum and position are not compatible, as measuring one affects the knowledge of the other.

The sixth postulate describes the time evolution of quantum states according to the Schrödinger equation, which is fundamental to quantum mechanics. The professor explains that if a system starts in a state of definite energy, the wave function will evolve in time by acquiring a phase factor, e^(iEt/ℏ). This results in stationary states, where the probability density remains time-independent. For arbitrary initial states, the system's state at a later time can be found by expanding the initial state in terms of energy eigenstates and applying the time evolution to each term. The professor assures students that the content following this point will not be on the exam, encouraging them to relax and gain an understanding of the broader applications of quantum mechanics.

The professor delves into the application of quantum mechanics to understand atoms, specifically the hydrogen atom. By solving the Schrödinger equation for the hydrogen atom, one can determine the allowed energy levels, which are quantized and described by the formula E = -13.6 eV / n^2, where n is an integer. These energy levels dictate the absorption and emission spectra of the hydrogen atom, which is how it is observed and studied. The professor humorously recounts the story of Schrödinger and the challenges he faced in solving the equation, highlighting the historical context of these scientific breakthroughs.

The explanation of how atoms, such as hydrogen, absorb and emit light is presented. The difference in energy between levels is given by the formula E_f - E_i = ℏw, which corresponds to the energy of a photon. The professor emphasizes that understanding the quantum states and their corresponding wave functions is crucial for calculating the rates of absorption and emission. The concept of degeneracy, where multiple states can exist at a given energy level, is introduced, and the role of electron spin in doubling the number of states is mentioned. The Pauli exclusion principle is highlighted as a fundamental principle that prevents two fermions from occupying the same quantum state, which has profound implications for the structure of atoms and the periodic table.

The treatment of identical particles in quantum mechanics is discussed. The professor explains that for identical particles, such as electrons, which are fermions, the wave function must be antisymmetric, meaning it changes sign if the particles' coordinates are swapped. This leads to the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state. In contrast, bosons, such as photons, can be in the same state and do not follow this restriction. The distinction between bosons and fermions is crucial for understanding the behavior of particles in various systems and is fundamental to the structure of the periodic table and the properties of atoms.