Postulati Flashcards
What is the equation for the time-dependent Schrödinger equation?
The state function satisfies the time-dependent Schrödinger equation, HΨ(q,t) = (-iħ)(∂Ψ/∂t), where H is the Hamiltonian operator.
What is the equation for a stationary state?
HΨ(q) = EΨ(q), where E is the energy of the system, The Hamiltonian operator is ‘-(h^2/2m)(∂^2/∂q^2) + V(q)’.
What is the function that contains all the information about our system?
The function of state or wave function, Ψ(q,t).
What is the value of a variable A in a given state?
The value of a variable A in a given state is given by the expectation value <a> = ∫ Ψ∗AΨdq or more simply <a> = 〈Ψ | A | Ψ〉.</a></a>
What is the expectation value of energy in a given state?
The expectation value of energy in a given state is given by <E> = <Ψ | H | Ψ>.</E>
What is the expectation value of energy in a stationary state?
In a stationary state, the expectation value of energy is a perfectly defined value equal to the eigenvalue.
What is the equation of motion for the expectation value of a quantity A?
The equation of motion for the expectation value of a quantity A is given by i~d<a>/dt = <Ψ | [A, H] | Ψ> + <Ψ | ∂A/∂t | Ψ>.</a>
What is the time scale for the variation of a quantity A?
The time scale for the variation of a quantity A is given by ∆t = ∆A(d<a>/dt)^-1.</a>
What is the meaning of the uncertainty principle between energy and time?
The uncertainty principle between energy and time implies that the lifetime of a stationary state is infinite, but the presence of perturbations (such as interaction with the environment) leads to a shortening of the lifetime.
What is a complete and closed set for all operators of the system?
A complete set means that any function Φ with the same boundary conditions as our eigenfunctions can be expressed as a linear combination of the eigenfunctions.
What is the meaning of the coefficients in the expansion?
The coefficients represent the overlap between Ψn and Φ, that is, how much of one component is in the other.
What is the analogy between the formalism in a vector space and the expansion of functions in terms of eigenfunctions?
The analogy is that the eigenfunctions of Hermitian operators can be considered as the fundamental vectors of the vector space, and the coefficients can be considered as the components of the vector.
What does the closure property mean?
The closure property means that if A is the operator associated with any dynamic variable of the system, then AΦ can also be expanded in the complete set.
What are the two important properties of complete sets of eigenfunctions?
The two important properties are that if two operators commute, they have a common complete set of eigenfunctions, and if A and B commute and have different eigenvalues, then the expectation value of B between the eigenfunctions of A is zero.
What is the probability that a measurement experiment will find the system in the state of maximum knowledge Ψk?
wk =| ck |^2=|< Ψk | Φ >|^2
