1 MODELING MATTER AT NANOSCALES 4. Introduction to quantum treatments Outline of the principles and the method of quantum mechanics

Why quantum mechanics?

Physics and sizes in universe Knowledge of “classical” physics was constructed on the grounds of observed regularities of the universe accessible to homo sapiens senses. The universe of very small particles (nano to femptoscopic scales) is not directly accessible and therefore natural laws derived from observables must be found and postulated on more general grounds.

Physics and sizes in universe Knowledge of “classical” physics was constructed on the grounds of observed regularities of the universe accessible to homo sapiens senses. The universe of very small particles (nano to femptoscopic scales) is not directly accessible and therefore natural laws derived from observables must be found and postulated on more general grounds.

Determinism? Determinism is a philosophy stating that for everything that happens there are conditions such that, given them, nothing else could happen. Either determinism and the identity of a physical object could not necessarily be a basis for a science that has no direct access to the facts tried to describe.

Determinism? Determinism is a philosophy stating that for everything that happens there are conditions such that, given them, nothing else could happen. Either determinism and the identity of a physical object could not necessarily be a basis for a science that has no direct access to the facts tried to describe.

“Concessions” Quantum mechanics must be based on probabilistic regards: Particles are considered as undistinguishable Systems can be defined by their average properties It means that several concepts developed for the universe at our scale are no longer necessary, as trajectories and exact speeds of identified objects.

“Concessions” Quantum mechanics must be based on probabilistic regards: Particles are considered as undistinguishable Systems can be defined by their average properties It means that several concepts developed for the universe at our scale are no longer necessary, as trajectories and exact speeds of identified objects.

“Concessions” Quantum mechanics must be based on probabilistic regards: Particles are considered as undistinguishable Systems can be defined by their average properties It means that several concepts developed for the universe at our scale are no longer necessary, as trajectories and exact speeds of identified objects.

“Concessions” Quantum mechanics must be based on probabilistic regards: Particles are considered as undistinguishable Systems can be defined by their average properties trajectoriesexact speeds It means that several concepts developed for the universe at our scale are no longer necessary, as trajectories and exact speeds of identified objects.

The state function of a system Any state of a given system can be associated to a periodic function:  = periodic function Such periodic functions use to be “wave functions”. Such periodic functions use to be “wave functions”.

The state function of a system Any state of a given system can be associated to a periodic function:  = periodic function Such periodic functions use to be “wave functions”. Such periodic functions use to be “wave functions”.

The state properties of a system There are linear operators that can be associated with physical properties of a quantum system  = Linear operator Operators are developed from classical expressions of the corresponding physical properties.

The state properties of a system There are linear operators that can be associated with physical properties of a quantum system  = Linear operator Operators are developed from classical expressions of the corresponding physical properties.

Vectors in the quantum space Either periodic functions defining a given state of a system and linear operators related with their physical properties are considered as vectors in the quantum space defined for the object system.

Eigenvalues and eigenfunctions The way to calculate or predict properties of a system defined by vectors is not with arithmetic, but linear algebra. Then, a property A of a system is obtained by calculating the a eigenvalue* of the  function when the corresponding  operator is applied: If it works,  is an eigenfunction** of the  operator. The eigenvalue is a scalar. * eigenwert ** eigenfunktion

Eigenvalues and eigenfunctions The way to calculate or predict properties of a system defined by vectors is not with arithmetic, but linear algebra. Then, a property A of a system is obtained by calculating the a eigenvalue* of the  function when the corresponding  operator is applied: If it works,  is an eigenfunction** of the  operator. The eigenvalue is a scalar. * eigenwert ** eigenfunktion

Eigenvalues and eigenfunctions The way to calculate or predict properties of a system defined by vectors is not with arithmetic, but linear algebra. Then, a property A of a system is obtained by calculating the a eigenvalue* of the  function when the corresponding  operator is applied: If it works,  is an eigenfunction** of the  operator. The eigenvalue is a scalar. * eigenwert ** eigenfunktion

How eigenfunctions of quantum systems are found? Wave functions that describe the energy of a k state of a quantum system can be obtained by solving the equation of eigenvalues and eigenfunctions of a Hamiltonian, which is a differential operator: This is known as the Schrödinger equation, in honor to Erwin Schrödinger.

How eigenfunctions of quantum systems are found? Wave functions that describe the energy of a k state of a quantum system can be obtained by solving the equation of eigenvalues and eigenfunctions of a Hamiltonian, which is a differential operator: This is known as the Schrödinger equation, in honor to Erwin Schrödinger.

Expectation values When an eigenfunction is not available for a given operator, the corresponding physical property can be approached by the calculation of a “matrix element” that gives an expectation value:

Algebra and quantum mechanics Any given wavefunction, as a vector, can be expressed in terms of a linear combination of other appropriate wavefunctions:

Algebra and quantum mechanics Operators with the same set of eigenfunctions commute: When two operators do not commute, their eigenfunctions are different, and the exact eigenvalue of both properties can not be obtained.

Algebra and quantum mechanics Operators with the same set of eigenfunctions commute: When two operators do not commute, their eigenfunctions are different, and the exact eigenvalue of both properties can not be obtained.

Algebra and quantum mechanics