**Matrix Mechanics**

In this lesson, we'll cover some of the fundamental principles and postulates of quantum mechanics. These principles are the foundation of quantum mechanics.

The eigenvalues are the values that you measure in an experiment: for example, the position or momentum of a particle. Because the eigenvalues are what you measure, it wouldn't make physical sense if the eigenvalue of an observable had an imaginary part. In this lesson, we'll prove that the eigenvalue of any observable is a real number.

In this lesson, we'll discuss how the spin of an electron can be measured by turning on a magnetic field.

The three operators—\(\hat{σ}_x\), \hat{σ}_y\), and \hat{σ}_z\)—are associated with the measurements of the \(x\), \(y\), and \(z\) components of spin of a quantum particle, respectively. In this lesson, we'll represent each of these three operators as matrices and solve for the entries in each matrix. These three matrices are called the Pauli matrices.

In this lesson, we'll derive an equation which will allow us to calculate the wavefunction (which is to say, the collection of probability amplitudes) associated with any ket vector \(|\psi⟩\). Knowing the wavefunction is very important since we use probability amplitudes to calculate the probability of measuring eigenvalues (i.e. the position or momentum of a quantum system).

In this lesson, we'll mathematically prove that for any Hermitian operator (and, hence, any observable), one can always find a complete basis of orthonormal eigenvectors.

**Schrodinger's Equation**

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.

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.

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.

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.

In this lesson, we'll discuss quantum dynamics.