Quantum Physics

Track 5: Quantum Dynamics

The main focus of quantum mechanics is the study of quantum dynamics, which aims to comprehend how a starting quantum state changes over time (see the conceptual docs on quantum computing for more info on Dirac notation). A quantum state is specifically sought given this beginning condition, an evolution time, and a description of the quantum dynamical system. It is helpful to step back and consider classical dynamics before moving on to explain quantum dynamics since doing so sheds light on how similar the two are in terms of fundamental properties.

Quantum dynamics can be described using exactly the same language. Any closed quantum system's dynamics are fully specified by the Hamiltonian, also known as total energy. The two ideas do, however, have some significant distinctions. In quantum physics, xx and pp are not just numbers as they are in classical mechanics. They don't even commute, in fact.

The proper mathematical term for these non-commuting objects is an operator, which, in situations where xx and pp may only accept a discrete range of values, coincides with the idea of a matrix. Therefore, in order to keep things simple, we'll suppose that our quantum system may be represented by vectors and matrices. Additionally, we demand that these matrices be Hermitian (meaning that the conjugate transpose of the matrix is the same as the original matrix). This ensures that the matrices' eigenvalues are real-valued, which is a requirement we apply to prevent imaginary numbers from being returned when we measure something like position.