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.