PERIODIC WAVEFUNCTIONS HAVE QUANTIZED EIGENVALUES OF MOMENTA AND ANGULAR MOMENTA

The wavefunction $$\psi(L,t)$$ is confined to a circle whenever the eigenvalues L of a particle are only nonzero on the points along a circle. When the wavefunction $$\psi(L,t)$$ associated with a particle has non-zero values only on points along a circle of radius $$r$$, the eigenvalues $$p$$ (of the momentum operator $$\hat{P}$$) are quantized—they come in discrete multiples of $$n\frac{ℏ}{r}$$ where $$n=1,2,…$$ Since the eigenvalues for angular momentum are $$L=pr=nℏ$$, it follows that angular momentum is also quantized.

Schrodinger's Time-Dependent Equation: Time-Evolution of State Vectors

Newton's second law describes how the classical state {$$\vec{p_i}, \vec{R_i}$$} of a classical system changes with time based on the initial position and configuration $$\vec{R_i}$$, and also the initial momentum $$\vec{p_i}$$. We'll see that Schrodinger's equation is the quantum analogue of Newton's second law and describes the time-evolution of a quantum state $$|\psi(t)⟩$$ based on the following two initial conditions: the energy and initial state of the system.

Time-Independant Schrodinger Equation: Free Particle and Particle in One-Dimensional Box

In this section, we'll begin by seeing how Schrodinger's time-independent equation can be used to determine the wave function of a free particle. After that, we'll use Schrodinger's time-independent equation to solve for the allowed, quantized wave functions and allowed, energy eigenvalues of a "particle in a box"; this will be useful later on as a qualitative understanding of the quantized wave functions and energy eigenvalues of atoms.

How initial states of definite energy change with time

In general, if a quantum system starts out in any arbitrary state, it will evolve with time according to Schrödinger's equation such that the probability $$P(L)$$ changes with time. In this lesson, we'll prove that if a quantum system starts out in an energy eigenstate, then the probability $$P(L)$$ of measuring any physical quantity will not change with time.

Quantum Dynamics

In this lesson, we'll discuss quantum dynamics.