To make sense of a non-polynomial function of an operator, you can interpret it as a Taylor series:Jean-Mathys du bois said:Summary:: how does the operator (N+1)^-1/2 α act on a |n> state of harmonic osciliator? N is the number operator N|n>=n|n> and α anihilation operator
I tried playing with the number's operator eigenvalues equation but couldn't get anywhere, can s/b help me out?
A ket state, denoted as |n⟩, is a vector in a complex Hilbert space that represents the state of a quantum system. It is part of the Dirac notation used in quantum mechanics, where kets are used to describe the state of particles, systems, or fields.
When an operator acts on a ket state, it transforms the state into another state, which can be either the same state or a different one. This operation is fundamental in quantum mechanics, as it allows us to calculate observable quantities and predict the behavior of quantum systems.
Common operators include the position operator (Ĥx), momentum operator (Ĥp), and Hamiltonian operator (ĤH). Each of these operators corresponds to a physical observable, and their action on ket states helps determine the properties of the quantum system being studied.
To calculate the result, you apply the operator to the ket state using matrix multiplication if the states and operators are represented in a specific basis. The resulting state can be expressed as a linear combination of basis states, providing insights into the system's new state after the operation.
Eigenstates are special ket states that remain unchanged, except for a scaling factor, when an operator acts on them. The corresponding scaling factor is called the eigenvalue. This relationship is crucial for understanding measurements in quantum mechanics, as measuring an observable associated with an operator will yield one of its eigenvalues, with the system collapsing into the corresponding eigenstate.